October 23, 2002 9:12 WSPC/124-JEE 00086 ...

October 23, 2002 9:12 WSPC/124-JEE

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Journal of Earthquake Engineering, Vol. 6, No. 4 (2002) 553–585 c Imperial College Press

SEISMIC DESIGN OF BASE-ISOLATED STRUCTURES USING CONSTANT STRENGTH SPECTRA

ARTURO TENA-COLUNGA Departamento de Materiales, Universidad Aut´ onoma Metropolitana Azcapotzalco, Av. San Pablo # 180, Col. Reynosa Tamaulipas, 02200 M´ exico, DF, Mexico [email protected] Received 11 July 2001 Revised 22 May 2002 Accepted 8 June 2002 This paper presents the concept of constant strength design spectra for the design of base-isolated structures; particularly those structures using isolators with a bilinear hysteretic behaviour when subjected to dynamic loading. The constant strength design spectra relate peak accelerations, velocities, displacements and effective isolated natural periods for bilinear systems with a given yield strength and post yield stiffness. Constant strength design spectra could be useful for the design of base isolators with bilinear hysteretic behaviour, as these devices can be designed for fixed yield strength and post yield stiffness. The concept of constant strength design spectra and its application for the design of base isolated structures is illustrated with case studies of specific structures. Keywords: Base isolation; inelastic spectra; constant strength spectra; seismic design; displacement design methods.

1. Introduction Base isolation has emerged as a viable structural option in seismic zones during the last decade as a consequence of concerted effort of researchers and practitioners worldwide, particularly in New Zealand during the 1970s and 1980s and in the United States and Japan since the 1980s. There have been extensive experimental and analytical studies on different types of base isolators that have allowed the development of design practices for specific base isolators and code procedures, such as those established in the Uniform Building Code (UBC) since 1991, the American Association of State Highway and Transportation Officials (AASHTO) since 1990, and the new International Building Code (IBC) of the United States recently published in April, 2000 [“International Building Code”, 2000]. It is the perception of the author that the growing interest in the use of base isolation in seismic zones of the United States seems to be related to the publication of these seismic code provisions; however, other researchers have different opinion [i.e. Kelly, 1999]. At present, Japan is the country that has the most base isolated structures in the world and the number of their applications has risen after the AIJ published recommendations for the seismic design of base isolation. Other earthquake-prone 553


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countries with a long-time interest in base isolation, such as New Zealand and Italy, have constructed several base-isolated projects for buildings and bridges using design methodologies proposed in those countries or adapted from the ones used in other leading nations. Many other nations that face earthquake hazard have conducted research studies and/or built base-isolated structures. It is clear that the implementation of seismic isolation technology in new or retrofit projects for buildings and bridges in countries like the United States and Japan is directly related to the availability of relatively simple design guidelines and/or the publication of code provisions. In order to promote the use of seismic isolation in other countries, one of the first steps is to provide the structural engineering communities with suitable design procedures. Therefore, in Mexico the first steps in the development of design provisions for seismic isolation were based upon UBC guidelines, New Zealand practice, and the regional seismicity of the country [i.e. Tena et al., 1997, Villegas-Jiménez and Tena-Colunga, 2000]; these developments are still under way. The Mexican effort is also considering alternative design procedures based upon the use of inelastic spectra, for the reasons explained below. As stated before [Tena-Colunga et al., 1997] the seismic design of base isolators is generally controlled by their maximum allowable displacement for dynamic stability rather than strength. This fact has been recognised before in the literature on base isolation. However, design practice of base isolation advocated by some building codes is currently based on a procedure where the design displacement is obtained indirectly from a given pseudo-acceleration spectra specified for conventional structures and an equivalent viscous damping concept to model the non-linear response of base isolators. The UBC code provisions [1991, 1994, 1997 versions] and the IBC code [2000 version] share this design philosophy. However, an “optimal” design of base-isolated structures would require having all relevant information at hand, including inelastic responses. In this regard, a pseudo-acceleration response spectrum gives all of the required information, as peak responses for velocities and displacements are algebraically related with those provided for accelerations if an equivalent viscous damping concept is considered to model the non-linear response of base isolators. Peak responses for displacements are important for the design of the isolation system, whereas peak accelerations are of particular interest for the design of the structure above. Acceleration versus displacement spectrum is increasingly used in the design of conventional structures and could also be useful for the design of base isolators. However, in this spectrum the corresponding periods are usually not included in the plot. Effective isolated natural periods are essential information for the design of base isolated structures. Thus, the use of an inelastic tripartite design spectrum [i.e. Newmark and Hall, 1982] would be potentially more useful, as the non-linear action of base-isolated structures is concentrated on the isolators, and this spectrum gives all the needed information directly in a single chart, as peak responses in acceleration, velocity and displacements for given periods are included. However, the inelastic design


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spectrum as presented by Newmark and Hall [1982] is based upon the concept of fixed displacement ductility demands, a condition that is not the best concept for base isolators. For a given excitation, a displacement ductility demand can be associated to different yield strengths; that is, the ductility demand is not unique, as illustrated by others [for example, Miranda, 1993]. Base isolators, particularly those with a bilinear hysteretic behaviour, can be designed for a fixed yield strength and post yield stiffness. Therefore, constructing constant strength design spectra (CSDS) as presented here could be useful for the design of base-isolated structures. The CSDS relate peak accelerations, velocities, relative displacements and effective isolated natural periods for bilinear systems with a given yield strength and post yield stiffness. Some authors have proposed the use of inelastic spectra, particularly for the preliminary design of sliding isolation systems. Among these efforts it is worth mentioning the inelastic spectra presented by Kartoum et al. [1992] and by Tsopelas et al. [1996] for the design of sliding isolation systems in terms of: (a) coefficient of friction, (b) period of isolation, (c) damping ratio of the isolation system, (d) period of the free standing pier, (e) damping ratio of the free standing pier and, (f) weight ratio between the pier and the deck. The inelastic spectra presented in these references can be considered CSDS, but they differ from those presented here in that: (a) they were developed for sliding systems with frictional behaviour, not for systems with bilinear hysteretic behaviour, and (b) peak displacement responses and normalised pier shears are presented on separate plots and not in a single tripartite plot. 2. Inelastic Response Spectra In the elastic range of response, a tripartite plot can be used to relate, in a single chart, the maximum dynamic response of a single degree of freedom system (SDOF) subjected to an earthquake excitation, in terms of Sa , Sv and Sd , for a given natural period T . However, in the non-linear range, the maximum responses in acceleration, velocity and displacement are no longer proportional. Nevertheless, a tripartite plot can still be used for non-linear systems, although it is worth noting that in this case, accelerations, velocities and displacements have to be read with reference to their corresponding axes. Based upon the study of the dynamic response of elastic-perfectly-plastic SDOF systems subjected to earthquake ground motions to achieve fixed displacement ductility demands (µ), Newmark and Hall [1982] presented the concept of a “modified response spectrum” (MRS). It can be observed in this spectrum that peak accelerations, displacements and velocities are not proportional, so different envelope curves have to be defined for each response quantity for a given value of a displacement ductility demand (µ). It can also be observed that there exist frequency (period) ranges where peak responses are controlled by the non-linear displacements, whereas in other zones, peak responses are controlled by other response quantities.


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3. Constant Strength Response Spectrum: Concept and Computation The constant strength design spectrum (CSDS) is basically a modification of the “modified response spectrum” (MRS) proposed by Newmark and Hall [1982], where the normalised yield strength (Vy /W ) of a SDOF system is fixed rather than the displacement ductility demand (µ). The CSDS is obtained from constant strength response spectrum (CSRS) computed individually for specific ground motions and a given hysteretic model. A constant strength design spectrum (CSDS) can be defined for any suitable hysteretic model. In this paper, the CSDS is defined for bilinear systems with a given yield strength and post yield stiffness (Fig. 1). Many commercial base isolators have this type of hysteretic behaviour, among others, rubber bearings and steel hysteretic dampers. To compute a CSRS for a given ground motion excitation, the following steps must be followed: (1) Define the parameters of the hysteretic model. For a bilinear system, define Vy /W , k1 and k2 /ki . (2) Define the equivalent viscous damping ratio for the structural system alone (ξ), the acceleration record, initial (Ti ) and final (Tf ) periods of interest, and the period increment (∆T ) for the computation of dynamic responses. The increments could be in arithmetic or logarithmic scales. The use of logarithmic increments is recommended. (3) Do Tj = Tj−1 + ∆T , Ti ≤ Tj ≤ Tf . (4) For each period of interest Tj .

D=

Fig. 1.

Design envelope curve for bilinear isolators that follow the restrictions of the UBC code.


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(a) Check that the integration time step (∆t) for the acceleration record is suitable for the computation of non-linear problems. A rule of thumb is to check that ∆t ≤ Tj /10. Otherwise, interpolate the ground motion record. A selective interpolation algorithm should be defined. There is no need to use very small integration time steps for long periods while they are needed for short periods (close to zero). (b) For each time step ∆t, solve the equation of motion of a non-linear SDOF system using a suitable numerical method (for example, Newmark-β method) given by: mx(t) + c(t)x(t) ˙ + k(t)x(t) = F (t)

(3.1)

where m is the mass of the system, c(t), k(t) and F (t) are respectively the damping coefficient of the system, the stiffness of the system associated to ¨(t), x(t) ˙ and x(t) are respectively Tj , and the effective load at time t; and x the acceleration, velocity and displacement at time t, as defined in the literature. (c) Save the maximum responses in acceleration, velocity, displacement and non-linear system force. Then, the effective isolated period TI of the system can be computed as follows: r Fj−max + Fj+max m TI = 2π ; keff = − (3.2) keff ∆j max + ∆+ j max − + − + where Fmax j Fmax j and ∆max j ∆max j are respectively the maximum negative and positive forces and displacements and keff is the effective stiffness of the non-linear SDOF system with an initial elastic period of interest Tj .

(5) Go to Step 3 until Tj = Tf . As it can be observed, the computation of a CSRS is relatively simple. A CSRS computed for a bilinear system with Vy /W = 0.10, k2 /k1 = 0.10 and ξ = 0.05 when subjected to the ground acceleration record UNIO-NS (Fig. 2) registered during the 1985 Michoacán earthquake is depicted in Fig. 3. It can be observed that: (a) peak non-linear responses in acceleration, velocity and displacement are not directly related and, (b) there are period ranges where the non-linear response is controlled

Fig. 2.

UNIO-NS acceleration record for the 19/09/85 Michoacan Earthquake.


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Fig. 3. CSRS for UNIO-NS acceleration record for bilinear isolators with design parameters Vy /W = 0.10 and k2 /k1 = 0.10.

by the displacements, but in other regions the velocities or the accelerations rule. These are also illustrated with the MRS presented by Newmark and Hall. It can also be observed that, for a period close to zero (T ≈ 0.0), the peak non-linear response in acceleration approaches the peak ground acceleration (dotted line, Table 1) and that the relative displacement tends to zero (dashed line), as it should also be expected for a non-linear system.


Table 1.

Some characteristics of the selected acceleration records typical of rock sites. Scale Factor (fsc ) to Match the Spectral Acceleration of Duration (s)

Event

Station

Record Strong Phase

amax (cm/s2 )

Zone D-I of MOC-93 for T = 2.24 s

E-W

N-S

E-W

N-S

71.8 48.9 59.1 52.6 59.6 62.7 63.6

20 10 5 16 5 25 7

162.0 138.0 114.0 89.0 81.0 147.0 120.0

101.0 138.0 157.0 116.0 102.0 163.0 125.0

2.241 2.468 12.909∗ 13.041∗ 7.224 3.695 4.369

2.141 1.570 7.282 6.722 4.989 2.124 3.668

20/09/85

AZIH

54.0

15

153.0

134.0

3.154

4.543

04/25/89

CPDR

37.8

5

93.0

103.0

7.642

8.492

∗Accelerogram

not included because does not meet the selection criteria.

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AZIH CALE PAPN PARS SUCH UNIO VILE


09/19/85

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4. Constant Strength Design Spectrum Constant strength design spectrum (CSDS) could be defined for the design of baseisolated structures. The following items have to be defined for a CSDS: (1) the hysteretic model of interest and its parameters, (2) a set of representative ground motion records for a given seismic zone and soil type and, (3) a criterion to define the design curves. These three items have to be carefully assessed if the CSDS is intended for seismic guidelines, recommendations or design procedures for specific building codes. In this paper, CSDS is defined for representative strong ground motions recorded in the Mexican Pacific Coast during recent strong earthquakes using a crude statistical criterion for base isolators with bilinear behaviour. The CSDS presented in this study were computed to assess their effectiveness for the design of base isolated structures, but they have not been developed yet to comply with the seismic code criteria of ruling Mexican seismic codes for the Mexican Pacific Coast. The specific criteria used for items 1 to 3 mentioned above are briefly described below. 4.1. Selected hysteretic model For the present study, a bilinear hysteretic model with a post to preyielding stiffness ratio k2 /k1 = 0.10 was selected (Fig. 1). This was done because this is a typical ratio for most laminated rubber bearings and laminated lead rubber bearings [i.e. Skinner et al., 1993], which are of particular interest to the author. Two normalised yield strengths were selected, Vy /W = 0.05 and Vy /W = 0.10, in order to cover a wide range where most elastomeric base isolators could be used in regions of severe ground shaking. 4.2. Selected acceleration records Typical accelerograms for the Mexican Pacific Coast recorded during recent earthquakes were selected for the present study. The following earthquakes were considered: (a) the Ms = 8.1 September 19, 1985 Michoac´ an earthquake, (b) the Ms = 7.6, September 21, 1985 aftershock for the Michoacán earthquake, (c) the Ms = 6.9, April 25, 1989 earthquake and, (d) the Mw = 8.0, October 9, 1995 Manzanillo earthquake. A total of 42 accelerograms for the horizontal ground motions for 15 different stations were available. A preliminary selection criterion was to include only records typical of rock sites not close to the epicentre; thus, a total of 12 stations and 34 accelerograms remained. All remaining records were then scaled in order to match the spectral acceleration for T = 2.24 s of the design spectrum for zone D-I of MOC-93 code [MOC-93, 1993, Tena-Colunga, 1999], as illustrated for some selected records in Fig. 4. The effective period of the isolated structure T = 2.24 s was arbitrarily selected as is an intermediate period in the period range 1.5 s ≤ TI ≤ 3.0 s where it is recognised in the literature that base isolation is most appropriate, as


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Fig. 4. Scaling procedure for some selected accelerograms to match the spectral acceleration for the design spectra of zone D-I of MOC-93 code for T = 2.24 s.

further explained in Sec. 4.4. When the scaling amplification factor (fsc ) required for some accelerograms was very large, the resulting peak ground accelerations for the scaled records were unrealistically high (for example, amax = 5.11g for the scaled N-S record of VIGA station of the 04/25/89 Earthquake). Thus, a final selection criteria for the scaled records was: (a) that scaled peak ground accelerations would not surpass amax = 1.2g and, (b) the scaling amplification factor will not surpass 10 (fsc ≤ 10). Therefore, only 16 accelerograms of 9 stations fulfilled all these requirements and they are identified in Table 1. The average peak ground acceleration of the 16 selected scaled records is amax = 0.558g. It is worth noting that the criteria specifically used in this study are somewhat arbitrary and could potentially be unsafe, depending on the seismic hazard for a given region. 4.3. Criterion to define design curves The criterion to define design curves for the CSDS was influenced by the selection and scaling of the ground motions and by the main objective of the present study, that was to assess the potential effectiveness of CSDS for the design of base-isolated structures, rather than defining CSDS associated to specific building codes. The definition of CSDS for specific building codes would require complete seismic hazard


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analyses, where further considerations should be made in the selection of ground motions, sizes of design earthquakes for different performance levels, exceedence rates, etc. In the present study, CSDS were defined for the mean plus one standard deviation (σ + Sdev ) of the 16 selected acceleration records. This criterion was selected, among other considerations, because it is less conservative than defining an envelope curve for the 16 selected acceleration records, noting that higher multiples for the standard deviation have lower rates of exceedence. Alternatively, other more complex and complete probabilistic criteria may be used. 4.4. Procedure to define specific CSDS For illustration purposes, this section presents the procedure used to define the CSDS for isolators with bilinear behaviour with Vy /W = 0.10 and k2 /k1 = 0.10 (Fig. 5), emphasising the constant strength design spectrum for displacements (CSDSD) and for accelerations (CSDSA). The procedure follows the general criteria described in previous sections and was used also for the other case study (Vy /W = 0.05, Fig. 6). The constant strength response spectra corresponding to σ + Sdev for the 16 selected acceleration records are depicted in Fig. 5. Peak responses below an effective isolated natural period of 1.5 s (TI < 1.5 s) are of little interest for the design of the base isolators, although they can be important for the

Fig. 5. CSRS for σ + Sdev and CSDS for bilinear isolators with design parameters Vy /W = 0.10 and k2 /k1 = 0.10.


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Fig. 6. CSRS for σ + Sdev and CSDS for bilinear isolators with design parameters Vy /W = 0.05 and k2 /k1 = 0.10.

design of the superstructure. Therefore, spectra are presented for the period range 1 s ≤ TI ≤ 10 s. In order to define the design envelopes for displacements (CSDSD, thick solid line) and accelerations (CSDSA, thick broken line) shown in Fig. 5, the following considerations were made. It is recognised in the literature that base isolation is most appropriate when the effective period of the isolated structure is in the range 1.5 s ≤ TI ≤ 3 s. Skinner et al. [1993] propose the lower limit and the upper limit is defined by the UBC code for using static design procedures and some options for dynamic design procedures [“Uniform Building Code”, 1997]. To the author’s knowledge, the highest effective period considered for a retrofit project with base isolators is the one considered for Los Angeles City Hall, close to 4 s [Youssef, 1996a and 1996b]. Therefore, the design envelopes were defined to reasonably cover the period interval 1.5 s ≤ TI ≤ 4 s, being less conservative for periods longer than six seconds (TI ≤ 6 s) or lower than 1.5 s (TI < 1.5 s), as it can be observed in Fig. 5. An attempt was made to define the envelopes with a few straight lines while still reasonably protecting the period range 1.5 s ≤ TI ≤ 3 s. The design envelopes should cover the response maxima in terms of acceleration (CSDSA) and displacement (CSDSD). The smoothed curves of Figs. 5–7 seem to contain anomalies for TI > 6 s (particularly the increments observed for displacements) that


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Fig. 7. CSDS for bilinear isolators with design parameters Vy /W = 0.05, Vy /W = 0.10 and k2 /k1 = 0.10.

probably reflect the ground motion sample and statistics, as well as the correction procedure used for the accelerograms. In fact, for long periods (TI around 15 s), displacements should tend to converge to the peak ground displacement [i.e. Chopra, 1995]. 4.5. Proposed CSDS Constant strength design spectra for displacements (CSDSD) and accelerations (CSDSA) for Vy /W = 0.05 and Vy /W = 0.10 are depicted in Fig. 7. It can be observed from the CSDSD shown that, for the period range of interest for most base isolation applications (1.5 s ≤ TI ≤ 3 s), it would be more convenient to design the isolators for lower normalised yield strength ratios (Vy /W = 0.05), as peak displacements are smaller for this envelope. However, one has to look also at: (a) the design curves for the acceleration transmitted to the superstructure (CSDSA), shown in the same figure or in a separate figure, as the design spectra for the maximum normalised base shear transmitted to the superstructure (Vmax /W ), depicted in Fig. 8, and (b) the displacement ductility demands associated with the definition of the curves corresponding to σ + Sdev for the 16 selected acceleration records, as shown in Fig. 9. It can be observed from Figs. 7 and 8 that the base shear transmitted to the superstructure decreases as the period increases, since the non-linear response of


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Fig. 8. CSDSA for design of the superstructure with design parameters Vy /W = 0.05, Vy /W = 0.10 and k2 /k1 = 0.10.

Fig. 9. Displacement ductility demands associated to the CSRS and CSDS for σ + Sdev shown in Figs. 5 and 6.


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systems (i.e. ductility demands) with long periods is smaller when subjected to ground motions for rock conditions. However, one can also observe that for 1.5 s ≤ TI ≤ 2 s, higher shear forces are transmitted to the superstructure for systems with Vy /W = 0.05 than for systems with Vy /W = 0.10. It can be observed from Fig. 9 that peak displacement ductility demands (µ) associated to the design curves for Vy /W = 0.05 are very high for the period range of interest (1.5 s ≤ TI ≤ 3 s), and that for the same period range µ is more reasonable for the design curves for Vy /W = 0.10. Then, one can conclude that higher shear forces are transmitted to the superstructure for systems with Vy /W = 0.05 than for systems with Vy /W = 0.10 in the period range 1.5 s ≤ TI ≤ 2 s because higher force increments are associated to the postyielding stiffness k2 as µ increases. It can also be concluded that the selection of the yield strength ratio for the isolation system (Vy /W ) that it is more adequate for a given structure depends on many factors, so it is not a good strategy to decide the selection of the characteristics of the isolation system based exclusively on the observation of the constant strength design spectra for displacements (CSDSD). The design spectra depicted in Figs. 8 and 9 are also important because they allow to define the period range where elastic responses are expected (the range where Vmax /W for the superstructure is lower than the target Vy /W for the isolators or µ is around one). This can help engineers to avoid poor designs, as high damping rubber bearings (HRB) and lead rubber bearings (LRB) should be designed to work in the inelastic range of response when subjected to moderate and strong earthquake loading.

5. Design of Base-Isolated Structures Using CSDS The design procedure involves an iterative process, which is described in detail in Tena et al. [1997]. The key steps are summarised below: (1) Select a trial effective period for the isolated structure (TI ), considering the following restrictions: 1.5 s ≤ TI ≤ 3 s TI ≥ 2T

(5.1) (5.2)

where T is the natural period for the fixed-base structure. (2) Select the normalised yield strength (Vy /W = β) for the isolation system, by selecting the minimum β value that would transmit the smaller shear to the superstructure (Figs. 7 and 8) with an acceptable displacement ductility demand µ (Fig. 9). (3) Obtain the maximum design displacement for the isolation system, D, from the CSDS of Fig. 7.


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(4) The total effective lateral stiffness for the isolation system can be computed as: keff =

keff . gTI2

(5.3)

(5) Propose the number of isolators to be used, Niso . (6) Compute the effective lateral stiffness for each isolator: keff =

keff . Niso

(5.4)

(7) Compute the yield shear force of each isolator: Vy =

βW . Niso

(5.5)

(8) Compute the ultimate shear force of each isolator: Vu = keff D .

(5.6)

(9) Check that each isolator satisfies the minimum mechanical requirements specified. The author suggest to adopt a recommendation of the UBC code that establishes that the effective stiffness of the isolation system at the design displacement should be greater than one third of the effective stiffness at 20 percent of the design displacement (Fig. 1). This recommendation is called here “UBC stiffness restriction”. This restriction is instrumental for the design of isolators and makes the design process iterative. Otherwise, if any unrestricted bilinear curve could be taken, there will be no need for an iterative procedure. According to Fig. 1, the UBC restriction is given by: 1 keff 2 3

(5.7)

1 Vy + k2 (0.2D − ∆y ) Vu ≥ D 3 0.2D

(5.8)

keff ≥ then:

Vu ≥

Vy + k2 (0.2 − ∆y ) Vy + k2 (0.2D − ∆y ) = . 0.6 α

(5.9)

An iterative procedure based upon the bisection method is proposed here by introducing an α variable (Eq. (5.9)) that satisfies the UBC stiffness restriction. It is proposed to limit α values in the following range in order to assure having reasonably flexible isolation systems under lateral loading: 0.5 ≤ α ≤ 0.6. The most appropriate value is that of the upper limit, because when α = 0.6 one obtains the most flexible isolator under the UBC stiffness restriction. And if k2 /k1 = 0.10 is selected, then k1 = 5keff , as it can be deducted from Fig. 1, Eqs. (5.7)–(5.10), as shown in detailed elsewhere [Tena, 1997]. Therefore, one should initially propose α = 0.6.


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(10) Compute the elastic stiffness (k1 ) and the post yielding stiffness (k2 ) of the isolators: k1 = 12.5keff (1 − α)

(5.10)

k2 = 0.1k1 .

(5.11)

(11) Compute the yield displacement for the bilinear isolators: ∆y =

Vy . k1

(5.12)

Care should be taken that the yield displacement obtained would assure that peak displacement ductility demands µ for the isolators are within reasonable bounds for the assumed α. It can be shown from Eqs. (5.6) and (5.9)–(5.12) that: ∆y =

(1.25α − 0.25)D . 11.25(1 − α)

(5.13)

Therefore, ∆y should be in the following range: 0.067D ≤ ∆y ≤ 0.111D .

(5.14)

(12) Compute the αc value associated to the preliminary design: αc =

Vy + k2 (0.2D − ∆y ) . Vu

(5.15)

(13) Analyse the obtained value of αc . If αc > 0.6, it is obvious that the method will not converge for the desired α values. Then, the process should be stopped and a smaller value of TI should be proposed and start again. If αc < 0.5, the method can or cannot converge in the proposed range of values for α, as the outlined procedure is based on a bisection method. Therefore, if after two consecutive iterations αc < 0.5, the iteration should be stopped and the solution might be to propose a higher TI and start again. (14) Compare αc with α, using a rational convergence criterion, for example: α c − 1 ≤ 0.01 . (5.16) α If this condition is met, the design is accepted and one can continue with Step 15. Otherwise, one should propose α = αc and repeat the procedure from Step 10. (15) Do a preliminary dimensioning of the isolators based upon a recognised method. There are many methods already available in the literature that can be used for this [for example, Skinner et al., 1993; Kelly, 1993; Tena, 1997; Naeim and Kelly, 1999], including design aids and tables from a given manufacturer. In this work, the following expressions based on those presented by


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Skinner et al. [1993] were used. For laminated rubber bearings of circular cross section, its diameter φ and its height h can be computed as: φ = 3D h=

πφ2 Giso 4keff

(5.17) (5.18)

where Giso is the shear modulus for the rubber. Similar simple expressions can be used for the design of laminated lead-rubber bearings (LRB) as presented elsewhere [Tena, 1997]. If the dimensions obtained are those of a very slender isolator, probably one is proposing more isolators than those needed to achieve the required lateral flexibility. Therefore, one solution is to propose fewer isolators, until an adequate slenderness aspect ratio (h/φ) is achieved, as it may be clear by analysing Eqs. (5.4) and (5.18). On the other hand, if the dimensions obtained are those of a very short isolator, it is likely that one is proposing fewer isolators than those needed to meet the required lateral flexibility. Therefore, one solution is to propose more isolators until an adequate slenderness aspect ratio (h/φ) is achieved. 5.1. Orthogonal effects The CSDS shown in Fig. 7 are design envelopes of peak responses for unidirectional input only. However, for the design of 3D structures (with or without base isolation systems) it is important to include multidirectional actions of the ground motions. It has been common practice in the design of structural systems to account for the simultaneous action of the two main horizontal components of the ground motion; this is what has been called “orthogonal effects”. The most popular and widespread rule used in seismic codes worldwide is the called “100% + 30% combination rule”, that basically states that: “requirements of orthogonal effects may be satisfied by designing such elements for 100% of the prescribed design seismic forces in one direction plus 30% of the prescribed design seismic forces in the perpendicular direction”. This 100% + 30% combination rule was proposed many years ago based upon studies with a reduced acceleration data set (relative to today’s standards) for lateral load combinations. Therefore, extrapolation of this rule to combine maximum displacements for the design of isolators may not have a justification other that its use is widely accepted in the structural engineering design community. FEMA-273 document under Sec. 9.2.4.5.C basically endorses the 100% + 30% rule to compute the maximum displacement of the isolation system [FEMA-273, 1997]. For the design of the isolation system using dynamic analyses, the 1997 UBC code requires an increase of 30% in the target design spectra to account for bilateral ground motions. As shown by Naeim and Kelly [1999], the resulting vector summations of the requirement of the UBC-97 Code and the 100%+30% rule (FEMA-273) are very different.


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In the opinion of the author, the questions to answer regarding orthogonal effects for the design of isolation systems are: (1) is the 100% + 30% combination rule still valid and reasonable to compute design displacements? and, (2) whether a rational rule for the combination of orthogonal effects can be independent of the structural period and the characteristics of the ground motion. In order to gain some insight on this topic, Tena-Colunga and Gómez-Sober´ on [2002] started a study where the ratio between peak displacements of the isolators when subjected to bi-directional seismic input (∆2D ) were compared with those obtained for unidirectional input (∆1D ) for symmetric and asymmetric systems. From the relatively small simulations done in Tena-Colunga and G´ omez-Sober´ on [2002] it is not clear how a constant 100%+30% combination rule can be justified for the design of base isolators in the selected period range.

6. Design Examples of Base-Isolated Structures Using CSDS The proposed CSDS were calibrated for the design of the base isolators for three low-rise school buildings and an eight-storey reinforced concrete (RC) building; these studies are presented in detail in Tena et al. [1997]. The school buildings were selected as they are benchmark structures for the author, as his research team has previously used other strategies for the design of base isolators for them [Tena-Colunga, 1996; Tena, 1997, Tena et al., 1997]. For these school buildings, a normalised yield strength ratio Vy /W = 0.10 was selected, as it was done in previous studies. The eight-storey RC office building was selected as an irregular structure where base isolation could be used. A normalised yield strength ratio Vy /W = 0.10 was also used for this building. For illustration purposes, the details of one of the school buildings and the eight-storey RC building, the design of the isolators, and the non-linear dynamic analyses conducted for these designs are presented in following sections. 6.1. EP2 school building EP2 is a four-storey school building 12.4 m tall (storey height of 3.1 m) with typical bay width of 3.5 m in the longitudinal direction (E-W) and 8.7 m in the transverse direction (N-S). Transverse infill unreinforced hollow concrete block masonry walls are placed every two bays. Transverse outer walls are reinforced concrete walls. Longitudinal unreinforced clay masonry walls do not run the entire story height, and they shorten the columns (Fig. 10). The main structural system consists of moment frames with 25 × 50 cm rectangular RC columns oriented in the N-S direction, 25 × 50 cm rectangular RC beams, and 10 cm thick RC slabs as floor system. Longitudinal reinforcement is the same for all columns (ρ = 0.046). Transverse reinforcement consists of closed hoops made of No. 3 bars placed at 5 cm (2E#3 @ 5 cm) at the columns ends in a 50 cm length and 2E#3 @ 25 cm in the remaining length. The longitudinal reinforcement supplied at the top and bottom of the beams is symmetric and varies from ρ = 0.0057 for the top storey beams to ρ = 0.0095


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N

stairs

(a) Plan View

(b) Elevation axis A

(c) Elevation axis B Fig. 10.

Plan view and elevations for EP2 school building.

for the first storey beams (ρ = ρtop = ρbottom ). Transverse reinforcement consists of 2E#3 @ 10 cm at the beams ends in a 50 cm length and 2E#3 @ 20 cm in the remaining distance. The specified compressive strength of the concrete (fc0 ) was 200 kg/cm2 (20.39 MPa) and the specified yield strength of the reinforcement steel (fy ) was 4200 kg/cm2 (428.1 MPa). The assumed compressive strength of ∗ the masonry (fm ) was 15 kg/cm2 (1.53 MPa) for the clay units and 20 kg/cm2 (2.04 MPa) for the concrete blocks, according to the masonry provisions of the 1995 Mexico’s Federal District Code. Different 3D elastic models for the EP2 building were made with ETABS assuming that the building was fixed at the base and using all represented modes that insure having at least 90% of the total modal mass acting in each main direction. The details of each model and their dynamic characteristics are described in Tena-Colunga [1996]. The natural period for the structure is T = 0.63 s when the longitudinal “non-structural” walls are included in the modelling. Some of the dynamic characteristics of the first two mode shapes are summarised in Table 2. The total weight for EP2 building is W = 1530 Ton (15 009.3 kN).


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Main dynamic characteristics for EP2 school Modal Mass (%)

Mode

Direction

Period (s)

E-W

N-S

Rotation

1 2

E-W N-S

0.63 0.12

88.35 0.00

0.00 85.54

0.00 0.00

6.2. ECO1 building ECO1 is an eight-storey, irregular office building composed of reinforced concrete (RC) waffle flat-slab frame system and peripheral RC shear walls. Typical plan views, elevations and gross dimensions of the structure are depicted in Fig. 11. The structure is irregular in plan and it has a major irregularity in frame 1 of Fig. 11, where there is no waffle-slab for bay A-B up to the roof level. The typical cross section for columns is square. The columns are 70 × 70 cm in the first two

A

B

C

A

D

B

C

4

4

6.10

6.10 3

3

8.54

8.54 2 6.10

2 1 7.83

8.24

8.24

(a) Typical plan

9th 8th

N

7.83

8.24

(b) Heliport plan

4.90

7th

6th

33.08

5th

4.03

28.1

4th

3rd

2nd 1st 7.83

8.24

8.24

(c) Elevation axis D Fig. 11.

(d) Elevation axis 1

Plan views and elevations for ECO1 building (dimensios in m).


Table 3.

Main dynamic characteristics for ECO1 building. Modal Mass (%)

Mode

Direction

Period (s)

E-W

N-S

Rotation

1 2 3

E-W N-S Rotation

1.01 0.68 0.40

67.59 0.23 1.73

0.14 71.26 0.45

2.07 0.16 67.36

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Table 4. Design process of the isolation system for school building EP2, considering unidirectional actions for the ground motion and the CSDSD of Fig. 7.

Ta (s) W (Ton) β = Vy /W D (cm) (Fig. 7) Niso Kefft (Ton/cm) keff (Ton/cm) Vy (Ton) Vu (Ton)

2.0 1530 0.10 21.2 10 15.39 1.54 15.30 32.63

Variable parameters

Iteration 1

Iteration 2

Iteration 3

α K1 (Ton/cm) K2 (Ton/cm) ∆y (cm) ∆y /D αc

0.6 7.70 0.77 1.99 0.094 0.522

0.522 9.20 0.92 1.66 0.078 0.548

0.548 8.70 0.87 1.76 0.083 0.545

Iteration 4 0.545 8.76 0.88 1.75 0.082 0.545


“Fixed” parameters

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storeys and 60 × 60 cm in the remaining stories. The floor system is a RC waffle flat-slab 5 cm thick with 16 cm wide main ribs measuring 30 cm in depth. The “waffles” are formed by hollow lightweight concrete blocks. The thickness of the concrete walls is 20 cm. Yield strength of reinforcement steel is fy = 4200 kg/cm2 (428.1 MPa) and compressive strengths for concrete are 300 kg/cm2 (30.58 MPa) for the two first stories and 250 kg/cm2 (25.48 MPa) for the upper stories. 3D elastic models for the building were made with ETABS, assuming that the building was fixed at the base and using sufficient modes to insure having at least 90acting in each main direction. Young modulus for reinforced concrete was taken according to the concrete norms from Mexican codes. The remaining modelling assumptions for structural and non-structural elements are described in Tena et al. [1997]. The dynamic characteristics for ECO1 building are reported in detail elsewhere and some of the main characteristics of the first three mode shapes are summarised in Table 3. The natural period for the structure is T = 1.01 s. Despite the irregularities, the mode shapes are just lightly coupled. The total weight for the building is W = 4793.5 Ton (47 024.2 kN).

Table 5. Summary of design parameters for the isolation system and the superstructure, considering unidirectional actions for the ground motion and the CSDSD (Fig. 7) and CSDSA (Figs. 7 or 8). EP2 Global design T (s) Ta (s) W (Ton) β = Vy /W D (cm) (Fig. 7) Niso Kefft (Ton/cm) α

ECO1

0.63 2.0 1530.0 0.10 21.2 10 15.39 0.545

1.01 2.5 4793.5 0.10 28.0 16 30.865 0.600

Parameters for the design of the superstructure Vs /W (Fig. 8) Vs (Ton)

0.154 235.6

0.135 647.1

Individual characteristics of the isolators (HRB) Vu (Ton) Vy (Ton) ∆y (cm) ∆M = D (cm) keff (Ton/cm) k1 (Ton/cm) k2 (Ton/cm) ∆y /D φ (cm) H (cm) Giso (kg/cm2 )

32.63 15.30 1.75 21.2 1.54 8.76 0.88 0.082 64 21 10.2

54.01 29.96 3.11 28.0 1.93 9.65 0.97 0.111 84 30 10.2


Table 6. Design process of the isolation system for school building EP2, considering bi-directional actions for the ground motion and the CSDSD of Fig. 7. “Fixed” Parameters

Variable Parameters

Iteration 1

Iteration 2

Iteration 3

Iteration 4

Iteration 5

α k1 (Ton/cm) k2 (Ton/cm) ∆y (cm) ∆y /D2D αc

0.6 8.21 0.82 3.11 0.085 0.484

0.484 10.59 1.06 2.41 0.066 0.513

0.513 9.99 1.00 2.55 0.070 0.506

0.506 10.14 1.01 2.51 0.069 0.508

0.508 10.10 1.01 2.52 0.069 0.507

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2.5 1530 0.10 28 36.4 6 9.85 1.64 25.50 59.76


Ta (s) W (Ton) β = Vy /W D (cm) (Fig. 7) D2D = D + 0.3D (cm) Niso Kefft (Ton/cm) keff (Ton/cm) Vy (Ton) Vu (Ton)

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6.3. Design of base isolators High damping rubber bearings (HRB) were designed for the EP2 and ECO1 buildings using the CSDSD depicted in Fig. 7 and the procedure outlined in Sec. 5. Vy /W = 0.10 was selected for the design of the isolation systems of both buildings. The key step in the proposed method is the selection of the design displacement D from the CSDSD depicted in Fig. 7. In order to calibrate the validity of the CSDSD of Fig. 7, only unidirectional action of the ground motion records should be considered. Therefore, the design procedure for the isolation system of school building EP2 under unidirectional action of the ground motions is illustrated in Table 4. The final designs of the isolation systems for EP2 and ECO1 buildings under unidirectional action of the ground motions are summarised in Table 5. It is also interesting to study how well does the 100% + 30% rule for orthogonal effects work for these particular structures. Therefore, the design procedure for the isolation system of school building EP2 using this combination rule for orthogonal effects is illustrated in Table 6. The final designs of the isolation systems for EP2 and ECO1 buildings under bi-directional action of the ground motions using the 100% + 30% combination rule are summarised in Table 7.

Table 7. Summary of design parameters for the isolation system and the superstructure, considering bi-directional actions for the ground motion and the CSDSD (Fig. 7) and CSDSA (Figs. 7 or 8). EP2

ECO1

Global design T (s) Ta (s) W (Ton) β = Vy /W D (cm) (Fig. 7) D2D = D + 0.3D (cm) Niso Kefft (Ton/cm) α

0.63 2.5 1530.0 0.10 28.0 36.4 6 9.85 0.508

1.01 2.5 4793.5 0.10 28.0 36.4 16 30.865 0.508

Parameters for the design of the superstructure Vs /W (Fig. 8) Vs2D /W = Vs /W + 0.3Vs /W Vs2D (Ton)

0.135 0.176 268.5

0.135 0.176 843.7

Individual characteristics of the isolators (HRB) Vu (Ton) Vy (Ton) ∆y (cm) ∆M = D2D (cm) keff (Ton/cm) k1 (Ton/cm) k2 (Ton/cm) ∆y /D2D φ (cm) h (cm) Giso (kg/cm2 )

59.76 25.50 2.52 36.4 1.64 10.10 1.01 0.069 110 58 10.2

70.22 29.96 2.52 36.4 1.93 11.86 1.19 0.069 110 50 10.2


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Different arrangements and numbers of HRB were considered in the design process. The number of HRB presented in Tables 4 to 7 corresponds to the best design of such isolators taking into account their dimensions. It is worth noting that the dimensions for the HRB of Tables 4 to 7 correspond to a preliminary design, as the actual dimensions of the HRB for a final design have to be adjusted to commercial sizes working together with a manufacturer. 6.4. Design of the superstructure The design of the structural system was not done as EP2 and ECO1 buildings involve the retrofit of existing buildings. However, the base shear coefficient for the design of the superstructure can be defined with the curves depicted in Figs. 7 and 8. The corresponding designs are summarised in Tables 5 and 7 for unidirectional and bi- directional action of the ground motions respectively. 6.5. Non-linear dynamic analyses The proposed designs for the isolation systems for EP2 and ECO1 buildings were tested with non-linear dynamic analyses of the base-isolated models when subjected to unidirectional and bi-directional acceleration input using the 3D-Basis program [Nagarajaiah et al., 1991]. Unidirectional input was used for the designs summarised in Table 5 whereas bi-directional input was used to evaluate the designs summarised in Table 7. Analyses for unidirectional input were conducted considering the E-W components as the “x component” and the N-S as the “y component”, orthogonal to both the short and long direction, one at the time. For these case studies, the acceleration records were scaled with the corresponding amplification factor fsc for each component identified in Table 1. For bi-directional input, an “x-y” quake is defined when the E-W component is acting in the long direction, whereas a “yx” quake is defined when the N-S component is acting in the long direction. In addition, for bi-directional input, the amplification factor fsc corresponding to the strongest component was used to scale both acceleration records. For example, for station UNIO, fsc = 2.124 associated to the N-S component (Table 1) was used to scale both the N-S and the E-W component. Results obtained from these analyses are summarised for EP2 school building in Tables 8 (unidirectional action) and 9 (bi-directional action) and for ECO1 building in Tables 10 (unidirectional action) and 11 (bi-directional action). In these tables, ∆i is the maximum dynamic displacement for a given isolator at an angle θ from the x axis, ∆M is the maximum allowable isolator displacement for dynamic stability as defined in Tables 5 and 7, ∆x max and ∆y max are the maximum relative roof displacements with respect to the isolation system in the x and y directions respectively, Vxe and Vye are the peak base shear forces transmitted to the structure in the x and y directions respectively, Vxi and Vyi are the peak shear forces developed for the isolators in the x and y directions respectively, and W is the weight for the structure.

Quake

∆x max (mm)

∆y max (mm)

∆i (cm)

∆i /∆M

Vxe /Vxi

Vye /Vyi

Vxe /W

Vye /W

AZIH CALE UNIO PAPN VILE SUCH

y y y y y y

19.0 14.8 21.8 21.7 31.4 24.9

0.1 0.0 0.1 0.1 0.2 0.1

12.62 11.42 11.53 20.30 20.49 16.35

0.60 0.54 0.54 0.96 0.97 0.77

0.83 0.73 0.92 0.81 0.99 0.89

0.71 0.75 0.90 0.86 0.73 0.70

0.135 0.114 0.144 0.168 0.176 0.165

0.003 0.001 0.001 0.001 0.000 0.001

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∆M = 21.2 cm, W = 1530 Ton = 15 009.3 kN. Table 9. Peak dynamic responses for the isolation project under bi-directional action of the ground motion for EP2 school building (TI = 2.5 s). Station

Quake

∆x max (mm)

∆y max (mm)

∆i (cm)

∆i /∆M

Vxe /Vxi

Vye /Vyi

Vxe /W

Vye /W


y-x y-x y-x x-y y-x y-x

15.8 12.9 16.7 12.4 24.3 20.0

0.6 0.5 0.6 0.9 0.6 0.6

14.79 16.79 14.34 25.13 28.22 17.37

0.41 0.46 0.39 0.69 0.78 0.48

0.86 0.74 0.84 0.86 0.99 0.90

0.72 0.71 0.71 0.71 0.71 0.71

0.113 0.101 0.112 0.090 0.168 0.134

0.077 0.065 0.078 0.125 0.084 0.079

∆M = 36.4 cm, W = 1530 Ton = 15 009.3 kN.

A. Tena-Colunga

Station


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Table 8. Peak dynamic responses for the isolation project under unidirectional action of the ground motion for EP2 school building (TI = 2.0 s).


Table 10. Peak dynamic responses for the isolation project under unidirectional action of the ground motion for ECO1 building (TI = 2.5 s). Quake

∆x max (mm)

∆y max (mm)

∆i (cm)

∆i /∆M

Vxe /Vxi

Vye /Vyi

Vxe /W

Vye /W


x y y y y y

42.9 50.5 59.2 71.7 69.0 77.4

5.0 5.3 6.9 8.9 8.4 9.1

15.92 17.39 16.41 29.57 23.14 18.88

0.57 0.62 0.59 1.06 0.83 0.67

0.89 0.89 0.95 0.78 0.90 1.07

1.07 0.92 0.91 1.03 0.97 0.95

0.103 0.105 0.109 0.120 0.137 0.130

0.018 0.017 0.018 0.015 0.020 0.021

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Station

∆M = 28 cm, W = 4793.5 Ton = 47 024.2 kN.

Station

Quake

∆x max (mm)

∆y max (mm)

∆i (cm)

∆i /∆M

Vxe /Vxi

Vye /Vyi

Vxe /W

Vye /W


x-y y-x y-x y-x y-x y-x

37.9 50.0 62.9 72.5 67.0 73.9

22.4 12.6 19.3 19.9 31.4 18.4

20.27 16.18 16.88 32.93 27.03 20.44

0.56 0.45 0.46 0.90 0.74 0.56

0.89 0.89 1.00 0.77 0.86 1.01

0.92 0.92 0.86 0.89 0.97 0.93

0.094 0.104 0.110 0.129 0.119 0.134

0.112 0.070 0.088 0.092 0.102 0.085

∆M = 36.4 cm, W = 4793.5 Ton = 47 024.2 kN.


Table 11. Peak dynamic responses for the isolation project under bi-directional action of the ground motion for ECO1 building (TI = 2.5 s).

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It can be observed from Tables 8 and 10 that the proposed designs for unidirectional action are adequate for all ground motions, as the quotient ∆ i /∆M is always less than one, except for station PAPN for ECO1 building, where the maximum allowable displacement is surpassed by 6%. However, the design procedure based on the CSDSD of Fig. 7 can be taken as satisfactory, considering the statistical criterion used to define such curves. CSDS shown in Fig. 7 were defined from a reduced set of ground motions using as statistical criterion the mean plus one standard deviation (associated with an accumulated probability density function equal to 0.8413 for a normal distribution). Improved results would have been obtained adding higher multipliers for the standard deviation. On the other hand, it can be observed from Tables 9 and 11 that the proposed designs for bi-directional input using the 100% + 30% combination rule are adequate for all ground motions for these particular structures, as the quotient ∆i /∆M is always less than one. Maximum responses for the isolators are related to the amplified motions for stations PAPN and VILE for both unidirectional and bi-directional input. Relative roof displacements are very small for both buildings for both unidirectional and bi-directional input as well. The effectiveness of the isolation system in reducing the base shear transmitted to the superstructure (Vxe /Vxi and Vye /Vyi ratios) is between 70% and 99% for EP2 building (Tables 8 and 9) and from 76% to 100% (Tables 10 and 11) for ECO1 building for most stations, but it can be even 7% higher as it happens for station SUCH and unidirectional input on the direction of interest (Table 10). Higher effectiveness in reducing the base shear transmitted to the superstructure has been observed for other buildings studied by the author [for example, Tena et al., 1997]. The maximum normalised dynamic base shear transmitted to the superstructure (Vxe /W ) for EP2 school building under unidirectional input (Table 8) obtained for stations VILE, PAPN and SUCH surpass the one obtained from ICDSA (VS /W = 0.154, Table 5), but the ones obtained for stations AZIH, UNIO and CALE are less than the one reported in Table 5. For ECO1 building, the normalised base shear obtained from CSDSA (VS /W = 0.135, Table 5) is only surpassed for station VILE under unidirectional input (Table 10). Therefore, for most stations the estimate of the maximum base shear from CSDSA is reasonable, considering the statistical criterion used to define such curves. On the other hand, the Vxe /W ratios for bidirectional input for buildings EP2 (Table 9) and ECO1 (Table 11) do not surpass the normalised base shear obtained from ICDSA for EP2 and ECO1 buildings using the 100% + 30% combination rule (VS /W = 0.176, Table 7). The non-linear dynamic analyses conducted for EP2 and ECO1 buildings suggest that the CSDS presented in Figs. 7 and 8 were useful and can be reliable for the design of base isolators, despite the shortcomings of the present study, among them, the use of a statistical criterion based upon the mean plus one standard deviation with a limited strong ground motion data base for defining the CSDS.


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Therefore, the concept of CSDS seems to be a promising tool for reliable design of base isolators with bilinear hysteretic behaviour, although a more rigorous procedure should be use to define design curves if CSDS are intended to be associated with a specific building code. 7. Summary and Conclusions This paper presents the concept of constant strength design spectra (CSDS) for the design of base-isolated structures, particularly those structures using isolators with a bilinear hysteretic behaviour when subjected to dynamic loading. The CSDS relate peak accelerations, velocities, displacements, and effective isolated natural periods for bilinear systems with a given yield strength and post yield stiffness. CSDS could be useful for the design of base isolators with bilinear hysteretic behaviour as these devices can be designed for a fixed yield strength and post yield stiffness. In this paper, CSDS were defined for representative strong ground motions recorded in the Mexican Pacific Coast during recent strong earthquakes using a crude statistical criterion for base isolators with bilinear behaviour. The CSDS presented in this study were computed to assess their effectiveness for the design of base-isolated structures, but they have not been developed yet to comply with code criteria of ruling Mexican seismic codes for the Mexican Pacific Coast. The development of CSDS associated to specific building codes would require complete seismic hazard analyses, where detailed considerations should be made in the selection of ground motions, sizes of design earthquakes for different performance levels, exceedence rates, etc., steps that will be taken in future works. The concept of CSDS and its application for the design of base isolated structures is illustrated in two buildings. Non-linear dynamic analyses conducted for EP2 and ECO1 buildings (and other structures, not shown) suggest that the proposed CSDS are useful and could be reliable for the design of base-isolated structures, despite the shortcomings of the study described in previous sections. Among these shortcomings, one can highlight the following: (a) using a simple statistical procedure with a limited strong ground motions data base, and (b) using the 100%+30% combination rule to account for bi-directional effects, as there are no specific studies yet devoted to propose more rational magnification factors due to this loading condition for base-isolated structures. Nevertheless, the concept of CSDS seems to be a promising tool for reliable design of base isolators with bilinear hysteretic behaviour. Further research is needed in order to define CSDS for specific building codes where, in addition to the items described above for seismic hazard analyses, a consideration must be made in order to incorporate the impact of bi-directional ground motions and the vertical component for the acceleration in a rational basis, based on specific studies devoted to base-isolated structures.


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Acknowledgements Financial support of Conacyt (National Science and Technology Council of Mexico) and Universidad Aut´ onoma Metropolitana is gratefully acknowledged. Constructive comments from anonymous reviewers for this paper were helpful to improve it and are gratefully acknowledged.

Appendix A. Notation The following symbols are used in this paper: amax = peak ground acceleration c(t) = damping coefficient at time t for a single degree of freedom system D = design displacement for the isolation system under unidirectional input D2D = design displacement for the isolation system under bi-directional input f = frequency of vibration fc0 = compressive strength of concrete fsc = scale amplification factor for the ground motions fy = yield strength of reinforcement steel F (t) = effective load at time t − Fmax j = the maximum negative force of the non-linear SDOF system with an initial elastic period of interest Tj . + Fmax = the maximum positive force of the non-linear SDOF system with an j initial elastic period of interest Tj . Giso = shear modulus for the elastomeric bearing k(t) = stiffness at time t of a single degree of freedom system kefft = total effective stiffness of the isolation system at the design displacement keff = effective stiffness of an isolator unit at the design displacement keff2 = effective stiffness of an isolator unit at 20% of the design displacement k1 = elastic (pre-yield) stiffness of a bilinear system k2 = postyielding stiffness of a bilinear system m = mass of a single degree of freedom system Ms = surface wave magnitude of an earthquake Mw = moment magnitude of an earthquake Niso = number of isolators to be used Sa = peak spectral acceleration Sd = peak spectral relative displacement Sdev = standard deviation Sv = peak spectral velocity T = period of vibration Tf = ending period of interest Ti = initial period of interest TI = effective period of the base-isolated structure


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Vs = maximum design shear force transmitted to the superstructure for the isolation system for unidirectional input Vs2D = maximum design shear force transmitted to the superstructure for the isolation system for bi-directional input Vu = ultimate shear force for an individual isolator Vy = yielding shear force for an individual isolator or the complete isolation system Vmax = maximum base shear transmitted to the superstructure by the isolation system Vxe = maximum base shear force transmitted to the superstructure, in the x direction Vxi = maximum shear force developed for the isolation system, in the x direction Vye = maximum base shear force transmitted to the superstructure, in the y direction Vyi = maximum shear force developed for the isolation system, in the y direction W = weight of a structural system x(t) = displacement at time t x(t) ˙ = velocity at time t x ¨(t) = acceleration at time t α, αc working variables β = normalised shear yield strength for the isolation system ∆i = maximum dynamic displacement for a given isolator in an angle θ from the x axis ∆max = maximum allowable displacement for base isolators for dynamic stability ∆M = maximum allowable displacement for base isolators for dynamic stability ∆t = integration time-step ∆x max = maximum relative roof displacement with respect to the isolation system, in the x direction ∆y max = maximum relative roof displacement with respect to the isolation system, in the y direction ∆y max = yield displacement for an isolator unit ∆− max j = the maximum negative displacement of the non-linear SDOF system with an initial elastic period of interest Tj . ∆+ = the maximum positive displacement of the non-linear SDOF system max j with an initial elastic period of interest Tj . µ = displacement ductility demand σ = statistical mean θ = measured angle (degrees) from the x axis ξ = equivalent viscous damping ratio.


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