EVOLUTIONARY MODEL OF VISCOELASTIC DAMPERS FOR STRUCTURAL ApPLICATIONS
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By Alessandra Aprile; Jose A. Inaudi/ and James M. Kelly3 ABSTRACT: Th effe~ts of tempe~ature on the energy dissipation of viscoelastic dampers for seismic mitigation 7 of structures are mvestigated. To Simulate the damper behavior, an evolutionary model is proposed to describe ~e dependence o~ ~e ~echanical propertie~ of th~ damper on the deformation frequency and the temperature mcre~e due to diSSipation. Thermorheologlcally Simple materials are considered and the influence of the deformatio~ frequency on the storage and loss moduli is modeled using fractional derivative operators. The effect of m~tenal temperature on the for~e-~eformation relation is modeled using the concept of evolutionary transfer func~0!1' and the proposed model IS Implemented using a step-by-step technique in the frequency domain. The p~edictions. of the proposed. model in the case of sinusoidal and seismic deformations show good agreement with e~penmental results. Fmally, the response spectra of single-degree-of-freedom structures with added viscoelastic ~pers and subjected to seismic excitation are computed using the proposed evolutionary model; the results obtained show that the thermal effect due to energy dissipation is not always negligible.
INTRODUCTION
A convenient way to improve the dynamic performance of a structure subjected to wind or earthquake loading vibration is to incorporate mechanical dampers to augment the structural damping. This damping increase yields a reduction in the expected structural damage through a significant reduction of the interstory drifts of the structure during the dynamic event. Although the developments in research and analysis techniques, paralleled by significant improvements and refinements of device hardware, make the mechanical dampers totally suitable for consideration in new or retrofit design, there are still relatively few applications to buildings (Mahmoodi 1969; Aiken and Kelly 1990; Tsai and Lee 1993; Inaudi et al. 1993; Inaudi and Kelly 1995; Lai et al. 1995; Shen and Soong 1995; Makris et al. 1995). Among a number of viable types of energy dissipation devices proposed, the viscoelastic dampers have found several successful applications for wind-induced vibration reduction of the tall buildings. Remarkable examples are the 110-story twin towers of the World Trade Center, in New York City, and the 73-story Columbia SeaFirst Building and the 60-story Number Two Union Square Building, both in Seattle. The implementation of viscoelastic dampers (VEDs) for seismic protection has been investigated only in the last few years (Zhang et al. 1989; Zhang and Soong 1992; Bergman and Hanson 1993; Hanson 1993; Tsai 1993; Chang et al. 1993; Kasai et al. 1993; Abbas and Kelly 1993; Chang et al. 1995; Munshi and Kasai 1995). An accurate model for the mechanical behavior of VEDs subjected to seismic loading must incorporate the variability of the material physical properties with the deformation amplitude, the excitation frequency, and the temperature conditions during dynamic service. Gemant, in 1936, suggested a fractional derivative constitutive relationship to model cyclic-deformation tests performed on viscoelastic material specimens. The Gemant model was 'Postdoctoral Res. Engr., Dipartimento Di Ingegneria, UniversitA Di Ferrara, Via Saragat, 1,44100 Ferrara, Italy. 'Res. Engr., EERC, Univ. of California at Berkeley, 1301 S. 46th St., Richmond, CA 94804. 3Prof., Dept. of Civ. Engrg., Univ. of California at Berkeley, 1301 S. 46th St., Richmond, CA. Note. Associate Editor: Sami F. Masri. Discussion open until November I, 1997. To extend the closing date one month, a written request must be filed with th~ ASCE Manager of Journals. The manuscript for this paper wa.s sUbrnt~ed for review and possible publication on March 21, 1996. ThIs paper IS part of the Journal of Engineering Mechanics, Vol. 123, No.6, June, 1997. ©ASCE, ISSN 0733-9399197/0006-0551-05601 $4.00 + $.50 per page. Paper No. 12973.
based on a fractional generalization of the Maxwell model. Later, many authors followed the fractional approach to viscoelasticity contributing new developments in the theory and widening of the application field (Caputo 1969; Smit and De Vries 1970; Bagley and Torvik 1983; Koeller 1984; Koh and Kelly 1990; Makris and Constantinou 1991). The use of a fractional operator to model the viscoelastic materials turns out to be very profitable, since it allows an excellent model accuracy using a small number (3 or 4) of parameters (three or four). From an analytical point of view, the mathematical complexity of the fractional formulation can be easily dealt with by using frequency-domain techniques. A crucial modeling aspect in viscoelastic materials is the degradation of mechanical properties due to the thermal effect rising during the dynamic event; indeed, experimental evidence shows that both of the storage and the loss moduli sensibly vary with the temperature of the material (Tsai 1994). To model the coupling between the mechanical properties and temperature, the writers have proposed an evolutionary constitutive relationship that models the material behavior using an evolutionary transfer function concept, given the material physical properties valued at the initial conditions (Aprile et al. 1994; Aprile 1995). The accuracy of the proposed analytical model is verified using dynamic testing data of VEDs tested by Blondet (1993) ~t th~ Civil Engineering Department of the University of CalIfor~la at B~rkeley. These devices are based on Polymer 109, a VIscoelastic material manufactured by 3M Corp. (St. Paul, Minn.). Us~ng a 3M database information on the storage and loss moduli of the material at different temperatures and at different frequencies, the proposed model is calibrated. The model is then used to predict the experimental results. Finally, single-degree-of-freedom (SDOF) systems with VEDs subjected to seismic excitation are studied: the equivalent period and damping ratio of the system are introduced and response spectra of the viscoelastic oscillator are computed using the proposed evolutionary model for two seismic events. EVOLUTIONARY MODEL
At constant temperature, the proposed model is based on the fractional standard linear solid model (Bagley and Torvik 1983), which is expressed as a(t)
+
aDQa(t)
= Go[e(t) +
bDQe(t)];
0 < ex < 1
(1)
where a = shear stress; £ = shear strain; and the four parameters Go, a, b, and IX must be evaluated from the available JOURNAL OF ENGINEERING MECHANICS 1 JUNE 1997/551
J. Eng. Mech. 1997.123:551-560.
experimental data. Also, DO. represents the fractional derivative of order a (Ross 1975) Daf(t)
=daf(t): = a dt
!!...
1
fO -
a) dt
r
0
0 only if (Bagley and Torvik 1983) Go ~ 0;
b > 0;
b a> 0; a
~
(10)
1
Eqs. (7)-(9) show that the adopted model describes the material mechanical properties as fractional power functions of frequency; actually, this model is able to fit experimental data with fair accuracy over a wide range of frequencies (Koh and Kelly 1990). Viscoelastic materials can be extremely sensitive to temperature changes; they can experience a real chemical-physical alteration for increasing temperature: from a glassy to a rubbery state, through a transition state, up to an irreversible chemical decomposition around 80-100°C. For increasing service temperature, both the stiffness and damping viscoelastic material capability decrease; besides, it is to be considered that
(1)
In the literature, what is expressed in (11) is called the evolutionary transfer function (Priestley 1965). By putting (11) into (4) and performing the inverse transform the material stress can be obtained
where i = the complex unit and S(iro)
= G*[jro, T(t)]
=2~ i~~ G*[iro, T(t)]E(jro)e
IW
'
dro
(12)
Eq. (12) can be evaluated when the temperature T(t) is known for each t. From the general heat transfer equation holding in thermomechanics, the following approximate energy balance equation can be derived if the effects of dilatation, free energy, and thermal conduction are neglected and the material has a large loss factor: aT(t) 1 ( ae(t) --Fd-at)--
at
at
c,p
(13)
where c, and p are, respectively, the material specific heat and mass density, and assumed to be constant since they are insensitive to the temperature change in the range of interest (i.e., 5-50°C). In Shapery's (1964) and other authors' opinion, the approximations adopted for (13) are proper for many useful applications of viscoelastic materials, mainly when viscoelastic layer small thicknesses and short duration loads are involved, like in the case of viscoelastic dampers in structures subjected to earthquake loading. By integrating (13) T(t)
=To + -CaP1
i' 0
ae(T) a(T) - dT aT
(4)
where To = temperature initial value. Kasai et al. (1993) have recently used (14) to estimate the temperature increase in viscoelastic dampers subjected to earthquake loading. The proposed model consists of (12) and (14). We further fake the thermorheologically simple material hypothesis (Schwarzl and Staverman 1952) and the consistent temperature-frequency equivalence principle (Jones 1980; Rogers 1983). Therefore, the temperature influence on viscoelastic material behavior can be translated into a frequency shift in accordance with a shift factor aT, as follows: (IS, 16)
where TR is the assumed reference temperature; and roR' consistent with it, is the reduced frequency. The shift factor aT can be evaluated using the following expression (Kasai et al. 1993), safely applicable at least for the temperature range that is of primary concern for the applications of interest aT(t)
TR)P =( T(t)
(17)
where the value of the parameter p must be empirically estimated. Using (15)-(18), (12)-(14) may be expressed through the following handier formulation:
552/ JOURNAL OF ENGINEERING MECHANICS / JUNE 1997
J. Eng. Mech. 1997.123:551-560.
a(t)
=J.... 21T
f
+~
_~
G*(jroR ) ITRE(iro)e'W' dro
(l8a)
/Oo,a,b,
T
a, p,
them. In Fig. 2 the main characteristics of the damper are schematically shown; the total shear area of the viscoelastic material is 1,858 cm 2 (288 sq in.) and the effective thickness of each pad is 2.54 cm (1 in.). The shear stress and strain are assumed to be constant within the pad (Units Equivalence to SI are: in. = 2.54 10- 2 m; kips = 103 lb = 4.448 kN; psi = lb/ sq in. = 6.895 103 Pa). During the tests carried out by Blondet, the specimens were subjected to harmonic excitation consisting of three cycles of displacement at a frequency of 0.5 Hz and at increasing maximum strain (10, 20. 50. and 80%) in order to estimate the values of G' and Gil. Then, the dampers were subjected to 10 cycles of harmonic displacements with a shear strain of 100%. also at 0.5 Hz, to represent conditions significantly more severe than those expected during the maximum credible earthquake. Finally, the first cyclic test (10% shear strain) was repeated after the damper had cooled down and it was verified that viscoelastic material could fully recover its mechanical properties after the test under extreme conditions. As a second test sequence, the dampers were subjected to dynamic load conditions corresponding to increasing levels of seismic demand. The displacement time histories used as command signals for the experiments were generated by a numerical simulation of the building response to recorded earthquakes, with the supplemental viscoelastic dampers. Thus. the building model was subjected to ground motion recorded during the Loma Prieta earthquake scaled respectively for the following seismic levels: service level earthquake (O.lg), design base earthquake (0.34g), maximum credible earthquake (O.4g). At last, some failure tests were conducted by subjecting a few YED specimens to increasing strain. Shear failure always occurred through the viscoelastic material at strain levels varying from 300% up to 400%. All of the tests were peformed at room temperature, from 23 to 25°C. As a preliminary step for the application of the proposed model addressed to the experimental results simulation, the parameters Go, a, b, and a of the model in (6) were evaluated; their magnitude was determined so that the G' and T) functions.
TOo TK, p, Ca, £(t), At /
1 to=O. T(to) = TOo
anto)=~j
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E(ic.o) = 1"[£(t)]
~
Ie = 1,2,3, ...
~ tk- tk-! + At
~ = o>antk.!) l+b(i ~)U . t O*(ICOi) = 0 0 t 1 + a (i COi)U o(t) = 1".J [O*(i~)E(ic.o)] at = o(tk), Et=£(tt) £t+!. £t.! Et2At 1 . ATt =-GtEtAt
pea
T(\) = T(\.!) + ATt
antk)=(~j
Ie 