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Engineering Structures 30 (2008) 3478–3488
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Effect of tuned mass damper on displacement demand of base-isolated structures Tomoyo Taniguchi a , Armen Der Kiureghian b,c,∗ , Mikayel Melkumyan c a
Tottori University, Tottori, 680-8552, Japan
b
University of California, Berkeley, CA 94720, USA
c
American University of Armenia, Yerevan, Armenia
article
info
Article history: Received 13 July 2007 Received in revised form 30 May 2008 Accepted 30 May 2008 Available online 2 July 2008 Keywords: Base isolation Displacement demand Far-field motions Near-field motions Optimal design Tuned-mass damper
a b s t r a c t The third author of this paper has previously proposed the installation of a tuned-mass damper (TMD) to reduce the displacement demand on a base isolated structure. The TMD consists of a mass-dashpot-spring subsystem that is attached to the isolated superstructure, analogous to a pendulum. The present paper examines the effectiveness of this scheme and determines optimal parameters for the design of the TMD. Both the base-isolated structure and the TMD are modeled as single-degree-of-freedom, linear oscillators. The optimal TMD parameters are determined by considering the response of the base-isolated structure, with and without the TMD, to a white-noise base acceleration. Such an excitation is representative of broadband ground motions having a nearly constant intensity over a duration several times longer than the period of the base-isolated structure. It is found that, under such an excitation, a reduction of the order of 15%–25% in the displacement demand of the base-isolated structure can be achieved by adding the TMD. Next, the responses of an example base-isolated structure with and without an optimally designed TMD to selected suites of far- and near-field recorded accelerograms are determined. The study shows that for far-field ground motions the effectiveness of the TMD is more or less similar to that predicted by the white noise model, whereas for near-field ground motions the effectiveness of the TMD is less, i.e. of the order of 10% or less. Reasons for this result are described. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Seismic base isolation has become one of the most effective technologies in protecting structures against destructive earthquakes. By providing flexibility in the base of the structure, the isolation system absorbs the bulk of the displacement demand of the earthquake, with the super-structure essentially displacing as a rigid body. Furthermore, base isolation drastically lowers the fundamental frequency of the system, putting it outside the dominant range of input frequencies and, thereby, reduces the acceleration at floor levels of the structure where sensitive equipment or nonstructural systems may be located. In doing this, the base isolation system itself undergoes a relatively large displacement. One important consideration in the design of the base isolation system is this displacement demand. Our interest in this paper focuses on a base isolation system made of laminated rubber bearings. The displacement capacity of such a system is directly related to its size and damping. One
∗ Corresponding author at: University of California, Berkeley, CA 94720, USA. Tel.: +1 510 642 2469; fax: +1 510 643 8928. E-mail addresses: t_tomoyo@cv.tottori-u.ac.jp (T. Taniguchi), adk@ce.Berkeley.edu (A. Der Kiureghian), mmelkumi@aua.am (M. Melkumyan). 0141-0296/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2008.05.027
option for reducing the displacement demand on the isolation system is to provide supplemental damping. This, however, may increase the in-structure accelerations [5]. In this paper we explore the possibility of using a tuned mass damper to reduce the displacement demand of the base isolation system. The third author has been instrumental in initiating the manufacturing of laminated rubber bearings in Armenia and using them for retrofitting of existing buildings or for new construction [8,10]. Today, more than 30 buildings in Armenia are built, retrofitted or under construction employing the base isolation technology, mostly using locally manufactured bearings made of neoprene—thus putting Armenia at a top rank in terms of the number of base-isolated buildings per capita. (The population of Armenia is around 3 million.) At the present time the manufacturing technology is capable of producing bearings with neoprene compounds with low or medium damping. However, manufacturing of large-size (i.e. larger than 60 cm in diameter), high-damping (larger than 10%) bearings remains a technological challenge in Armenia. For this reason, there has been an interest in exploring alternatives for reducing the displacement demand of the base-isolation system. With the above motivation in mind, the third author has proposed a scheme to reduce the displacement demand of a base-isolated structure by installing a tuned-mass damper
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Fig. 2. Idealized model of base-isolated structure with TMD. Fig. 1. Schematic of base-isolated structure with TMD.
(TMD) (denoted as ‘‘dynamic damper’’ in [7,9]) attached to the isolation floor. The effectiveness of an appended mass-spring system in reducing the dynamic response of a structure has been known for a long time [1,12,11]. Numerous investigations and implementations of this idea for fixed-base buildings have been made (see, e.g., [4,13,6,3]). In such buildings, the TMD is usually placed in an upper floor in order to experience a larger acceleration for efficiently mobilizing itself and absorbing the energy in the system. Usually a large portion of the floor or the entire floor of the building must be devoted to the TMD. In the case of a base-isolated structure, since the maximum relative displacement occurs at the level of the bearings, the TMD must be attached immediately above the isolation system. It can be provided as a mass-spring subsystem attached either above or below the isolated floor of the building, as shown in Fig. 1. One can imagine various functions for such a subsystem, e.g., an exercise room, a swimming pool, parking space, utilities room, as long as the mass remains relatively constant in time and large displacement can be accommodated. As described by Melkumyan [9], the proposed TMD scheme also has the advantage of increasing the capacity of the base-isolated building against overturning forces. This paper aims at examining the effectiveness of the proposed TMD scheme and determining the set of optimal parameters for its design. We assume the base-isolated structure can be modeled as a single-degree-of-freedom (SDOF) oscillator. This essentially assumes that the superstructure acts as a rigid body, which is a reasonable assumption for a base-isolated structure. The TMD is modeled as a SDOF oscillator as well, which is attached to the base-isolated structure as an appendage. After formulating the equations of motion and defining key parameters, we consider the stationary response of the base-isolated structure with and without the TMD when the system is subjected to a whitenoise base acceleration. Such an excitation is representative of a broadband ground motion having a nearly constant intensity over a duration several times longer than the period of the baseisolated structure. The simplicity of this excitation model allows us to determine the effectiveness of the TMD in terms of a few key parameters. This then leads to the identification of optimal parameters (mass, frequency, damping) of the TMD for a given base-isolated structure. It is found that the optimally designed TMD can effect a reduction of 15%–25% in the displacement demand of the isolators. Next, the responses of an example baseisolated structure with and without an optimally designed TMD to selected suites of far- and near-field recorded ground motions are determined. The investigation shows that for far-field ground motions the effectiveness of the TMD is more or less similar to that predicted by the white-noise model. Furthermore, the same level of reduction is achieved in the acceleration response. For near-field ground motions, on the other hand, the effectiveness of the TMD is less, i.e., of the order of 10% or less. The reason has to do with the short, pulse-type nature of near-field ground motions.
The practical implementation of a TMD in a base-isolated building obviously involves additional considerations, such as provision of space for the displacement of the TMD and cost. These considerations, admittedly important ones, have not been addressed in this paper. 2. Equations of motion Consider the combined system consisting of the base-isolated structure and the TMD, as shown in Fig. 1. We assume the base-isolated structure alone behaves approximately as a SDOF oscillator having an effective mass mp , a natural frequency ωp , and a damping ratio Îśp , where the subscript p refers to ‘‘primary’’. We also assume the TMD by itself behaves approximately as a SDOF oscillator with an effective mass ms , a natural frequency ωs , and a damping ratio Îśs , where the subscript s refers to ‘‘secondary’’. The combined system consisting of the base-isolated structure (the primary subsystem) and the TMD (the secondary subsystem) is a 2DOF system, as shown in an idealized form in Fig. 2. It is known (see [2]) that such a composite primary–secondary system is generally non-classically damped, even when the individual sub-systems are classically damped. Hence, to properly model the system, account must be made of the non-classical damping nature of the combined system. The equations of motion of the combined system is described by MuĚˆ + Cu̇ + Ku = −M1 xĚˆg (t )
(1)
where u(t ) = [up (t ) us (t )] is the vector of displacements relative to the ground, xĚˆg (t ) is the ground acceleration and T
M=
mp 0
0 , ms
2Îśp ωp mp + 2Îśs ωs ms C= −2Îśs ωs ms
2 ωp mp + ωs2 ms K= âˆ’Ď‰s2 ms
−2Îśs ωs ms , 2Îśs ωs ms âˆ’Ď‰s2 ms 1 , 1= . 1 ωs2 ms
(2)
For the subsequent analysis, it is useful to introduce the mass ratio ms Îł = (3) mp and the tuning parameter
ωp − ωs (4) ωave where ωave = (ωp + ωs )/2 is the average frequency. The mass β=
ratio describes the size of the TMD; we consider values in the range γ = 0.01 to 0.10. The tuning parameter describes the proximity of the natural frequencies of the two sub-systems. It is well known that the TMD is more effective when γ is large and β is near zero. Results are presented below in terms of these two parameters, as well as the individual system parameters defined earlier.
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the combined system, a frequency-domain approach using the frequency-response matrix (FRM) (instead of the usual modal superposition approach) is used. The FRM of the system in (1) is given by H(ω) = âˆ’Ď‰2 M + iωC + K
−1
.
(6)
Let ÎŚxĚˆg xĚˆg (ω) denote the power spectral density (PSD) of the ground acceleration. The PSD matrix of the response vector u is then given by
8uu (ω) = H(ω)M1ÎŚxĚˆg xĚˆg (ω)1T MT H(ω)∗T
where the superposed asterisk denotes the complex conjugate. As measures of the responses of interest, we consider the meansquares of the displacements up (t ) and us (t ) of the primary and secondary subsystems, respectively. These are given by
Fig. 3. Frequencies of the combined system.
3. Frequencies of the undamped combined system
Ďƒ = 2 p
As mentioned earlier, the combined system is non-classically damped. Nevertheless, it is insightful to examine its undamped frequencies. Analytically solving the eigenvalue problem â„Ś 2 M8 = K8 for the undamped system, we obtain the following expressions for the two natural frequencies of the combined system.
1
â„Ś1 , â„Ś2 = √
2
Âą
ωp2 + (1 + γ ) ωs2
12 q 2 ωp2 + (1 + Îł ) ωs2 − 4ωp2 ωs2 .
(7)
(5)
Fig. 3 shows plots of the two frequencies of the combined system, normalized with respect to the frequency of the primary sub-system, as a function of the tuning parameter for different mass ratios. The ratio of sub-system frequencies, ωs /ωp , is also shown for reference. It can be seen that for large negative β values, i.e., for ωp ωs , the first mode of the combined system has a frequency that is close to but lower than that of the primary sub-system and the second mode has a frequency that is close to but greater than that of the secondary sub-system. Conversely, for a large positive β value, i.e., for ωs ωp , the second mode of the combined system has a frequency that is close to but greater than that of the primary sub-system, while the first mode has a frequency that is close to but smaller than that of the secondary sub-system. It is reasonable to expect that for these values of β , the mode of the combined system that has a frequency close to that of the primary sub-system dominates its response in the combined system. For β values close to zero, i.e., near perfect tuning, the two modal frequencies of the combined system are symmetrically positioned relative to the frequency of the primary sub-system. In this case, the two modes tend to equally contribute to the responses of the primary and secondary subsystems in the combined system. As we will shortly see, the TMD is most effective in reducing the displacement demand on the baseisolated structure when β is in the range 0.1–0.2. In this range, the second mode of the combined system dominates the response of the base-isolated structure. It is also notable that the frequencies of the combined system move further apart from the sub-system frequencies and from each other as the mass ratio, γ , increases. 4. Stochastic dynamic analysis To examine the effectiveness of the TMD in reducing the displacement demand of the base-isolated structure, we first consider the stationary response of the combined system to a zero-mean, broadband stationary stochastic base acceleration. To properly account for the non-classical damping nature of
Z
+∞
Όup up (ω)dω
(8)
Όus us (ω)dω
(9)
−∞
Ďƒs2 =
Z
+∞ −∞
where ÎŚup up (Ď&#x2030;) and ÎŚus us (Ď&#x2030;) are the diagonal elements of 8uu (Ď&#x2030;). Closed form expressions for the elements of the 2 Ă&#x2014; 2 matrix H(Ď&#x2030;)M are derived in Appendix. These are used in (7) to derive expressions for ÎŚup up (Ď&#x2030;) and ÎŚus us (Ď&#x2030;). Numerical integration is then used to evaluate (8) and (9). To investigate the effect of the TMD in reducing the displacement demand of the base-isolated structure, we consider the response ratio Ď&#x192;p /Ď&#x192;p0 , where Ď&#x192;p0 denotes the root-mean-square of the response of the base isolated structure without the TMD. In order to make the results independent of the specifics of the input ground motion, the excitation is assumed to be a white-noise process having a constant PSD ÎŚxĚ&#x2C6;g xĚ&#x2C6;g (Ď&#x2030;) = ÎŚ0 . This model is a good approximation for broadband earthquake ground motions and lightly damped structures. For this model, it is well known that 2 Ď&#x192;p0 = Ď&#x20AC; ÎŚ0 /(2Îśp Ď&#x2030;p3 ). Furthermore, the response ratio Ď&#x192;p /Ď&#x192;p0 is independent of the intensity ÎŚ0 of the white noise. In fact, as can be verified from the expressions in Appendix, the ratio Ď&#x192;p /Ď&#x192;p0 only depends on the mass ratio, Îł , the tuning parameter, β , and the damping ratios Îśp and Îśs of the two subsystems. Figs. 4(a)â&#x20AC;&#x201C;(d) show plots of the response ratio Ď&#x192;p /Ď&#x192;p0 as a function of β for the primary damping ratio Îśp = 0.05 and secondary damping ratios Îśs = 0.05, 010, 0.15 and 0.20, respectively. Three curves are shown in each plot for the mass ratio values Îł = 0.02, 0.05 and 0.10. A fourth curve with diamondshaped markers is also shown, which is described below. First consider Fig. 4(a), which is for Îśp = Îśs = 0.05. As one would expect, the effectiveness of the TMD in reducing the displacement demand on the base-isolated structure depends on both the mass ratio, Îł , and on the tuning parameter, β . For a fixed Îł , the optimal TMD occurs at a positive β value, which corresponds to a frequency of the TMD that is smaller than the frequency of the base-isolated structure. Thus, contrary to the notion of a â&#x20AC;&#x2DC;â&#x20AC;&#x2DC;tunedâ&#x20AC;&#x2122;â&#x20AC;&#x2122; mass damper, the optimal reduction in the response of the baseisolated structure occurs not at β = 0 but for 0 < β . The locus of these optimal points for all Îł values is plotted in the figure (line with diamond markers) and is called the â&#x20AC;&#x2DC;â&#x20AC;&#x2DC;design curveâ&#x20AC;&#x2122;â&#x20AC;&#x2122;. Note that the diamond marks on the design curve indicate Îł values from 0.01 to 0.10 at increments of 0.01. It is seen that the optimal value of the tuning parameter β moves towards greater positive values with increasing mass ratio. For example, at Îł = 0.05, the optimal value of the tuning parameter is βopt = 0.10, which using (4) corresponds to the optimal TMD frequency of Ď&#x2030;s = 0.905Ď&#x2030;p , whereas at Îł = 0.10 the optimal value of the tuning parameter is βopt = 0.18, which corresponds to Ď&#x2030;s = 0.835Ď&#x2030;p . At these
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(a) ζs = 0.05.
(b) ζs = 0.10.
(c) ζs = 0.15.
(d) ζs = 0.20.
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Fig. 4. Reduction in the seismic demand of the base-isolated structure caused by the TMD for ζp = 0.05.
mass ratios, the TMD reduces the demand on the base-isolated structure by about 21% and 22%, respectively. On the other hand, the reduction in demand for the optimal TMD with mass ratio γ = 0.02 is 17%. This confirms the well known result that increasing the mass ratio increases the effectiveness of the TMD (e.g., see [11]). Note, however, that even a mass ratio of γ = 0.01 or 0.02 provides considerable reduction in the demand. Figs. 4(b)–(d) show results similar to those shown in Fig. 4(a) but for secondary damping ratios ζs = 0.10, 0.15 and 0.20, respectively. It is seen that significant improvement in the effectiveness of the TMD is achieved by increasing its damping ratio from 0.05 to 0.10. However, further increase in the damping ratio of the TMD provides marginal improvement or even diminishes its effectiveness (e.g., compare the curves for γ = 0.02 for increasing ζs ). It appears that the TMD damping ratio ζs = 0.10 is a good choice if the damping ratio of the base-isolated structure is ζp = 0.05. Figs. 5(a)–(d) show results similar to those described above but for the primary damping ratio ζp = 0.10. It is seen that, compared to the case of ζp = 0.05, the effectiveness of the TMD is reduced to no more than 5%–15%. Melkumyan [7] anticipated this effect by suggesting the use of the TMD in conjunction with low-damping rubber isolation bearings. The design curves in Figs. 4 and 5 can be used to select the frequency of the TMD, for given frequency of the base-isolated structure, the mass ratio and the two damping ratios, to achieve the maximum reduction in the displacement demand on the baseisolated structure. It is important to observe, however, that the curve for each fixed γ has a relatively flat bottom. Therefore, if a small error is made in estimating the frequencies of the primary or secondary subsystems, the reduction in the response of the baseisolated structure will not be greatly affected. Roughly speaking, relative variations δωp and δωs in the two frequencies lead to the
q
absolute variation δβ ∼ = 2 δω2 p + δω2 s in the tuning parameter. For example, if the primary frequency is known within a 4% error
and the secondary frequency is known p within 2% error, then
the estimated error in β is around 2 0.042 + 0.022 = 0.09. This formula can be used to estimate the range of variations in the tuning parameter for given uncertainties in the sub-system frequencies. In summary, the following conclusions can be derived from the above analysis of the stochastic response: (a) The effectiveness of the TMD in reducing the displacement demand on the base-isolated structure increases with increasing mass ratio, provided the TMD is optimally tuned; (b) The optimal TMD always has a frequency smaller than the frequency of the base-isolated structure; (c) The TMD is more effective for a lightly damped isolation system; (d) For the damping ratio ζp = 0.05 of the base-isolated structure, a good choice for the damping ratio of the TMD is ζs = 0.10; higher TMD damping ratios do not significantly improve the effectiveness of the TMD; (e) A reduction of 20% or greater in the displacement demand of the base-isolated structure with damping ratio ζp = 0.05 can be achieved by use of an optimally designed TMD having a damping ratio of ζs = 0.10 and mass ratio 0.05 ≤ γ ; (f) Variations in the order of 2%–3% in the frequencies of the base-isolated structure and an optimally designed TMD have a negligible influence on the effectiveness of the TMD. Several of the above conclusions, including (a), (c) and (f), and the fact that a higher damping for the TMD beyond a certain level does not increase its effectiveness, are confirmations of well known results for TMDs in fixed-base buildings. The above analysis assumes that the TMD responds within its elastic limit. Since the TMD is nearly tuned to the primary subsystem, it may experience a large response, which may put it beyond its yield limit. To investigate this possibility, we examine the response ratio σs /σp , which is only a function of the parameters γ , β , ζp and ζs . Fig. 6 shows this response ratio as a function of
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(a) ζs = 0.05.
(b) ζs = 0.10.
(c) ζs = 0.15.
(d) ζs = 0.20. Fig. 5. Reduction in the seismic demand of the base-isolated structure caused by the TMD for ζp = 0.10.
Fig. 6. Amplification in the displacement of the TMD for ζp = 0.05 and ζs = 0.10.
the tuning parameter β for ζp = 0.05, ζs = 0.10 and the mass ratio values γ = 0.02, 0.05 and 0.10. It appears that, depending on the tuning parameter and the mass ratio, the response of the TMD can be 2–4 times larger than that of the base-isolated structure. This finding clearly calls for a careful design of the TMD and the space it occupies in the building in order to accommodate the large displacement demand. Alternatively, one may allow the TMD to dissipate energy through hysteretic action. This potentially beneficial effect is not considered in the present study. 5. Time history analysis The analysis in the preceding section employed a stationary white-noise process as a model for the ground acceleration. This is convenient, since for this model the effectiveness of the TMD can be assessed with the least number of system parameters and without involving any parameters that characterize the input excitation. However, one may question whether this idealized model of the ground motion accurately describes the effectiveness of the TMD, since real earthquake ground motions are neither
stationary nor have a uniform spectral content, as represented by the white-noise model. It is well known that, in order for the TMD to be effective, it is necessary that the energy input into the system be gradual so that there is time for transfer of energy from the primary system (the base-isolated structure) into the secondary system, the TMD. This suggests that the non-stationary nature of the ground motion may have a strong influence on the effectiveness of the TMD. For this reasons, it was decided to examine the effectiveness of the TMD by using time-history analyses with selected recorded accelerograms. The stand-alone base-isolated structure considered for this analysis has a frequency of ωp = π rad/s (2.0 s period) and a damping ratio of ζp = 0.05. The TMD is assumed to have the damping ratio ζs = 0.10, the mass ratio γ = 0.05, and the stand-alone frequency ωs = 0.910π rad/s (2.2 s period), which corresponds to the optimally designed value of the tuning parameter, βopt = 0.094, as can be seen in Fig. 4b. For this system, a response ratio of 0.75 (i.e. a reduction of 25% in the displacement demand) is expected from the analysis with the white-noise excitation. The recorded accelerograms are selected from the PEER strong-motion database at http://peer.berkeley.edu/smcat/index.html. To properly account for the non-classical damping nature of the combined system, direct numerical integral of the equations of motion in (1) is carried out. The second-order Runge–Kutta algorithm is used for this purpose. Considerable attention was given to the characteristics of the selected ground motions, which are listed in Table 1. Since the analysis is linear, and the ratio of responses with and without the TMD is of interest, the intensity of the motion is immaterial. However, the temporal evolution of the motion and its frequency content are important considerations. As proxies for these characteristics of the recorded motions, we selected the distance of the recording site from the fault rupture and the local site condition. The distance from the fault rupture tends to influence the nonstationary character of the accelerogram. In
IMPVALL H-DLT352
Imperial Valley 1979/10/15 23:16
KERN TAF111
Kern County 1952/07/21 11:53
IMPVALL H-E08140
Imperial Valley 1979/10/15 23:16
CHICHI TCU129-W
TABAS TAB-TR
Chi-Chi, Taiwan 1999/09/20
Tabas, Iran 1978/09/16
KOBE KJM000
LOMAP LGP000
Loma Prieta 1989/10/18 00:05
Kobe 1995/01/16 20:46
NORTHR NWH090
Northridge 1994/01/17 12:31
Near-field ground motions
CHICHI TCU047-N
Chi-Chi, Taiwan 1999/09/20
KOBE KAK090
LOMAP SFO090
Loma Prieta 1989/10/18 00:05
Kobe 1995/01/16 20:46
NORTHR 116090
Record component
Northridge 1994/01/17 12:31
Far-field ground motions
Earthquake
C
C
B
C
?
C
B
B
D
C
C
B
USGS site class.
Table 1 List of earthquakes and computed response ratios
3.00
1.18
0.60
3.80
6.10
7.10
41.0
33.01
26.4
43.6
64.4
41.9
Distance (km)
0.852
1.01
0.821
0.602
0.563
0.583
0.178
0.413
0.345
0.351
0.329
0.208
PGA (g)
121.4
60.0
81.3
54.3
94.8
75.5
17.5
40.2
27.6
33.0
27.9
10.3
PGV (cm/s)
94.58
50.15
17.68
32.32
41.18
17.57
8.99
22.22
9.6
19.02
6.03
2.67
PGD (cm)
Accelerogram (g)
71
97
93
92
91
92
76
60
74
72
96
94
91
91
92
75
61
80
78
92
90
80
103
Acc. (%)
107
Disp. (%)
Response ratio
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(a) Northridge.
(b) Loma Prieta.
(c) Imperial Valley.
(d) Kobe.
(e) Chi-Chi.
(f) Kern county.
Fig. 7. Displacements of the base-isolated structure with (solid line) and without (dashed line) the TMD for far-field ground motions (Unit: m).
particular, near-field ground motions tend to contain a long-period pulse, the ‘‘fling’’, which directly results from the dislocation at the fault, whereas far-field ground motions tend to have a fairly smooth transition of temporal and spectral contents. To account for these effects, two sets of ground motions were selected: nearfield ground motions with distances ranging from 0.6 to 7.1 km from the closest point on the fault rupture, and far-field ground motions with distances ranging from 26.4 to 64.4 km from the fault rupture. The local site condition influences the frequency content of the ground motion. For the selected records, the site condition is characterized by the USGS classification. Among the records considered, 4 have classification B (shear-wave velocity in the range 360–750 m/s), 6 have classification C (shear-wave velocity
in the range 180–360 m/s), and one has classification D (shearwave velocity less than 180 m/s). One record has no classification. For each record, the ratios of the maximum absolute displacement and maximum absolute acceleration responses of the base-isolated structure with the TMD relative to the corresponding responses of the stand-alone base-isolated structure are computed. These response ratios along with other characteristics of each earthquake record are listed in Table 1. Figs. 7 and 8 respectively show comparisons of the computed displacement responses of the baseisolated structure with and without the TMD for the six farfield ground motions and six near-field ground motions, and Figs. 9 and 10 show similar results for the computed acceleration responses.
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(a) Northridge.
(b) Loma Prieta.
(c) Imperial Valley.
(d) Kobe.
(e) Chi-Chi.
(f) Tabas.
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Fig. 8. Displacements of the base-isolated structure with (solid line) and without (dashed line) the TMD for near-field ground motions (Unit: m).
The following observations can be made from the computed response ratios listed in Table 1 and the time-history results in Figs. 7â&#x20AC;&#x201C;10: It is first noted in Table 1 that in all but one case (far-field Northridge record) reductions in the displacement and acceleration responses of the base-isolated structure are achieved by adding the TMD. Secondly, the response ratios are nearly the same for the displacement and acceleration responses for each ground motion. This indicates that the TMD has similar influences on the displacement and acceleration responses of the baseisolated structure. Thus, the reduction in the displacement demand is not achieved at the expense of increasing the acceleration response, as may be the case with other alternatives, such as the provision of supplemental damping [5]. Thirdly, no correlation is observed between the response ratios and the site classifications.
Evidently, the site classification is too crude a measure to have a direct relation with the effectiveness of the TMD. For the far-field ground motions (the first six rows in Table 1), the response ratio is around the expected 75% for four out of the six records. Among these records, the ones of Imperial Valley, Kobe and KernCounty earthquakes have nearly stationary characters during their respective strong motions phases. The ChiChi earthquake has a distinctly nonstationary behavior. However, the largest pulses in the accelerogram occur after a period of gradual increase in the intensity of the motion (see Figs. 7 and 9 bottom left). This gives opportunity to the TMD to be mobilized and absorb energy from the primary structure. For the far-field Loma Prieta record, the response ratios are 0.90 and 0.92. The small reduction in the response can be attributed to the fact that the peak
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(a) Northridge.
(b) Loma Prieta.
(c) Imperial Valley.
(d) Kobe.
(e) Chi-Chi.
(f) Kern county.
Fig. 9. Accelerations of the base-isolated structure with (solid line) and without (dashed line) the TMD for far-field ground motions (Unit: g).
responses to this record are primarily due to a large acceleration pulse happening around 11s (see Figs. 7 and 9 top right). Since this large acceleration pulse occurs early in the excitation, the effectiveness of the TMD is similar to the case of near-field ground motions described below. The most puzzling result is that of the far-field Northridge record, which appears to have a fairly stationary character during its strong-motion phase, but the effect of the TMD in this case is a slight amplification of the response of the base-isolated structure. This case is further examined below. For the near-field ground motions (the last six rows in Table 1), the reductions in the displacement and acceleration responses of the base-isolated structure are less than 10% in all but the case of Tabas record. We attribute this to the fact that, with the exception
of Tabas, all these motions contain large acceleration pulses early in their time histories, which produce the peak responses of the base-isolated structure (see Figs. 8 and 10). As mentioned earlier, this is a typical characteristic of near-field ground motions, which are directly affected by the fault slip with little influence from the dispersive effect of waves traveling long distances. For such records, there is no time for transfer of energy from the baseisolated structure into the TMD and, hence, the TMD does not become fully mobilized to achieve its effectiveness, as predicted by the stationary analysis in the preceding section. Although this phenomenon is generally known, we have not found a quantitative analysis of its effect in the TMD literature, particularly in relation to near-field ground motions. In contrast to the other near-field
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(a) Northridge.
(b) Loma Prieta.
(c) Imperial Valley.
(d) Kobe.
(e) Chi-Chi.
(f) Tabas.
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Fig. 10. Accelerations of the base-isolated structure with (solid line) and without (dashed line) the TMD for near-field ground motions (Unit: g).
records, since large acceleration pulses in the Tabas record occur in the middle of the record, there is enough time for the TMD to be mobilized to achieve its effectiveness (see Figs. 8 and 10 bottom right). In this sense, the Tabas record behaves like a far-field ground motion, while the far-field Loma Prieta record behaves like a nearfield ground motion. We now return to examine the case of the far-field Northridge record. As mentioned earlier, this record has a nearly stationary strong-motion phase, yet the response ratios for both the displacement and acceleration responses are greater than 1. That is, for this ground motion, attaching the optimally designed TMD actually enhances both the displacement and acceleration demands on the base-isolated structure. To understand the reason
for this behavior, we examine the response spectra of the selected far-field ground motions. Fig. 11 shows the 5% damping response spectra for these motions, all normalized by their values at 0.5 Hz, which is the frequency of the base-isolated structure without the TMD. A thicker line highlights the spectrum for the Northridge record. It can be seen that the Northridge record has a sharp dominant peak at around 0.4 Hz frequency, which has a much larger value than the spectral displacements at all other frequencies. Thus, the energy in this motion is mostly concentrated around 0.4 Hz frequency. As can be verified in Fig. 3, this frequency happens to coincide with the first undamped modal frequency of the base-isolated structure with the TMD. Thus, by adding the TMD, the first mode of the combined system, i.e. the mode dominated
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Fig. 11. Normalized response spectra of far-field ground motions for 5% damping.
by the displacement of the TMD, is subjected to this large spectral amplitude, resulting in an amplification of the response of the base-isolated structure relative to its response without the TMD. In other words, in this case, because of resonance of the TMD with the input excitation, the TMD further excites the base-isolated structure instead of absorbing its energy. It is noted that none of the other response spectra in Fig. 11 has this particular feature. The response ratios predicted based on stochastic analysis shown in Figs. 4 and 5 represent ensemble averages. The effectiveness of the TMD for individual realizations of the stochastic ground motion would, of course, vary around these averages. Therefore, in examining the time-history results, it is appropriate to consider the averages over the selected samples. Considering all six samples, the average response ratios for the far-field ground motions are 0.81 for both the displacement and acceleration responses. If the Northridge record is not included, the averages reduce to 0.76 and 0.77, respectively, which are in line with that predicted by the stochastic analysis. For near-field ground motions, the average response ratios are around 0.89 for both displacement and acceleration responses. If the Tabas record is excluded, both averages are around 0.93. It is clear that the TMD is not effective in reducing the demand on the base-isolated structure for near-field ground motions. 6. Conclusions The effectiveness of a TMD to reduce the seismic demand on a base-isolated structure is investigated. Using stochastic dynamic analysis based on a white-noise model of the ground motion, the optimal parameters of the TMD that maximally reduce the seismic demand on the base-isolated structure are determined. This investigation reveals that, depending on the mass, damping and frequency characteristics of the TMD, the displacement demand on the base isolated structure can be reduced by 15%–25%. It is shown that the TMD is more effective for lightly damped isolators. Furthermore, the effectiveness of the TMD increases with its mass, but not necessarily with its damping. To account for the nonstationary and non-white nature of ground motions, a series of time history analyses with far- and near-field recorded ground motions are carried out. The analyses show that for far-field ground motions the effectiveness of the TMD is in concordance with the predictions of the stochastic analysis, except for one particular record, which happens to have a sharp spectral peak in resonance with the TMD. Importantly, the TMD
produces virtually identical reductions in the displacement and acceleration demands of the base-isolated structure. For near-field ground motions, the effectiveness of the TMD is no more than 7%–10%. The reason is that for such motions the peak response usually is due to a large pulse early in the record, so that sufficient time is unavailable for the TMD to be mobilized. One may question the viability and cost-effectiveness of installing a TMD with a mass ratio as large as 5% or more of the building mass to effect a reduction of no more than 15%–25% in the displacement demand of a base isolation system. However, if the TMD is designed as an integral part of the base-isolated building and it serves a useful function, then such a scheme may prove to be both beneficial and economical. We note that, even though the TMD may experience large displacements, these motions will have low frequency and, hence, will be tolerable by humans and certain equipment (similar to wind-induced motions in top floors of tall buildings). In any case, the results presented in this paper provide valuable information to any engineer contemplating the use of a TMD in a base-isolated building. Appendix The product of the frequency response matrix and the mass matrix in (7) is given by H(ω)M
" ωs2 − ω2 + 2iζs ωs ω = ωs2 + 2iζs ωs ω D 1
# ε ωs2 + 2iζs ωs ω (A.1) ωp2 + εωs2 − ω2 + 2iω ζp ωp + εζs ωs
where D is D = ω2 ω2 − ωp2 − ωs2 (1 + ε) − 4ζp ζs ωp ωs + ωp2 ωs2
+ 2iω ωp ωs ζp ωs + ωp ζs − ω2 ζs ωs (1 + ε) + ζp ωp . (A.2) References [1] Den Hartog JP. Mechanical vibrations. New York (NY): McGraw-Hill; 1947. [2] Igusa T, Der Kiureghian A. Dynamic characterization of two degree-of-freedom equipment-structure systems. J Eng Mech ASCE 1985;111(1):1–19. [3] Inaudi J, Kelly JM. Mass dampers using friction-dissipating devices. J Eng Mech ASCE 1995;121. [4] Kaynia AM, Veneziano D, Biggs JM. Seismic effectiveness of tuned mass dampers. J Struct Eng ASCE 1981;107(8):1465–84. [5] Kelly JM. The role of damping in seismic isolation. Earthq Eng Struct Dyn 1999; 28:3–20. [6] Khachian EE, Melkumyan MG, Khlgatyan ZM. Method of seismic protection of multistoried buildings. In: Proceeding of UNESCO international seminar on Spitak-88 earthquake. 1989. [7] Melkumyan MG. The state-of-the-art in structural control in Armenia and proposal on application of the dynamic dampers for seismically isolated buildings. In: Proceedings of the third international workshop on structural control for civil and infrastructure engineering, 2000. [8] Melkumyan MG. State-of-the-art on application, R&D and design rules for seismic isolation of civil structures in Armenia. In: Proceedings of the 8th world seminar on seismic isolation, energy dissipation and active vibration control of structures. 2003. [9] Melkumyan MG. First application of the dynamic damper in the design of seismically isolated dwelling house. In: Lesch R, Irschik H, Krommer M, editors, Proceeding of the third European conference on structural control. vol. I, Mini symposium M6. 2004. [10] Melkumyan MG. Current situation in application of seismic isolation technologies in Armenia. In: Proceedings of international conference on the 250th anniversary of the 1755 Lisbon earthquake, 2005. [11] Reed FE. Dynamic vibration absorbers and auxiliary mass dampers. Shock and vibration handbook, vol. 1. New York (NY): McGraw-Hill, Inc.; 1961 [Chapter 6]. [12] Suzuki T. Response of elevated water tanks (2nd report). J Arch Institute Japan 1950;6:29–32 [In Japanese]. [13] Warburton GB. Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Eng Struct Dynam 1982;10: 381–491.