Engineering Structures 30 (2008) 3610–3618

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Engineering Structures

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Nonlinear behavior of dynamic systems with high damping rubber devicesA. Dall’Asta a,∗, L. Ragni ba ProCAm, Dipartimento di Progettazione e Costruzione dell’Ambiente, Università di Camerino, Italyb DACS, Dipartimento di Architettura Costruzione e Strutture, Marche Università Politecnica delle Marche, Italy

a r t i c l e i n f o

Article history:Received 14 August 2007Received in revised form31 May 2008Accepted 2 June 2008Available online 18 July 2008

Keywords:High Damping RubberDissipating devicesDynamic behaviourRheological model

a b s t r a c t

TheHighDamping Rubber (HDR) iswidely used in seismic engineering and,more generally, in the passivecontrol of vibrations. Its constitutive behaviour is quite complex and is not simply non-linearwith respectto strain but also shows a transient response during which material properties change (Mullins effect). Anumber of recent works were dedicated to analyzing and modelling the material behaviour. The presentwork intends to study the consequences of such non-linear behaviour in the dynamic response of S-DoF systems where the restoring force is provided by dissipative devices based on the HDR (structuralsystem with dissipative bracings and isolated systems). Preliminary analyses under harmonic forces andimpulsive excitations were carried out in order to separately characterize stable and transient responses.Finally, the response under seismic inputs with different intensities was studied. Results show that theMullins effect may play an important role in the seismic response and the dynamic properties of thesystem change significantly for seismic events with different intensities.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In the field of seismic engineering the rubber with enhanceddissipating properties, usually known as High Damping Rubber(HDR), is extensively adopted in bearings for the seismic isolationof bridges or buildings [1], and is also used for dissipatingdevices in order to increase stiffness and the energy dissipationcapacity of structures [2–5]. With respect to other types of damperdevices, based on elasto-plastic or viscous materials, the HDR-based damper seems to be a promising energy dissipating devicebecause no permanent strains occur after seismic events andmoreover it permits dissipating energy even for small lateraldisplacements produced by wind or minor earthquakes.Some difficulties in the use of this kind of dissipating device de-

rive from its complex dynamic behaviour which, makes it difficultto evaluate the behaviour of equipped structures accurately and togive design indications. More specifically, the material behaviouris strongly non-linear and both stiffness and damping proper-ties vary with the amplitude of strain and depend on the strainrate [6,7]. Furthermore, the presence of filler added to the naturalrubber, makes the response of the HDR strain history-dependentand causes a transient behaviour in which stiffness and damp-ing change remarkably. The phenomenon, usually known as the‘‘Mullins effect’’ or ‘‘scragging’’, is a consequence of the damageof the microstructure, that occurs during the process [8,9]. Recent

∗ Corresponding author. Tel.: +39 320 7985790.E-mail address: andrea.dallasta@unicam.it (A. Dall’Asta).

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.06.003

studies [10] showed that the transient response is related to themaximum strain attained by the material and is influenced by thestrain rate. The initial properties of the material may be howeverrecovered (healing effect) and the healing times depend on thema-terial considered and on the temperature [10]. The rubber studiedin [11] showed a rapid recovery of a large part of the material stiff-ness even if the complete recovery may take several months. Con-sequently seismic analyses of structures endowed with HDR de-vices should be performed with virgin material properties, even ifsome scragging process were applied to the devices, and a modelwith damage should be adopted. The influence of scragging on theseismic response is also evidenced in [12]where the bidimensionalresponse of an isolated bridge is analyzed.The present work intends to analyze the dynamic response of

single degree of freedom systems in which the restoring force isgiven by HDR devices, by using a unidimensional model basedon virgin properties of the rubber and including the scraggingphenomenon, previously developed by the authors [11]. The aimof these numerical analyses is to evidence characteristic aspectswhich can be of interest in the structural design under seismicactions and which cannot be described by simpler models, suchas linear visco-elastic or elasto-plastic, usually used to simulatethe HDR-based devices behaviour [13,14]. The analyses considera range of the shear strain from 0.0 to 2.0, which seismicdissipating devices usually undergo. Three different ratios betweenmass and stiffness have been considered in order to study therubber response in different dynamic situations, spanning fromvibrations with long period, which furnish information aboutisolated systems, to vibrations with short period, which furnish

A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618 3611

Fig. 1. Mullins effect: different strain amplitudes (a) different strain rate (b).

information about the rubber response in dissipating bracesinserted within deformable frames.In particular, Section 3 investigates the harmonic behaviour of

the dynamical systems subjected to sinusoidal forces. The resultsrefer to the stable, post-transient, response and furnish informa-tion regarding the influence of non-linear behaviour of the HDR onthe dynamic response of the system, once theMullins effect is over.The following section is otherwise oriented to highlight the

influence of the Mullins effect that influences the initial part of thedynamic response. For this purpose, the system behaviour underan impulsive initial input is studied.Finally, the last section is dedicated to the analysis of the system

subjected to seismic inputs. The analyses show that the Mullinseffect may influence and change the system response in the caseof similar seismic events (similar frequency content andmaximumdisplacement attained). Furthermore, the influence of all thenon-linear phenomena on the response under seismic events ofdifferent intensities is analyzed in order to show how the dynamicproperties of the system change by varying the input intensity.

2. Dynamical system

2.1. HDR model

The model used to simulate the HDR response is a rheologicalunidimensional model able to describe the transient response ofthe rubber, depending on both the strain rate and the maximumstrain attained, as evidenced by experimental tests reported byFig. 1. In the model the state of the material is furnished by theshear strain γ , defined as the ratio between the shear displacementand the thickness of the rubber, and by a set of internal variablesαi describing the inelastic response and the Mullins effect. Thetangential stress may be derived from the free energy per unitvolume ϕd (γ ;αi) by the relation

τd =∂ϕd

∂γ(1)

whereas the dissipated power per unit volumewdmay be obtainedfrom the derivativewith respect to the internal variables (repeatedindexes denote summation, superposed dot denotes time deriva-tive)

wd =∂ϕd

∂αiαi. (2)

The stress deriving from a strain history may be determinedonce the initial state and the process η = γ are known, on the

basis of the nonlinear functions gi (γ , η;αi), which describe theevolution of the internal variables:[γαi

]=

[η

gi (γ , η;αi)

]. (3)

The expressions of the free energy and those of the evolutionfunctions adopted in the following analyses are reported andcommented in the Appendix.

2.2. S-DoF system

The S-DoF (Single-Degree of Freedom) dynamical systemconsidered consists of a mass m and an HDR-based dissipatingdevice that furnishes the restoring force. It is assumed that alinear relation, defined by a constant c , exists between the massdisplacement u and the shear strain of the device rubber γ = cu/h,where h is the thickness of the rubber layer. The value of c dependson the geometry of the connection between the device and mass.The restoring force per unit mass fd can be expressed in theform

fd =cAmτd (4)

where A is the area of the HDR layer in the device. The state ofthe system is consequently described by the vector x = [u, v;αi]where v is the velocity of the mass. The evolution law has thefollowing form:

x =

[ uvαi

]=

v

−fd(cuh, cv

h;αi

)+ fe

gi(cuh, cv

h;αi

) = A (x) (5)

where fe is the external force per unit mass. It may be useful toobserve that the following non-linear resultsmay also be extendedto cases different from those considered here. The equation ofmotion can be rewritten by remembering that u = γ h/c and bydividing each term by the thickness h. The equation assumes theform

x =

[γγαi

]=

γ

−cAmhτd (γ , η;αi)+

feh

gi (γ , η;αi)

(6)

consequently the same strain history may be observed in all thosecases where the two ratios cA/mh and fe/h did not vary (e.g. non-linear response does not vary by doubling fe if both A and h are alsodoubled).

3612 A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618

Fig. 2. Harmonic response of case b: maximum displacements, maximum forces, dissipated energy and maximum free energy.

3. Harmonic analysis

In this section, a harmonic analysis is carried out in orderto investigate the nonlinear response of the system subjected toperiodic external force with varying frequency and amplitude.In the tested range of external actions the system shows a

transient response and attains a stable behaviour after a certainnumber of cycles. The response related to an external action withperiod T is considered to be stable at the instant t when thedifference between the state history observed in the last periodand the state history observed in the previous period is sufficientlysmall. More precisely, once the two state functions x1 (t) =x (t − T + s) and x2 (t) = x (t − 2T + s) have been defined on thesame interval s ∈ [0, T ] and a positive real constant ε has beenchosen, the response is considered to be stable if ‖x1 − x2‖ < ε.In the application it was assumed that ‖y (s)‖ = max |y (s)| , s ∈[0, T ].Having the aim to characterize the response under seismic

events acting on the system where the state variables are nullinitially, the analyseswere performedby assuming x = 0 for t = 0.The external force has the expression

fe (t) = f0 sin (2π t/T ) (7)

where f0 is the amplitude of the force per unit mass and T is itsperiod. In particular three cases corresponding to three values ofstiffness are considered in the analyses. The intermediate case bwas obtained by a rubber layer with an area A = 78 200 mm2and a thickness h = 10 mm; a mass of 105 kg was considered.The other cases a and c were obtained with a device area fourtimes larger and smaller, respectively. Such values were chosenin order to obtain dynamical systems that show the displacementpeak for external force periods of about 0.5 s (case a), 1.0 s (case b)and 2.0 s (case c). The chosen stiffness values make it possible tostudy different situations referable to different structural systemswhere devices are introduced in order to reduce seismic effects,like frames with dissipating bracings (T = 0.5–1.0 s) or isolatedstructures (T = 1.0–2.0 s). The external force periods considered

in the analyses span from 0.3 to 4.0 s, which is the range of interestin seismic response. Finally, the constant c = 1 was assumed inthe analyses.For each case, the maximum intensity of the external force f0m

was calibrated to provide a maximum value of shear strain (γ =u/h) equal to 2.0, which is an usual maximum strain value in thedesign since larger strains cause a strongly hardening behaviourof the rubber. Other results were also evaluated for smaller valuesof the maximum strain by considering different intensities of theexternal force, equal to 0.75f0m, 0.5f0m and 0.25f0m. The constant cof Eq. (4) was taken as being equal to 1.0.Fig. 2 reports the maximum values of displacement (u), the

restoring forces (fd) the energy Wd dissipated in a cycle and theextreme values Φe attained by the free energy in the periodicresponse. The diagrams are in a non-dimensional format and wereobtained by dividing displacements, forces, energies and periodsby reference values defined with the criteria indicated below.The reference displacement uref and the reference force fdref arethe value of the maximum displacement and the value of themaximum device force attained with the external force f0m. Thereference period Tref is the period at which these maximum valuesare reached. The reference energy value is given by

Eref =12fdrefuref. (8)

The results refer to case b, forwhich f0m = 37 kN, uref = 20mm,fdref = 1.20 N kg−1 and Tref = 1.0 s. By observing Fig. 2 it isevident that the main displacement peak occurs at a period (Tm)which decreases by decreasing the force intensity. This is a typicalbehaviour of softening systems even if the behaviour seems tobecome weakly hardening for the largest value of the input force,as is usual in the large strain range. The amplitude of the mainpeak increases nonlinearly with the intensity of the input force, asmay be seen in the same figure. This trend is mainly controlled bythe dissipation properties of the system, which worsen for largestrains.

A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618 3613

Fig. 3. Displacement time histories and force–displacement diagrams at the two peaks exhibited by the system: main peak (a), secondary peak (b).

The maximum displacement rapidly changes for periods whichare slightly larger than Tm, although multiple responses were notobserved in the cases analyzed. Predictably, unstable behaviourmay be exhibited by the system for larger strains. It can be alsoobserved that for every value of the input force the response curveshave a secondary peak, whose period is about 1.8 times that of themain peak.In order to analyze the transient response and the shape of

the stable loop of the system subjected to harmonic loads anumber of analyses in the time domain were reported. Specificallytwo periods were considered: T1 = Tm and T2 = 1.8Tm with theinput force f0m. The analysis results, displacement vs time andforce vs displacement, are reported by Fig. 3. In the case ofT1 = Tm the displacements are approximately sinusoidal and arestrongly amplified. In the case of T2 = 1.8Tm the displacementhistory becomes periodic but is no longer sinusoidal. In this casedisplacement contributions with higher frequencies arise.Similar trends were observed in the other cases a and c , even if

a number of changes occur. A comparison may be obtained fromFig. 4 where the maximum values of displacements and shearstresses (τd = fd/A) of the previous case b are reported togetherwith results of case a, which is stiffer and attains maximum valuesfor about Tm = 0.5 s, and case c , which has a lower stiffness andattains maximum values for about Tm = 2 s. The main differencesregard the secondary peak and stress intensity, which increasefor stiffer system as a consequence of the increment in thestrain rate.In order to furnish synthetic information and to permit

a numerical comparison between different situations, threeparameters were chosen to describe the system response. The firstparameter

G =fdmumA

(9)

is the ratio between the device force and displacement at the in-stant when the system attains the maximum displacement in theperiodic response. It may be interpreted as an approximated esti-mation of the material stiffness (shear modulus) at the maximumdisplacement and is strongly dependent on the period Tm.

The other two parameters are derived from energy quantitiesand describe the dissipative properties of the system. The firstparameter is the damping coefficient, which may be defined, inanalogy of linear systems, as

ξ =Wdm4πΦem

(10)

where Wdm is the dissipated energy and Φem is the peak ofthe energy stored in the system during a period. The previousparameter (ξ ) furnishes information on the ratio betweendissipated and stored energy and will be evaluated, as usual, forthe period at which the system response is at its maximum value,changing case by case.Sometimes, a different parameter ξe (equivalent damping

coefficient) in which Φem is substituted by Eref, was adopted tostudy rubber behaviour [13–15]. This latterwas not used because ithas no physical meaning, since Eref does not measures the internalenergy of the material.In order to describe the dissipation rate and, more generally,

the global ability of the system to reduce the effects induced by anexternal input, another parameter

λ =2πξTm

(11)

is introduced. This was obtained by dividing the previousparameter by the duration of the time interval on which thedissipated energy is evaluated (average dissipated power). Theconstant term 2π was added in order to make it similar to thecoefficient describing the exponential decay in linear systems.Table 1 reports the results evaluated at Tm and for the cases andthe external force levels considered previously.Predictably, by comparing the results referring to the same case

(a, bor c), itmaybe observed that the shearmodulus of thematerialstrongly increases when the maximum strain decreases. On thecontrary, the damping coefficient does not change significantlyby reducing the external force (the dissipated energy increasesin the same way as the free energy) whereas the dampingrate coefficient augments as the strain decreases (the energy isdissipated more rapidly). On the other hand by comparing results

3614 A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618

Fig. 4. Displacements and forces obtained for the three studied cases a, b and c .

Fig. 5. Free vibration of case b: time histories of displacement (a) and restoring force (b).

Table 1Parameters describing the harmonic response

γ f0 Tm G ξ λ

(N kg−1) (s) (N mm−2) – (s−1)

Case a

2.00 1.66 0.47 0.86 0.26 1.961.12 1.24 0.40 1.01 0.22 2.700.58 0.83 0.34 1.31 0.21 3.100.21 0.41 0.30 1.66 0.22 3.78

Case b

2.00 0.37 1.00 0.75 0.29 0.941.04 0.27 0.82 0.93 0.24 1.360.54 0.18 0.72 1.23 0.22 1.510.19 0.09 0.61 1.56 0.25 2.00

Case c

2.00 0.08 2.08 0.67 0.27 0.440.96 0.06 1.68 0.88 0.23 0.660.49 0.04 1.48 1.15 0.22 0.750.17 0.02 1.26 1.48 0.27 1.02

referring to similar strain amplitudes, it is possible to observethat by decreasing the period of the input force (Tm) the shearmodulus slightly increases, whereas the damping coefficient maybe considered constant. Considering all the cases analyzed, thevalues of the shear modulus span from 0.67 to 1.66, whereas therange variation of the damping coefficient is from 0.21 to 0.29.

4. Impulsive excitation

The Mullins effects mainly influences the system behaviourin the initial path of the response under external force. In orderto analyze a dynamic situation strongly affected by the transientphenomena related to theMullins effect, the systemwas subjectedto initial conditions consisting of an initial velocity v0, to which aninitial value of kinetic energy (E0 = 0.5mv20) is associated, whilethe other state variables were taken equal to zero.The three previously defined cases were considered and the

initial velocitywas calibrated in order to attain the samemaximum

displacement adopted in the previous section (harmonic analysis)and corresponding to the limit shear strain, γ = 2.0. Other resultswere also evaluated by reducing the external input and assumingthe following values for the initial velocity: 0.75v0, 050v0, 0.25v0.Fig. 5 describes the time histories of displacement and the

device force for the intermediate case b. Here again, the resultsare presented in a non-dimensional format. The same values ofthe reference displacement, device force and time interval wereadopted in order to simplify the comparison with the results ofthe harmonic analysis. Results of case b were obtained with v0 =200 mm s−1.Comparing, the cyclic response and the free vibration motion

corresponding to the same limit strain γ = 2.0, it is evidentthat the free system vibrates more rapidly at the initial stageand the maximum value of the device force is notably higher.In other words, the system is remarkably stiffer. A more directunderstanding of the differences in the stable and transientresponse for similar values of maximum strain may be obtainedby Fig. 6 where the device stress–strain graphs of the two analyses(harmonic and impulsive) are traced together for two differentvalues of the external input multiplier. The differences are moreremarkable for large strains, which are also related to a large strainrate. The graph seems to show that the transient response is alsomore dissipative. Here again, similar trends were observed in theother cases a and c.The initial behaviour of the system may be characterized on

the basis of the motion observed in the initial time interval[0, t∗], assuming t∗ as the instant at which displacement returnto zero. The same three parameters used in the harmonic analysiswere evaluated. In particular the stiffness G was evaluated usingEq. (9), the damping coefficient was evaluated from the energydecay, likewise in linear systems. The expression is

ξ√1− ξ 2

=ln E (t∗)− ln (E0)

2π(12)

A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618 3615

Fig. 6. Comparison between stable and transient response for maximum strain γ = 2 (a) and γ = 1 (b).

Table 2Parameters describing the transient response

γ v0 T ∗ G ξ λ

(mm/s) (s) (N mm−2) – (s−1)

Case a

2.00 410.0 0.21 1.19 0.25 3.791.52 307.5 0.20 1.12 0.23 3.620.97 205.0 0.19 1.20 0.20 3.370.46 102.5 0.17 1.41 0.20 3.75

Case b

2.00 200.0 0.43 1.09 0.24 1.871.51 150.0 0.42 1.00 0.22 1.710.97 100.0 0.39 1.07 0.20 1.630.45 50.0 0.36 1.16 0.20 1.81

Case c

2.00 94.0 0.88 0.97 0.23 0.841.51 70.5 0.87 0.89 0.22 0.790.94 47.0 0.80 0.97 0.19 0.750.44 23.5 0.74 1.16 0.20 0.86

Fig. 7. Energy decay (case b and γ = 2).

where

E(t∗)= E0 −Wd

(t∗). (13)

The trend of the energy decay is shown in Fig. 7 for the studiedcase b and the initial velocity v0. The third parameter, describingthe exponential decay, was evaluated by the expression

λ =πξ

t∗(14)

which is similar to the previous Eq. (11) while taking into accountthat, in this case, the system describes approximately half asinusoid during the time t∗. Table 2 reports the results evaluatedfor t∗ and for the cases and the external force levels consideredpreviously.By comparing results referring to the same case (a, b or c) it

may be observed that in all of the cases analyzed, especially forlarge strains, the values of the shear modulus related to similarstrains, are significantly larger than those obtained in the harmoniccase. On the other hand the values obtained for the equivalent

Table 3Parameters describing the seismic response (case b)

Accelerogram γ F G Ei Wdv/Ei WdM/Ei(kN) (N mm−2) (kN mm)

1 2 175 1.11 364 0.76 0.242 2 129 0.82 1384 0.88 0.123 2 164 1.05 577 0.85 0.154 2 167 1.07 646 0.84 0.165 2 152 0.97 364 0.78 0.226 2 141 0.90 594 0.84 0.167 2 163 1.04 557 0.83 0.17

damping coefficient are not very different from those obtainedfrom the stable response. This means that stiffness and dissipativeproperties are proportionally enhanced.Observing the dissipation rate coefficients, it is however

possible to observe that the system dissipates more quicklyin the transient response with respect to the stable response,especially for large values of strain (when theMullins effect ismoreimportant).

5. Seismic excitation

The aim of this section is to study how and how much non-linear phenomena previously analysed may affect the seismicresponse. In particular the influence of the accelegram type andaccelegram intensity on the system response is investigated.

5.1. Influence of the accelerogram type

Seven ground motions were chosen from the earthquakesavailable on the European Strong-motion Database (http://www.isesd.cv.ic.ac.uk) related to a ground type C (soft soil). Eachaccelerogram has a different PGA and spectrum shape. Coherentlywith the approach of the previous sections, whose aim is to furnisha characterization of the non-linear system at particular valuesof the rubber strain, each accelerogram was scaled in order toobtain the maximum shear strain γ = 2. The accelerogramsobtained have about the same spectral ordinate at the period ofthe system considered as reported by Fig. 8, with reference tothe case b. Table 3 reports the values of the force, the previouslydefined parameter G, the input energy and the ratio betweenthe dissipated energies (energy WdM dissipated by the damageinduced byMullins effect and the energyWdv dissipated by viscousphenomena) and the input energy (Ei).Even if the accelerograms were calibrated to produce the same

maximum displacement, the maximum forces observed are quitedifferent from each other. Their values span from 175 kN, in thecase of accelerogram 1, to 129 kN in the case of accelerogram2. This is mainly due to the Mullins effect which makes theresponse dependent on the particular history considered. The two

3616 A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618

Fig. 8. Acceleration spectra of scaled accelerograms.

deformation histories are reported by Fig. 9a. It may be observedthat the former exhibits the largest strain at the initial part ofthe history while the latter attains maximum values only after anumber of lower cycles were traced. As a consequence, the stressof the former case are more strongly influenced by the Mullinseffect and the force is higher, as may be observed in Fig. 9b where

the stress–strain diagrams of the two cases are reported. Theinfluence of the Mullins effect may also be analyzed by observingthe trends of energyWdM dissipated by the damage induced by theMullins effect and energy Wdv dissipated by viscous phenomena.The progress of these energy contributions are reported by Fig. 9ctogether with the input energy Ei. It is evident that in the formercase the damage dissipation due to the Mullins effect occursmainly in the initial response, whereas the damage progress moreuniformly in the latter case.Obviously the values of the parameter G are also different

for each accelerogram and span from the values obtained withthe stable response to those obtained by studying the transientbehaviour.The same maximum force trend may be observed in the other

cases. In case a the forces span from 646 kN to 745 kN, whereas incase c these span from 30 kN to 39 kN.Themean values of the force, the parameter G, the input energy

and the ratio between the dissipated energies and the input energy,related to all the cases analyzed, are reported by Table 4. It shouldbe noted that the part of the energy dissipated by theMullins effectis significant in all the cases, especially in the case c and the meanvalues obtained of G are closer to those obtained considering thetransient response.

Fig. 9. Displacement time histories (a), stress–strain diagrams (b), energy time histories (c).

A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618 3617

Table 4Mean values of parameters describing the seismic response

γ F G Ei Wdv/Ei WdM/Ei(kN) (N mm−2) (kN mm)

Case a 2 692 1.106 3383 0.85 0.15Case b 2 156 0.997 641 0.84 0.16Case c 2 36 0.921 128 0.82 0.18

Table 5Results for different levels of seismic input (case b)

β γ F G Ei Wdv/Ei WdM/Ei(kN) (N mm−2) (kN mm)

0.25 0.3 37 1.577 19 0.74 0.260.5 0.8 73 1.166 113 0.76 0.230.75 1.4 110 1.004 276 0.77 0.231 2 156 0.997 641 0.84 0.16

5.2. Influence of the accelerogram intensity

Finally, a comparison between solutions at different levels ofpeak ground acceleration was carried out in order to examine theresponse by varying seismic events. Calculations were repeated byconsidering the previous set of accelerograms scaled by the values0.75, 0.50, 0.25. By reducing the input, different counteractingphenomena which make the difference with respect to previousanalyses, may be foreseen. In fact, the stable cyclic responsebecomes stiffer when strain decreases. On the other hand smallseismic inputs involve a lower strain rate so that the viscous forceand Mullins effect should consequently be lower.Table 5 reports the average values of maximum forces and

displacements observed in case b; the results refer to the fourdifferent levels of external input and β denotes the scalingfactor. The results show that the system becomes stiffer forlower inputs because the ratio between force and displacementincreases. In addition the Fourier transform of the solutionpermits evidencing the variation of the system’s sensitivity withrespect to the frequency. Displacements obtained for the set ofaccelerograms considered were transformed and the functionU (β;ω), obtained from the average transformation values, wasevaluated for different values of the scaling factor β . Fig. 10 reportsthe modula of the scaled functions Us (β;ω), obtained by dividingU (β;ω) by its norm

∥∥U∥∥ (β) = √∫R

∣∣U (β;ω)∣∣2 dω. (15)

Diagrams show that dynamic properties of the system stronglychange by varying the seismic input level, in particular it becomesmore sensitive to high frequencies. The functions Us (β;ω) have aunitary norm and a numerical comparison between these may beattempted by introducing the norm of the difference

ds (β1;β2) =

√∫R

∣∣Us (β1;ω)− Us (β2;ω)∣∣2. (16)

The following values were obtained between the response atthemaximumseismic input and the other inputs: ds(1.00; 0.75) =0.14, ds (1.00; 0.50) = 0.28, ds (1.00; 0.25) = 0.38. Similar trendswere observed in case a and case c.

6. Conclusions

The behaviour of the dynamic systems with HDR restoringforce was studied in three steps: analysis of the response underharmonic excitation to characterize the stable behaviour, analysisunder impulsive excitation to characterize the initial transient

Fig. 10. Modula of the scaled function versus frequency (case b).

response and analysis under seismic excitation. The input levelwas assigned in order to characterize the behaviour at assignedvalues of the maximum strain and three ratios between mass andHDR device were considered to span cases of interest in structuraldesign. Some synthetic measures of stiffness and dissipativeproperties were introduced to permit a comparison between theresults.The harmonic analysis revealed that the maximum displace-

ment function exhibits a shape characteristic of softening systemsin the neighbourhood of its maximum values and a secondary re-sponse peak arises for periods about 1.8 times the period at whichthe maximum value is reached. Global stiffness and dissipativeproperties depend on the input level.The study of the initial response under impulsive excitation

showed that the system vibrates more rapidly. The response is no-tably stiffer and more dissipative as a consequence of the Mullinseffect. Stiffness and dissipative properties are proportionally en-hanced.The analysis under seismic input evidenced that the response

is strongly different from the stable behaviour and remarkablydepends on the particular strain history. Furthermore, the dynamicproperties change with the input level and different sensitivityof the system with respect to input frequency content may beobserved for accelerograms with different PGA.As expected, the dynamics of HDR-based systems are quite

complex in that all the phenomena characterizing the materialbehaviour, the transient response and the non-linear dependenceon strain and strain rate, strongly influence the dynamic behaviorof the systems and the more important design parameters, suchas maximum displacements and forces. This aspect cannot beneglected in seismic design of dissipated or isolated structures andshould be considered in elaborating simplified approaches used inanalysis and design procedures.

Acknowledgement

The present work was financed by the Italian research grantReLUIS project – D.P.C. (task 7).

Appendix

The following expression of the free energy per unit volumewasadopted for the rubber:

ϕd (γ , ξ) = [1+ αm (1− α4)]ϕe (γ )+Ev12(γ − α1)

2

+Ev22(γ − α2)

2+ (1− α5)

Ev32(γ − α3)

2 (A.1)

3618 A. Dall’Asta, L. Ragni / Engineering Structures 30 (2008) 3610–3618

Table A.1Numerical parameters used in the free energy expression

fe(γ ) (N mm−2) Ev1 (N mm−2) v1 (N mm2 N−1 s−1) η1(γ , α4) (N mm−2)

0.029γ 5 + 0.082γ 3 + 0.29γ 2.56 0.078 0.179+ 0.127(1− α4)+ 0.047|γ |

Ev2 v2 η2 αm E3v v3 η3 ζe ζv(N mm−2) (N mm2 N−1 s−1) (N mm−2) (N mm−2) (N mm−2) (N mm2 N−1 s−1) (N mm−2)

0.447 26 0.025 1.5 0.256 2.23 0.025 1.0 0.2

and the expression of stress may be obtained by means of Eq. (1);it has the following form

τd = fe (γ )+ Ev1 [γ − α1]+ Ev2 [γ − α2]+ αm (1− α4) fe(γ )

+ Ev3 (1− α5) [γ − α3] . (A.2)

A rheological model consisting of the sum of two contributionswas adopted. The former contribution is described by constitutivelaws that do not change during strain history. The stress is providedby the sumof a nonlinear elastic contribution fe (γ ) = dϕe (γ ) /dγand two nonlinear viscoelastic contribution described by theconstants Ev1, Ev2 and two internal variables (dashpot strains)α1, α2 related to different relaxation times.The latter furnishes the transient response (Mullins effect)

which degenerates as the strain history progresses. The stress isprovided by the sum of a contribution which is proportional toelastic stress fe (γ ) (αm is the proportional constant) and a vis-coelastic contribution described by the constant Ev3 and the relatedinternal variableα3. The other internal variablesα4 andα5 increaseduring deformation and control the degrading response.The evolution laws of internal variables have the following

expressions

α1 =

[|γ |

η1 (γ )+ ν1

]Ev1 [γ − α1] (A.3)

α2 =

[H (−γ γ )η2

|γ | + ν2

]Ev2 [γ − α2] (A.4)

α3 =

[H (−γ γ )(1− α5)

|γ |

η3+ ν3

]Ev3 (1− α5) [γ − α3] (A.5)

α4 = ζe |γ | (0.5 |γ | − α4) if α4 < 0.5 |γ | (A.6)α4 = 0 if 0.5 |γ | ≤ α4 ≤ 1 (A.7)

α5 = ζv |γ | (1− α5) (A.8)

where H denotes the Heaviside function (H (x) = 1 if x > 0 andH (x) = 0 if x ≤ 0).Previous constant Evi, the function fe (γ ), η1 (γ ) and parameters

ν1, η2, ν2, η3, ν3, ζe, ζv depend on the rubber compound. For the

high damping rubber considered, the values assigned to the allnumerical parameters and the expression of function fe (γ ) arereported in Table A.1. These parameters were calibrated directlyfrom the experimental tests results [11] and refer to a rubbercompound with medium stiffness.

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