Non-linear dynamic modelling and design procedure of FV spring-dampers for base isolation

Engineering Structures 23 (2001) 1556–1567 www.elsevier.com/locate/engstruct

Non-linear dynamic modelling and design procedure of FV springdampers for base isolation Stefano Sorace

a,*

, Gloria Terenzi

b

a b

Department of Civil Engineering, University of Udine, Via delle Scienze 208, 33100 Udine, Italy Department of Civil Engineering, University of Florence, Via di S. Marta 3, 50139 Florence, Italy Received 19 February 2001; received in revised form 22 June 2001; accepted 22 June 2001

Abstract A viable non-linear dynamic design procedure for fluid viscous spring-dampers used in base-isolation systems of building structures is presented herein. The procedure consists of a preliminary design phase developed for a single-degree-of-freedom idealisation of the structural system, and in a final verification phase. The damping coefficient values ensuring achievement of the basic performance levels hypothesised in the design problem are estimated at the first stage. An explicit relationship interpolating the damping coefficient demand curves traced out as functions of damper loss factor, and dynamic input frequency composition and amplitude, is proposed as the basic design equation to this aim. This relationship, derived from previous analytical studies, is further validated herein against the results of experimental forced-vibration tests conducted on a steel frame mock-up. A first demonstrative application of the procedure is then carried out on a simple building with very stiff superstructure. Doubled calibration of the damping coefficient under normative-generated and pulse-type input ground motions, and evaluation of the influence of the pre-stress load imposed on several types of fluid viscous devices in the manufacturing process, are included. Selected case studies are discussed in the accompanying paper.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Base isolation; Supplemental damping; Non-linear dynamic design analysis; Fluid viscous devices

1. Introduction Fluid viscous (FV) devices represent an advanced technology in the field of seismic protection and vibration control of buildings. Operating on the principle of the flow of special compressible fluids through orifices, these elements are characterised by very high cyclic-fatigue life, compact size in comparison to reaction force capacity, and a damping force component that is out of phase with displacement. Concurrent with the remarkable energy dissipation levels that can be attained by means of properly conceived orifice shapes and locations, all these features ensure high mechanical performance when utilised both within passive [1,2] and semi-active [3] control systems. Furthermore, the possibility of freely calibrating the behaviour of FV devices

* Corresponding author. Tel.: +39-432-558050; fax: +39-432558052. E-mail address: [email protected] (S. Sorace).

between the two limit conditions of pure springs, or pure dampers, prompts their use for a variety of protective strategies (base isolation combined with supplemental energy dissipation; damped bracings; damped cables; etc). Currently, four general design procedures are recommended by earthquake regulations for new buildings [4], and guidelines for rehabilitation of existing ones [5], with regard to structural systems with seismic protective elements: linear static and dynamic ones; and non-linear static and dynamic ones. All these procedures are typically framed within the context of performance-based design, and try to extend the basic criteria defined for traditional structures to base-isolated and supplementally damped buildings. Linear techniques are admitted only for limited and almost uniformly distributed response non-linearities, a satisfactory schematisation of which can be obtained by referring to the “linearised” viscous damping measures suggested in the same documents, as well as in other recent studies dealing with this topic [6]. Non-linear static (NLS) “push-over” methods allow

0141-0296/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 1 ) 0 0 0 6 3 - 3

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exploration of the post-yielding response of the superstructure, so as to add building performance levels characterised by non-negligible damage states to the elastic response-related levels that basically govern the definition of design objectives. Concerning application to the class of velocity-dependent FV spring-dampers, a limitation of the NLS method, as well as of linear approaches, is the energy-dissipation equivalence by which the “linearised” damping coefficient — utilised to reduce the 5%-damped response spectrum ordinates — is quantified. This equivalence, which equates the energies dissipated in a vibration cycle by the actual nonlinear system and by the equivalent linear one, does not account for the rate-sensitivity effects in the dynamic behaviour of these elements. Furthermore, the actual structural velocity, rather than the spectral pseudo-velocity commonly employed for static seismic calculation, should be adopted to achieve substantial convergence to the exact (i.e., dynamic time-history) numerical solution. Both limitations can be overcome by static procedures properly conceived for velocity-dependent devices, such as the one proposed in [7,8] for FV dissipaters. Based on a normalised damper capacity referred to a “power consumption” approach, the procedure directly considers the rate of energy dissipation instead of energy, and transformation of pseudo-velocity into actual structural velocity. Improved correlation to dynamic analysis results is ensured by these modifications [7]. For practical applications, an empirical equation has also been provided in [7] to estimate preliminarily the amount of equivalent viscous damping added by the FV protective system. This facilitates the single-degree-of-freedom (SDOF) idealisation withstanding push-over analysis. Considerable computational effort is however needed to apply this procedure, as well as less sophisticated nonlinear static design methods, since the following steps need anyway to be developed: (a) a complete modal analysis to reduce the actual multi-degree-of-freedom (MDOF) system to the idealised SDOF scheme; (b) nonlinear push-over calculation and transformation of the relevant response curve into the acceleration—displacement response spectrum format, when the capacity-spectrum method [4,5] is employed; (c) for the general case of a multi-objective design approach, final verification of dynamic performance under the maximum considered earthquake, in order to assess that for this extreme level of action damage to structural members is within acceptable limits, and the protective system does ensure sufficient operation capacities [8]. It can thereby be noted that, although NLS procedures should in principle represent an easier design approach compared to non-linear dynamic (NLD) ones, they nevertheless require the use of practically the same structural analysis programs (points (b) and (c)), thus involving the same level of expertise by the professional structural designer. Based on these observations of NLS methods, it was

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deemed of interest to define a workable NLD procedure that, without entailing greater effort of analysis, has the advantage of giving a unified approach to the preliminary and final-verification design phases. The basic tool of the method proposed in this paper is a design equation that directly provides the damping coefficient as a function of loss factor, and dynamic input amplitude and spectral composition. Formulated on the basis of previous analytical studies [9,10], and further validated herein with experimental tests, this equation allows easy selection of the damping coefficient range ensuring the desired mechanical performance of the protective system. After summing up the analytical relations of the computational model utilised to simulate the behaviour of FV devices, as well as the relevant coefficient tuning on experimental basis; and the most significant trends of seismic response obtained from a parametric investigation developed by the same model, the procedure contents are detailed, and a simple application discussed. The mutual influence among the pre-load optionally imposed on several types of FV elements, base-isolation period, and damping coefficient is also analysed. This procedure — which, to the authors’ knowledge, represents the first proposal of a self-consistent NLD design approach to the class of non-linear viscous devices — is applied to two actual case studies in a companion paper [11]. 2. Analytical model of FV spring-dampers 2.1. Model formulation The equation of motion of a SDOF system subjected to a dynamic action f(t), with a mass m connected to a FV device with combined hysteretic and “spring” properties (according to the Kelvin—Voigt rheological model sketched in Fig. 1) is:

Fig. 1.

Schematic of SDOF system incorporating FV spring-damper.

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mx¨ (t)⫹Fd(t)⫹Fe(t)⫽f(t)

(1)

where x¨ (t)=response acceleration. The Fd(t) damping and Fe(t) elastic reaction force components can be expressed as: Fd(t)⫽csgn(x˙(t))|x˙(t)|

a

(k1−k2)x(t) R k1x(t) 1/R 1+ F0

冋 | |册

Fe(t)⫽k2x(t)⫹

(2) (3)

where c=damping coefficient; sgn(•)=signum function; |•|=absolute value; a=fractional exponent (less than 1); k1, k2=translation stiffness of the response branches situated below and beyond the pre-load threshold, respectively; F0=pre-load imposed in the manufacturing phase; and R=integer exponent. The non-linear viscous hysteretic response defined by (2), as well as the physical meaning of parameters k1, k2, and F0 are schematically shown in Fig. 1. Combination of terms (2) and (3) allows simulation of the actual behaviour of a variety of FV devices [12,13], i.e., single- or double-acting; ranging from pure dampers to pure spring elements; and with or without pre-load. Concerning the pre-load, applied to ensure re-centring capabilities after termination of a dynamic excitation, as well as to avoid device operation under low input actions, the following empirical relation can be used to calculate F0 in the design phase: Fmax F0⫽ n

the SA spring-damper, in the latter element. A vertical pin sliding within two slots of the intermediate and outer casings alternately joins the two chassis to the inner housing (under tension), or allows them relative motion (under compression) [10]. This produces an identical response capacity of DA devices in both senses of motion. Repeated series of quasi-static and dynamic cyclic tests were conducted on the two types of spring-dampers under displacement-controlled sinusoidal motions. Typical reaction force—displacement loops obtained from tests on one medium-sized SA and one small-sized DA device are illustrated in Fig. 2. The response curves obtained by fitting relations (2) and (3) against test results are also plotted in this figure, together with a schematic of the two types of devices, and show a satisfactory correlation between experimental and analytical data. The a exponent values calibrated for the two devices are equal to 0.11 (SA) and 0.18 (DA), that is, near the lower and upper bounds of the [0.1, 0.2] overall range identified for the class of silicone-fluid FV elements [12].

(4)

where Fmax=maximum reaction force generated by the device under quasi-static loading (whose value is provided by the manufacturer). Recommended n values range from 7 to 2, with the basic choice fixed at 5. Different n values could be used by assuming n as an explicit design variable. Numerical integration of Eq. (1) (where f(t)=⫺ mu¨g(t) in the case of seismic action, u¨g(t) being the ground acceleration time-history) is conducted by the implicit—explicit algorithm originally proposed in [14], and modified in [9] to introduce the non-linear term (2) into the iterative solution scheme. This method proved to be accurate and quickly converging in previous studies of SDOF and MDOF systems characterised by the reaction force components (2) and (3), under harmonic and seismic input motions [9,10]. 2.2. Experimental calibration of the model The experimental surveys concerned a series of single-acting (SA) and double-acting (DA) silicone-fluid pre-loaded FV devices in ordinary production [10]. The main difference between a SA and a DA element is the addition of two coaxial casings around the basic core of

Fig. 2. Experimental and analytical response loops obtained from selected tests on (a) SA and (b) DA FV devices.

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As regards the remaining model parameters, a value of 5 for the coefficient R was derived from the fitting process. Since this result is on average valid for all types of FV devices tested in the previous phases of this research [10], and considering that R is a shape factor that does not influence the energy content of response, the assumption R=5 was held throughout the entire numerical investigation. Regarding a, two parallel analyses were conducted with the limit values pointed out above (0.1 and 0.2) and they provided very similar responses [9]. In view of this, only the outcome of a=0.1 is presented, for brevity’s sake, within the synthesis reported in the next section.

3. Seismic response of SDOF systems incorporating FV spring-dampers Twenty-one SDOF oscillators — representative of base-isolated systems for which the superstructure is stiff enough to be regarded as a rigid mass — with vibration periods Tis varying between two limits of interest for application of the base-isolation technique to building structures (1.5 to 3.5 s), were examined. These oscillators were subjected to: (a) 15 Italian real seismic records, derived from the Friuli 1976, Campania–Lucania 1980 [15], and Umbria–Marche 1997 earthquakes, all scaled at a peak amplitude Ai=0.35 g; (b) an ensemble of eight artificial signals generated from the design spectrum of Eurocode 8 [16], for B-soil class, 5% damping factor, and Ai=0.35 g; and (c) the cycloidal pulse function defined by the following relations [17]: u¨ g(t)⫽wpnp cos (wpt) 0ⱕtⱕTp

(5)

u˙ g(t)⫽np sin (wpt)

(6)

ug(t)⫽

0ⱕtⱕTp

np np ⫺ cos (wpt) 0ⱕtⱕTp wp wp

(7)

where wp=2p/Tp, vp=0.7 m/s, and Tp=3.2 s, which resembles the typical near-source forward-and-back pulse recorded at El Centro Array #5 station during 15 October 1979 Imperial Valley earthquake. Accelerograms (a) and (b) are widely representative in terms of spectral composition, and duration of the stationary part of ground motion time-histories (as also underlined by relevant Housner intensities, normalised to Ai, that vary from 0.108 s2 to 0.518 s2). The pulse function is a remarkably severe input for systems with medium-tohigh isolation periods, purposely introduced in the global investigation to assess possible operation limit-conditions of the protective system. This function produces the same effects as other near-source pulse-type models, like the family of forward-and-back pulses generated from blind-thrust fault simulations — utilised in [18] for extensive design analyses of base-isolated structures —

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which only differ for a parabolic, rather than sinusoidal shape of the displacement pulse. The scaling factors adopted for the three types of signals were aimed at simulating in all cases earthquakes with associated magnitude M=6.5. The response spectral curves — for a=0.1 and c increasing from 10 to 120 kN·(s/mm)α, in steps of about 8 kN·(s/mm)α — obtained by plotting the peak absolute acceleration Ab,max as a function of the vibration period in base-isolated conditions Tis, are plotted in Fig. 3. The seismic inputs used to obtain the curves are: the two real ground motions giving rise to the lowest (1997 Assisi EW main shock, Umbria–Marche earthquake) and the highest (1980 Calitri NS main shock, Campania–Lucania earthquake) seismic demands; the artificial accelerograms (for which the response is expressed in mean terms over the entire set of eight signals); and the near-source pulse. Fig. 4 presents the spectral curves in terms of the maximum base displacement db,max, as a function of Tis, calculated for the same ground motions. The following remarks derive from the observation of Figs. 3 and 4: 1. Considerable differences emerge, both in terms of acceleration and displacement, when passing from the least to the most demanding inputs. The maximum Ab,max and db,max values never exceed 0.21 g, and 30 mm for the Assisi earthquake, while at the same time they reach about 0.45 g, and 360 mm for the artificial signals; 0.6 g, and 450 mm for Calitri; and 0.45 g, and 1000 mm for the cycloidal pulse. This scattering highlights the critical role of direct dynamic analyses, as well as of the input ground motion selection, in the design of base-isolated structures equipped with non-linear viscous dampers. 2. Ab,max increases rather than decreases over the entire Tis range for the Assisi earthquake by increasing c. For the artificial accelerograms and the Calitri earthquake, this “inverted” effect is observed for Tisⱖ2.3 s and cⱖ30 kN·(s/mm)α. For the Calitri earthquake, which also presents a second “crossing” point among the various curves in proximity to Tis=1.7 s, there is quite an amplified response for c⬍30 kN·(s/mm)α. In the case of the simple displacement-pulse, the maximum acceleration always decreases by increasing c, for all Tis values. Except for a narrow sub-range in the proximity of the lowest limit of the Tis range considered (Tis=1.5⫺1.7 s), where Calitri locally gives rise to Ab,max=0.6 g, nearly coincident peak Ab,max values are observed in the remaining zones of the spectral curves for the same real ground motion, artificial accelerograms, and the pulse function. At the same time, whereas similar shapes of the entire curves are observed in the first two cases — which allows the Eurocode 8-generated signals to “cover”, in mean terms, even the response to the most demanding real motion — a decidedly wider amplification zone of the

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Fig. 4. Fig. 3. inputs.

Absolute acceleration spectral curves for selected seismic

Base displacement spectral curves for selected seismic inputs.

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same curves is seen for the cycloidal pulse. This results from the resonance-like response effects induced by this action. As a consequence, although absolute maxima over the complete Tis range are comparable, the relative maxima in terms of acceleration for Tisⱖ2.2 s, and the c demands for design, are directly determined by the pulse function. 3. For all inputs, the base displacement always decreases by increasing the damping level. Moreover, db,max is substantially independent from Tis for cⱖ30 kN·(s/mm)α, for both real and artificial accelerograms. In the case of the near-source pulse, db,max is found to be a growing function of Tis, for all c values. Also for db,max, when Tis exceeds 2.2 s the highest demand is caused by the pulse function. The maximum allowable value to implement reasonably sophisticated connections for the piping, ducts, and plumbing crossing the base-isolation plane can be fixed at around 400 mm. Therefore, the response is generally constrained within this limit by adopting just the lowest damping coefficient values for each input. The only exception is for the case of the cycloidal pulse, for which rather low c values (c⬇50 kN·(s/mm)α) are required. 4. As a consequence of the previous points, and also by taking into account the results for the remaining ground motions — all included among the documented ones — no actual response benefits are obtained by increasing c over low-to-medium levels (i.e., over c⬇60 kN·(s/mm)α). On the other hand, very low c values may cause remarkable amplification effects for particular Tis sub-ranges and ground motions.

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energy. As it is known, h represents the most suitable damping parameter for non-linear viscous systems, since it is based on measurable physical quantities. This also applies for this class of dissipaters [9,19]. The “equivalent effective damping ratio” beff used by FEMA 273 guidelines [5] as well as by the majority of technical standards, is in fact directly related to h, from which it only differs by a factor of 2, i.e., h=2beff. The design equation was derived from an extensive analytical study on non-linearly damped SDOF systems [9] subjected to harmonic input actions. This equation, which determines the c values that provide the same h factors as equivalent (i.e., with equal mass and stiffness) linear viscous SDOF systems in steady-state response conditions, has the following form: c(h,j,Ai)⫽c0.1(h,Ai)

(1+J)m [(1−j2)2+h2]m

(9)

Although limited to SDOF systems, these results clearly underline the need for precise criteria capable of determining the proper choice of coefficient c in the design phase. A self-consistent design procedure, providing explicit c calculation, is proposed for this purpose in the next section.

where c0.1 is the damping coefficient calculated as the value to obtain h under the action of a sinusoidal input with amplitude Ai, and frequency 1/10 of the fundamental vibration frequency fis of the system in base-isolated conditions; and q, m are tuning coefficients. The values of q and m obtained by interpolating the results of the analytical study [9] are reported in Table 1 for a=0.1 and 0.2, as well as for various h levels. The prediction capabilities of (9) are shown in Fig. 5, where the c/c0.1 curves calculated for h, equal to 0.2 and 0.5, and a=0.1 and 0.2, are superimposed to the numerical points representing the series of c values (normalised to c0.1) that were used in the equation of motion (1) to obtain the same h values. Once the target h is fixed, expression (9) provides a quick evaluation of the extreme c value that can theoretically be demanded to the damper by imposing the resonant condition j=1, whatever the type of deterministic or random dynamic action with peak amplitude Ai. For design application or performance assessment purposes, this condition should be conservatively assumed when

4. Design procedure for FV spring-dampers

Table 1 Calibrated values of coefficients q and m

4.1. Formulation of the procedure

a

h

q

m

0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 ⱖ0.6 ⱖ0.6

0 0 ⫺j/8 ⫺j/8 +j/8 ⫺j/8 +j/8 ⫺j/8 +j/4 ⫺j/4 +j/4 ⫺j/4

0.45 0.38 0.44 0.35 0.44 0.30 0.44 0.30 0.44 0.35 0.44 0.35

The procedure is based on an explicit design equation that allows estimating the maximum c demand for any given frequency ratio j (i.e., the ratio of the harmonic load frequency to the oscillator frequency), peak input acceleration Ai, and damping level. The last term is quantified by the loss factor h: h⫽

Ed 2pEe

(8)

where Ed is the dissipated energy, and Ee is the stored

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reasonably satisfactory model in terms of global response parameters when the ratio of Tis to the fundamental vibration period in fixed-base conditions Tfb is no lower than 4. The SDOF model can be easily generated by means of readily available computer programs such as [20–22], which already include spring and dashpot elements capable of reproducing the behaviour of FV spring-dampers. Based on this SDOF representation, the design procedure steps are formulated as follows:

Fig. 5. c/c0.1 curves derived from Eq. (9), and response points obtained from direct numerical integration.

analysing the effects of pulse-type ground motions. Nevertheless, these inputs generally give rise to a lower transient response than the one induced by a sinusoidal input action with the same amplitude. On the other hand, for recorded or artificial accelerograms, the assumption j=1 is too conservative in any case, since under-resonant responses are normally induced in actual structural systems. Hence, a shifted j value should be adopted for estimating c, namely j should be calibrated on the basis of the spectral composition of the input acceleration time-histories. For typical base-isolation configurations (Tis ranging from 2 to 4 s, i.e., fis=0.25⫺0.5 Hz) as well as for typical spectral compositions of real and normative-generated artificial ground motions — whose main frequency content is on average included in the [0.8, 5] Hz range — values no lower than 1.5 should be selected for f. However, for the lowest h levels (hⱕ0.2), such an assumption can lead to quite low c estimates, since (as highlighted by the h=0.2 graphs in Fig. 5) the c/c0.1 ratios calculated for j=1.5 are reduced in these cases, at least by a factor of 4 compared to the peak c/c0.1 value corresponding to j=1. This factor even reaches 6 for h=0.1 and a=0.1. When imposing a target loss factor hⱕ0.2, it is therefore preferable to consider j=1.2⫺1.3, in order to avoid such remarkably reduced c estimates in the analysis conducted with real and artificial input motions. For higher h levels, and especially for hⱖ0.5, j can instead be directly calibrated on the assumed fis value and the lowest significant frequency in the spectral composition of the design accelerograms. By way of example, if the lowest frequency is approximately equal to 0.8 Hz, and fis is fixed at 0.4 Hz, a value j=2 can be adopted. Starting from these observations, the preliminary phase of the design procedure is carried out by referring to a SDOF structural model, for which the superstructure is idealised as a rigid mass, according to the scheme in Fig. 1. It is known that such an assumption provides a

1. Assumption of Tis, and thus of k2=4p2·m/T 2is, where m is the global superstructure mass. k1 is generally taken as at least 15 times k2, so as to ensure a nearly rigid behaviour of the device below the pre-load threshold F0; the latter term is initially evaluated by (4), with n=5; 2. Assumption of a reference h value (named h0), to be tentatively calibrated on the basis of the structure characteristics and of the hazard level hypothesised for the design earthquake, summed up in terms of peak acceleration Ai (more than one hazard level is established when dealing with a multi-objective design problem). Relatively low h0 levels, i.e., h0ⱕ 0.3, are often sufficient in the case of quite rigid superstructures, whereas medium loss factor values (h0ⱖ0.5) are deemed appropriate for more deformable buildings [10]; 3. Assumption of the reference j value for c computation, to be directly calibrated with the spectral composition of the artificial and real accelerograms adopted, with the exception of h0ⱕ0.2, in which case j=1.2⫺1.3 must be set; and equal to 1, for nearsource pulses; 4. Calculation of c0.1, to determine the “anchor point” of curves (9); 5. Calculation of c by (9), for the two different types of input. The mean c value coming from the artificial accelerograms (which are the reference design ground motions from a normative viewpoint) and from significant real records is adopted as the result of this stage of the design analysis (named cdes). On the other hand, the c value derived from the pulse function sets the limit over which no further benefits can be obtained in terms of global response parameters, for the pre-fixed h0 value, in nearsource conditions. Therefore, the pulse function-related c value quickly quantifies the greatest damping capacity that should at most be assigned to the FV device to reach adequate protection also against this highly hazardous type of motion. The seismic performance with the cdes value is subsequently checked in the final design phase, where the behaviour of protected buildings with a not very stiff superstructure is evaluated by means of a more accurate structural model, under the action of the same input sig-

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nals utilised in the preliminary design stage. A series of structural and non-structural performance limits, consistent with the design objectives as well as with the characteristics and use of the buildings, will be selected in the verification phase to definitely assess the response capacities of the devised protective system and the superstructure. A flow-chart of the entire design procedure is shown in Fig. 6 for a quick visualisation of its basic steps. 4.2. Experimental check of design Eq. (9) Before applying Eq. (9) in practice, it has been further validated against the outcome of an experimental forcedvibration survey conducted on the half-scale steel frame

Fig. 6.

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shown in Fig. 7. A detailed view of one of the two identical double-acting spring-dampers placed at the frame base is provided in the same figure. The experimental analysis consisted of a series of tests developed by varying the frequency of the harmonic input load applied to the ground floor by a high-power dynamic exciter, over a wide range, as documented in detail in [23]. Among other results, these tests produced the set of response points plotted in Fig. 8, where the analytical curve (9) was calibrated against test results (a=0.18, and h=0.5). Test points and analytical prediction appear to be well correlated along the entire frequency range explored during testing (corresponding to jⱖ0.6).

Flow-chart of the design procedure.

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Fig. 8. Response points obtained from the experimental tests and corresponding c curve by Eq. (9) normalised to c0.1.

to be conducted, for the Eurocode 8-generated artificial accelerograms and the cycloidal pulse defined by (5), (6) and (7). Tp=2.2 s was identified as the value determining the most demanding effect in relation to the imposed Tis period. The artificial signals and the pulse function were scaled in this case to the peak amplitudes Ai,1=0.25 g and Ai,2=0.12 g, respectively, in order to simulate events with magnitude M=6.1, as conservatively hypothesised for the low-seismicity site where the building is situated. The design analysis included only this basic level of seismic action, for which substantially undamaged post-earthquake conditions of structural and non-structural members were required. Based on the characteristics of the adopted devices, a value of 0.2 was assigned to the exponent a. The main steps of the design are summarised below with the same sequence as in the previous section.

Fig. 7. Global and detailed views of the tested steel frame.

5. First application of the procedure A compact older masonry building, characterised by a low fixed-base vibration period Tfb=0.126 s, is examined herein for a simple demonstrative application of the procedure. The retrofitting scheme preliminarily formulated in [24] for this building consisted of adopting four double-acting FV devices for each side in plan, and a base-isolated period Tis=2 s. The devices were assumed to be placed in symmetrical positions with respect to the centre of gravity, so as to avoid torsional effects under horizontal loads. This allowed a single plane analysis

1. Starting from the assumed Tis=2 s value, a value of k2=3.928 kN/mm was derived for each device, with m=1592 kg as the total mass of the building. At the same time, k1=15 k2=58.92 kN/mm was computed, whereas n was fixed at 5; 2. h=0.1=h0 was tentatively selected; 3. In view of the low h0 value hypothesised, j=1.2=ja was established for the artificial accelerograms ( j=1=jp was adopted by default for the near-source pulse); 4. c0.1=25.2 and 9.4 kN·(s/mm)α were derived from Ai,1 (in mean terms) and Ai,2, respectively; 5. Based on the Ai,1 and Ai,2 values, the two c(h0,f,Ai,j) curves in Fig. 9 were drawn by means of (9). Nearly coincident c estimates came out from the two computations in this case, i.e., c(h0,ja,Ai,1)=55 kN·(s/mm)α, and c(h0,jp,Ai,2)=53 kN·(s/mm)α. Thus, cdes=c(h0,ja,Ai,1)=55 kN·(s/mm)α was adopted as the outcome of the preliminary design phase.

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Fig. 9.

c coefficient calculation according to the design procedure.

The analyses were then repeated by introducing cdes in the SDOF model, which was also utilised — in parallel with a sophisticated finite element model of the building described in detail in [24] — within the final verification phase. The absolute acceleration and base displacement response time-histories are reported in Fig. 10 (in the case of the artificial accelerograms, they correspond to the most demanding signal, named “EC8-AS3”). These graphs show a satisfactory performance, since both Ab and db peak values do not exceed 0.14 g, and 70 mm

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for Eurocode 8-signals; and 0.17 g, and 100 mm for the cycloidal pulse input. More specifically, the calculated peak accelerations are lower than the values capable of activating potential mechanisms of crack diffusion along the main lateral load-resisting masonry walls. This aspect was assessed by a separate analysis carried out by the finite element model, which produced practically coincident results in terms of Ab,max and db,max [24]. At the same time, the very limited maximum base displacements obtained are consistent with the requirement of avoiding damage to adobes and ornamental elements included in this building [24], as well as to service lines crossing the foundation plane. Even though no explicit limitation on the response parameters was assumed for assessing the performance of the retrofitted building (considered as a “historic” one) the aforementioned data confirmed the h0 value initially selected — and thus the corresponding cdes estimate — as the final design choice. A further check was done, aimed at evaluating the influence of pre-load for this case study. Fig. 11 shows the (Ab,max⫺c) and (db,max⫺c) response curves obtained for the system studied by varying n within the reference range [2, 7] mentioned in Section 2.1. In these graphs, the curves for the case without pre-load, denoted by the symbol F0=0, are also drawn. As a general outcome of this analysis, a remarkable influence of n emerges in correspondence with the lowest portion of the considered c range, i.e., for cⱕ30 kN·(s/mm)α. Within this sub-

Fig. 10. Absolute acceleration and base displacement time-histories for cdes=55 kN(s/mm)α, obtained from the most demanding artificial signal (scaled at Ai,1) and the cycloidal pulse (scaled at Ai,2).

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Fig. 11. Absolute acceleration and base displacement spectral curves obtained from the artificial signals (mean values) and the cycloidal pulse, by varying pre-load for the case study system.

range, non negligible response amplifications are noted, compared to the basic design choice n=5, for the lowest values of this parameter (n=2,3), for the artificial accelerograms; and for the highest ones (n=6,7), for the nearsource pulse. In order to guarantee optimum control of the seismic response, this suggests to confirm the value of 5 as the preferential design value for n. Concerning the supplementary c sub-range (cⱖ30 kN·(s/mm)α), a substantial convergence in terms of acceleration is reached for the various n values (with the only exception being n=2, for the artificial signals). Moreover, for cⱖ60 kN·(s/mm)α, that is immediately over the selected cdes value, the acceleration response becomes nearly independent from the damping coefficient, for both types of input motions. Reaffirming the trends emerged from SDOF analyses in Section 3, no higher acceleration (and thus base shear) reduction capabilities can be attained by increasing c beyond these levels. With regard to base displacements, slightly higher benefits could only be derived under the action of cycloidal pulse. Even in this case, for each n choice, starting from c⬇50 kN·(s/mm)α db,max is nevertheless constrained within the limits of 150–200 mm, which ensures minimal or no damage to infrastructures. The observations above highlight that for each n, cdes provides an effective balance between high performance of the protective system and limited size — and costs — of incorporated devices.

6. Concluding remarks A self-consistent dynamic design procedure was proposed for the class of FV spring-dampers incorporated in base-isolation systems of building structures, to overcome the main limitations that hinder the use of this type of analysis in design practice. These include the absence of precise criteria for pre-estimating the seismic damping demand, the burdensome computation required, and the considerable level of expertise needed to develop it. Based on an explicit design equation and initial SDOF idealisation of the problem, the procedure easily calculates damping coefficient values ensuring the sought energy dissipation capacities of FV devices, and can therefore be applied by utilising the same computer programs as required for non-linear static methods. Moreover, whereas the latter methods generally include an additional check with a dynamic analysis, the procedure presented herein has the advantage of providing a unified approach to the preliminary and final verification design phases. The case study examined in this paper, belonging to the simplest class of buildings with a very rigid superstructure (for which final verification can often be limited to quick controls on base acceleration and displacement by means of the initial SDOF model), showed the potential of the procedure in locating the optimal damping coefficient range under normative-generated and

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pulse-type input motions. The influence of the pre-load optionally imposed on highly-performing types of FV devices was also evaluated, by varying its tuning coefficient within the proposed choice range. This case study and the parametric assessment of protective capacities of the considered technology highlighted that its highest performance is attained starting from rather moderate damping coefficient values (from which compact device sizes and favourable cost-to-benefit ratios derive). The trends which have emerged from this research, as well as the effectiveness of the design procedure, are further investigated in an accompanying paper [11], where reference is made to more complex case studies, represented by a reinforced concrete and a five-storey steel building.

Acknowledgements The work presented in this paper, as well as in the accompanying paper, was partially sponsored by the Italian Ministry of University and Scientific–Technological Research (MURST) with a grant on the Project “Seismic retrofit of monumental buildings by base isolation and innovative materials”. The authors express their gratitude for this support.

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