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Stochastic response of secondary systems in base-isolated structures
Probabilistic Engineering Mechanics 7 (1992) 91-102
Stochastic response of secondary systems in base-isolated structures G. Juhn, G.D. Manolis & M.C. Constantinou Department of Civil Engineering, State University of New York, Buffalo, New York 14260, USA
(Received September 1990; accepted September 1991) In this work, the stochastic response of secondary systems attached to a baseisolated structure undergoing random ground motions is examined. It is assumed that the properties of this combined structural system are deterministic, while the ground motions are described by a filtered white noise model. The only nonlinear component of this structural system is its base isolation mechanism, which is linearized by using equivalent linearization. Also, a substructuring algorithm is developed which requires the dynamic properties of the individual, fixed-base components of the structural system. Both stationary as well as nonstationary cases are considered and comparisons are made with the results of Monte-Carlo simulations to ascertain the validity of this methodology. The example studied herein is a six-storey steel building frame with a base isolation system consisting of sliding bearings and restoring force springs. For this example, spectra are constructed that account for primary-secondary system interaction and depict the effect of variations in the base isolator's structural parameters and in the mass and location of the secondary system on the latter's root-mean-square (RMS) accelerations.
Only a limited number of experiments have been performed on shaking tables because of the high costs associated with such work. In this respect, the pioneering work of Kelly and his associates 5'6 with scaled steel frames containing secondary systems and under fixed base or base-isolated conditions must be mentioned. Some full-scale tests have also been conducted for nuclear power plant equipment and electrical equipment. Finally, experiments involving secondary systems alone have also been conducted using pseudodynamic testing. A detailed review of experimental methods for secondary systems can be found in Manolis et al. 7 Understanding the dynamic behavior of secondary systems is the first step in devising ways to protect them. This is very important in view of the fact that most secondary systems perform critical functions. A novel way that has recently emerged is base isolation, which is an aseismic design strategy with its objective being reduction of the transmission of motions from the ground to the structure. This is achieved by lowering the fundamental frequency of the structure to values that are below the dominant frequencies of the ground excitations and/or by dissipation of energy through inelastic action. A properly designed base isolation system protects not just the superstructure, but also its contents. A comprehensive review of base isolation systems can be found in Kelly. 8 A cursory classification of base
INTRODUCTION The behavior of secondary systems and critical equipment in conventional structures and in industrial facilities was reviewed by Chen & Soogn. 1 The two basic approaches for analyzing structures containing secondary systems are the floor response spectrum approach and the combined system approach. The former method, in its original form, ignores primary-secondary system interaction and consequently gives results that are valid under certain conditions only. The combined system approach, which includes perturbation methods, complex modal analyses, substructing concepts, etc., is more general but requires lengthier computations. What makes this problem difficult to analyze, besides its topological complexity, is the mismatch between the mass of the primary structure and that of the secondary systems, the existence of non-classical damping, and the possibility of inelastic deformations under intense ground shaking. Recent advances 2-4 have addressed the aforementioned problems, with attempts to include interaction effects in the floor response spectra as well as to improve numerical techniques for combined system analyses. Probabilistic Engineering Mechanics 0266-8920/92/$05.00
© 1992 Elsevier Science Publishers Ltd. 91
92
G. Yuhn, G.D. Mano[is, M.Co Constantinou
isolators includes rubber beatings with a lead core, 9 laminated rubber bearings, ~° the earthquake barrier system] ~ and various systems based on friction such as the alexisismon system, ~2 the resilient-friction base isolator, ~ Teflon sliding bearings, ~<~s and the friction pendulum system] 6 The basic components of a base isolator are: (a) A restoring element, such as a spring or a rubber bearing, that provides centering force capabilities. The restoring element may exhibit hysteretic material behavior resulting in energy absorption capabilities. (b) A sliding element, such as a Teflon bearing, that alters the transmision of motions across its ends. Sliding elements may exhibit either a constant (Coulomb) or a velocity-dependent friction coefficient. (c) Additional energy absorption capabilities provided by viscous dampers, yielding beams or similar devices. Base isolation has the advantage that all nonlinearities exhibited by the combined structural system are concentrated at the base level. A comparative study of six representative base isolation systems with respect to the 1940 El Centre and the t985 Mexico City earthquakes appears in Suet d. ~7Other recent analytical work includes examining the distortion in the high frequency content of the secondary system response caused by assuming ctassicaI instead of complex modes for the base-isolated structure, TM identifying the primary mode of the baseisoIated structure as controlling the response of high frequency secondary systems, ~ and studying the effect of yielding in a fixed base structure as a means of protecting the secondary systems9 All the work previously mentioned has been for a deterministic input. Among the earlier work to ~ook at the case of stochastic input is that of Singh & Wen, 2'~who employed the Markov vector approach to examine the nonstationary seismic response of a two degree-of-freedom (DOF) primary-secondary structure model. As far as base isolated structures are concerned, Ahmadi 2~- coe.sidered the response of a sliding structure to stationary as well as filtered white noise. Constantinou & Tadjbakhsh23 also used stochastic concepts in studying the optimum design of a friction-type base isolator supporting shear buildings. ~In later work, Constantinou & Tadjbakhsh ~4 employed equivalent linearization in conjunction with damper-type isolators whose hysteretic behavior was modeled by a nonlinear differential equation. This equiw alent tinearization technique was subsequently extended to cover friction-type elements, such as sliding bearings. ~ Also recently, L i n e t M.26"2v considered the stationary respor~se of base-isolated buildings subject to randorn earthquake excitations whose spectral content is modelled by a Kanai-Tajimi and a high-pass fiker. For the idealized case of a single DOF secondary system
attached to a single DOF primary str~ct~re where t"~eti~ components exhibit hysteretic ~.onlineari:ies~ igusa 2~ developed analytical expressior~s asiv~g the perturbation method in conjunction with equivalent 12nearizatien tha!-: showed a significant reduction, of the system response when it was subjected to a wide-band input. For the same problem, the n°americat results of Pires29 indicate a s:~5o. stantiai reduction of the secondary system accetera~oss in the vicinity of resonance due to energy dissipation, b ~ the forces transmitted to the secondary system increased ir~ the presence of inelasticity a~ higher freqaer~cies~ Finally, ~wan & Paparizos 3° studied the response e~ a nonlinear single DOF system ~.n temas of ~ts sewer spectral density fianction and concluded that equivalent ~i~aearization works well %r a system where friction: is supplemented by. a rest©ring spring. The present work focuses on the dynam~.c response o! secondary systems in a base-isolated st,~cmre azbjectec to random ground motions. ~n particular, ~he eq"oAvale~t lineariza~iola procedure for a base isolation system ses~. sisting of a sliding bearing component ~hat exhibits velocity-dependent friction plus a series of res~o£ng helical springs is discussed, aIor~g with modeiing aspec-~s concerning the combined structural system and the inp~a~ motions. The response statistics of 'the combined s~ruc~ tare augmented by the stochastic ground motion modei are obtained by solving a Liapunov matrix equatiOrio These results~ along with Monte-Carlo simuIations, for both stationary and nonstationary cases~ are ~he~ presented for a six-story s~cructura] system Fiu~all?~ a series of parametric studies are performed ;o ascer~xdn the effect of changes in the base isolator ann secondary system structural parameters on ~he RMS accderations of secondary systems attached to the structarai system and to estaNish the degree of passive pro~ection e~?re4 to the secondary systems by the base isolator
NUMERICAL MODELING ConnNeed sg~uct~are The combined structure includes the primary str~ctare.~ the secondary systems, and the base isolation mechanism° We focus on a mukioDOF frame with a number ef single D©F secondary systems attached to rt at va~cus locations, plus a single D©F base isolator. This discrete parameter, combined structure is shown in Fig 1 a~adits equations of motion are given beIow as
where
{x}
}
-
{xj
i
Stochastic response of secondary systems in base-isolatedstructures
former model requires the additional variable z(t) for its proper description that will also be subsequently discussed. As usual, in eqns (1) and (2), dots indicate time derivatives, g is the acceleration of gravity, superscript T denotes transposition while summation E extends over all superstructure DOF. It is obvious that a lumped mass representation is assumed, with MT being the total mass of the system. Finally, Appendix I gives explicit expressions for the case of a six-story frame containing three secondary systems attached to the top floor. Introduction of modal transformation at this point is advantageous because it requires the dynamic properties of the individual components rather than those of the combined structure. This advantage is important since most experimental identification procedures for structures are done at the component level. 7 In particular, the generalized co-ordinates {q} of the combined structure are given in terms of the physical co-ordinates {x} as 31
Z,Z f,¢ t",.
XXZ J,,
93
48 "
{X}: {Xb {Xp}I : [~b [•p] ]tqb t = [~b]{q}
'l
(a)
{qp}
{xs} )
[¢j J {, {q~} J
(3)
FRAME
'1
SLIDING BEARINGS 5HAK ING TABLE
(b)
Fig. 1. (a) Six-story frame with attached secondary systems and (b) base isolation mechanism.
MT~b + Cbkb + KbXb + #oMTgZ
Cb = (2~bOgb + ce)MT and
Kb
[M*]{~} + [C*]{~t} + [K*]{q} + {e*} =
-{F*}5~g
(4)
Expressions for the above system matrices, as well as for the modal participation factors {F*} and vector {e*} can be found in Appendix 2 for the six-story structure example.
for the superstructure and
= --MT~g -- E M)~, i where
where ~bb = (MT) -1/2, [q~p] are the eigenvectors of the fixed-base primary structure normalized with respect to the mass matrix [Mp], and [q~s] is a diagonal matrix containing the secondary system mass terms (Ms1)-1/2, (MJ-1/2, and (M~3)-1/2. By substituting eqn (3) in eqns (1) and (2) and pre-multiplying by [~b]T, the following modal equation is obtained
= oMT} (2)
for the base isolator. In the above, all displacements x are relative to the base except for xb that is relative to the ground, and subscripts p, s, b and g stand for primary structure, secondary system, base isolator and ground, respectively. Furthermore, M, C, and K stand for mass, damping and stiffness, respectively. Square brackets denote matrices, curly brackets denote vectors and absence of this symbolism denotes scalars. Also, Ogband {b respectively are the natural frequency and damping ratio, while ce and #o are damping and friction coefficients resulting from equivalent linearization of the base isolator that will be subsequently discussed. We will distinguish between a viscoplastic (VP) model and a constant sliding (CS) model for the base isolator. The
Base isolation The base isolation system consists of Teflon sliding bearings with restoring force capability provided by linear springs. The equation of motion of the base isolator, once it has been placed inside a multi-story structure, is eqn (2). The following two models have been proposed 25 for base isolation systems of the frictional type: (a) a model based on the theory of viscoplasticity which is capable of accounting for stick-slip conditions, and (b) a model based on the assumption that sliding is continuous across the Teflon sliding bearing interfaces. It should be noted that both models are capable of accounting for a velocity-dependent friction coefficient #(-tb), a fact which has been experimentally observed for Teflon sliding bearings. 15 In particular, the following expression has been found 14to best describe the friction
94
G. guhn, G.D. Mano[is, M . C . Constantinou
coefficient: #(2b) =
by
# ~ , - Dr exp ( - ~12~1)
(5)
tn the above, #~,~ is the coefficient of friction at high velocity, Dr is the difference between # ~ and the coefficient of friction at very tow velocity, and ~ is a constant dependent on the bearing pressure and on the conditions that exist at the sliding surfaces. Coulomb friction, where # is a constant, is recovered as a special case by setting Dr to zero in eqn (5). in the viscoplastic (VP) model case, the dimensionless variable z(t) o f e q n (2), whose range is [ - 1, 1], obeys the following constitutive equation proposed by Wen ~ for modeling hysteretic behavior =
- - ( V l , b l Z M ~-~ + ~ & t z ! ° -
A&)/y
(6)
where I3, ?, and A are dimensionless constants characteizing the shape of the sliding isolator's hysteresis toop, y is the yield displacement of Teflon and n controls the mode of transition of the model into the inelastic range. Constantinou e t a [ . ~4 found that eqn (6) satisfactorily reproduces the behavior of Teflon sliding bearings provided that A = t and 13 + 7 = 1. For this combb nation of parameters, the model described by eqns (5) and (6) collapses into a viscoplastic model. Finally, plots of the variation of the friction coefficient with vdocity and of the loops produced by friction force versus sliding displacement, obtaining from experiments as well as from numerical simulations, can be found in Refs ]4, 15 and 25. An important property of Teflon sliding bearings is reducton of the friction force mobilized across the sliding interfaces as the sliding velocity decreases. This means that tess force is needed to maintain relative motion as the superstructure's sfiding slows down. As a resu!L relative motion continues with almost complete absence of stick-slip episodes, a fact that has been confirmed in shake table tests of model structureSo 3~ Accordingty, the continuous sliding (CS) model for friction is well justified and can be recovered from the VP model by simply replacing z(t) in eqn (2) by sgn (kb), where sgn is the sign function. The CS model has Been found 2s to produce good results when compared with Monte-Carlo simu!ations, except for sliding bearings which exhibit high friction. Furthermore, both VP and CS models should be examined because they provide upper and lower bounds to the simulation results for the base isolator responseY As can be seen by reference to eqn (2), the method of stochastic equivalent linearizatiort 32 has been used in order to obtain approximate statistics for the response of the base isolation system and consequently of the combined structural system. The equivalent damping (G) and friction (be~) coefficients appearing in eqn (2) are obtained by minimizing the mean square difference between the original nonlinear equation of motion of the base isolator with the restoring force due to friction given
for the VP model or by f(kb)
-- ff(3~b)Jl~ygsgn txw
•* ic for the CS model and the linearized equanor. ~sem. :~e. eqn (2)° The resulting expressions for the VP model are gben by
c% =
E[f(J%)kb]/E{22] "
and
~
{9)
1 - h[f,Xb)Zl/E[z-k i g
#e
with E being the expectauon opera,or. By assuming ~ha; 2b and z of the linearized system are jointly Gaussian. zero mean processes, eqns (9) resuk in /
x exp (cdE[22]/2
xi/2
o er~c@E[xb!' /~, 2 ) / ~ ! z ~
>
and #~" =
Vm~. -- Dr exp @2E[2~]/2) erl% G[~'Ex~]:"a,,, Z;~, /
where erfc is the complementary error eqn (6) is Hnearized as
funcuon
C~2b - Kez
=-
&]so,,
~~~'~
where the equivalent damping and stiffness coe~iciems C~ and Ke, respectively, are given by the ~%Jlowing expressions for n = 3: (3~2[z j E l x b z j E t x b j
-
=
I
-
E[&z]~/s[mg]~/:)+ 29 k ~ )
<
and
;
(ECz2]E[k2U2
~ " "~--kXb: ~
'd
4- 6fl
E[kbz]g[z2] L'; j (i~2!
The CS model can be recovered from me above o } % disregarding eqn (1 i) and se~t~ '~ n g #o - 0 plus re~iaci~.g .. by
x
2 ~..g exp(0-5~ 2 or,) ~nc(~%)~!
/
:~
'in eqn (2), where £, is the standard deviatior~ of c~
13'
95
Stochastic response of secondary systems in base-isolated structures
recast as a first-order matrix differential equation in the form
Ground motions
In order to represent strong ground motions as a stochastic process, the usual Kanai-Tajimi (K-T) filter is supplemented with a high-pass (H-P) filter that removes singularities at zero frequency. This approach has been recently used in Lin et al. 26 and results in the following form for the power spectral density function of the
{I~} = [A]{Y} + {e}qJ(t)f(t)
(16)
In the above
{y}T = [{0}T, {q}V, z, 2g, Xg, 2f, xr] {ey = [{O} T, {O} T, 0, 0, 0, - 1, 0] and
[A] =
-- [M*]- 1[C@],
- [M*I-'[K*],
- [ M * ] - ' {e*},
[M*]-' {F*} (2{rcor,
(or,
-- 2~g cog,
-CO~)
[I]
[o1
{o}
0
0
0
0
{--Ce(/)b, {0} T}
{oF
-Ko
0
0
0
0
{o} T
{O} T
0
{O} T
o 0
1
0
0
0
(off
{oy {o} T
0
0
-- 2~fcof
-CO~
{O} T
{o} T
o
0
0
1
0
ground accelerations
&(co) = I1 - (±~212 + 4¢2g(~gg) co2
2~fOgf
-CO~ 2~%
[S] = [A][S] + [S][A]T + [B]
\cog/J
.so \~fl J (14) In the above, So is the power spectrum of white noise. What eqn (14) implies is that white noise is being filtered through the K-T filter (subscript g) and the H-P filter (subscript f) as 2~gcog2r + (D~Xf)
and
-
co2g
(17)
The covariance matrix of the response {Y}, Sij = E[Y~Yj], can be obtained by solving the following Liapunov differential equation of order N x N (N = 25 for our example)
1 + <~ ~
2 = 2~g "{- 2~gcog2g + (.OgXg
-
I
(15)
2r + 2(rcor2f + co~xf = ~ ( t ) f ( t ) respectively, where f(t) is a white noise process with intensity 2rcS0 and ~(t) is a deterministic time envelope function. Values chosen for the above parameters are So = 71"0cm2/s 3, COg = 3"47Hz, ~g = 0"59, COr= 0.50Hz, and ~r = 1.0, which correspond to a magnitude six ground motion o n r o c k . 26
where [B] is a matrix with all elements equal to zero except BN-~,N-~ = 2rcSoW2(t). Since [S] is symmetric, eqn (18) can be written as a system of N ( N + 1)/2 firstorder nonlinear differential equations that can be solved using standard techniques such as the Runge-Kutta or Adams-Moulton methods. The stationary case ([S] ~--0, BN_I, N 1 = 2zcS0) results in a system of N ( N + 1)/2 nonlinear algebraic equations that can also be solved using standard techniques such as Newton's iteration. All these techniques, however, require prior vectorization of [S]. There are other avenues open for solving eqn (18). It is possible, for instance, to work in the frequency domain as done by Lin et al. 26 The approach which was adopted here was to use complex modal analysis34 so as to diagonalize the covariance matrix. This first requires finding the eigenvalues 2i, the right eigenvectors [@R], and the left eigenvectors [q)e] of eqn (16). By introducing generalized co-ordinates {Q} and using the complex modal transformation { Y} = [@"] {Q}, eqn (16) can be decoupled to read as {{)} - [2]{Q} =
NUMERICAL SOLUTION The system of linearized differential equations (4) is augmented by the ground motion equations (15) and
(18)
{F}~(t)f(t)
(19)
where {F} = [@L],{e}. Consider now stationary white noise excitation (~(t) = 1.0). The stationary solution of eqn (19) is {Q(t)} = fo [H(u)] { r } f ( t - u)du
(20)
G. Yuhn, G.D: Manolis, M.Co Constantinou
96
Table L Prh~ary structure proFerties under fixed-base conN~io1~s
Mode
!
% (Hz)
2
1o17 3.88 5-0 5.0 0"068 0-I 70 0.138 0-301 0"200 0-271 0"25t 0"098 0-289 - O-t27 0-314 -- 0-301 AlI diagonal elements equal to 56-6 x
~ (%)
[Mp] (kg)
3
4
5
o5
6.64 5-0 - 0-256 - 0-232 0:077 0-3i I O"t35 - 0.264 t03
9-52 5.0 - 0-357 0-037 0°305 - 0-102 - 0°230 0 o166
] 2°40 5.0 0°237 ,0.273 0"073 0"227 -- 0-32~ O-136
?4°46 5.0 - 0 ~t6i 0~264 --0-310 0:280 - 0-198 0~065
where [H(u)] is a diagonal matrix whose ith component is hi(u) = e '~'~, with u > 0o Since {Q(t)} is a complex random response, the correlation function matrix of
{Q(0} is [Re(z)] :
E[({Q(t)} {Q*(t + r)}T] 2rcS0{F} {F*} ~ fo [Hiu)][H*(u + ~)]T du
:
(20 where * denotes complex conjugate. An element of
[RQ(~)] is t~CO
R~(Z) :
2zSoGi] J0 e;~e;¢(~+*)du
-
2rcs0a~je
V~ /()o, + )~)
(22)
with G~j being an element of matrix [G] = {F} {F*} v. The covariance matrix of the generalized co-ordinates {Q(t)} is s~
=
R~(o)
:
-2~soc,/(< + g )
(23)
Finally, the covariance matrix of the co-ordinates { Y(0} can be obtained as
s ] = [o ~1[sq [¢~*F
(~)
To revert to the physical co-ordinates [SX], eqn (3) needs to be used in conjunction with the above equation for the modal part {q} of response vector {Y}. It should be noted that matrix [A] contains coefficients Ce, K~, #~ and C~ from the tinearizafion process that are functions o f the response covariance and are thus unknown. This necessitates the use of an iteration scheme where: (a) initial values of these coefficients are assumed; (b) [S x] is cap culated based on those initial guesses; (c) the coefficients are computed via eqns (i0) and (12) using the results of the second step; and (d) the iterations are terminated when the values o f the coefficients converge. The same procedure is also applicable to the nonstationary response. The starting point is to write {Q(0}
:
~ [H(~ -- ~)] {F}~F(z)f(r)dz
(25)
in lieu of eqn (20). Then, the procedure for calculating the correlation function matrix IRe(t, z)] follows along the lines previously described.
NUMERICAL RESULTS The dynamic response o f the secondary sysmms attached to a base-isolated, six-story frame (see Fig. I) in ~he fo~:~ of RMS acceleration spectra is now examined. The numerical values assigned to the dynamic properties of the six-story frame under fixed-base condkions arm ~:o the base isolation mechanism are ~isted in TuNes ! and 2~ respectively° These properties were extrapolated from ar~ experimental program, 7n4~:sn3 where a base i~otation system using Teflon sliding bearings and restoring hekicat springs, originally detached from and subsequently attached to a quarter-scale six-story stee! frame contain. ing secondary systems, was tested on a shake m b b . The RMS response spectra were obtained by s o b i n g eqn (16~ via the complex mode approach previously discussed° The Monte-Carlo simuiations, 3s which are based on the so!ution of eqns (4) and (5), were obtained by generating 200 realizations of the input process and using ensemNe averages. Since an increase from 200 to 400 sampbs did not appreciably change the resuRs, the Monte-Carlo simulations are considered to be the exact solution. At firs% ~ Is necessary
Table 2. Base isolator #repert[es Nmax
(%)
Df
~
rd)b
~b
~
":
/{!
$2
(%~ (s/cm) (Hz) (%)
High f , iction case !7"0 10"0 3.22
0-5
0-0
0-t
0'9
!..0
3
0 25
Intermediate friction case 1.9 9-27 0"24 0"5
0~0
0,J
0-9
i-S
J
0.25
Low friction case 5.72 4 - 8 5 0'20
0"0
0"i
0~9
-g
b
0~25
0-5
Note: Total mass MT = 3774 x 103kg~
Stochastic response of secondary systems in base-&olated structures
Table 3. Error (as %) in the viscoplastic and constant sliding models versus base friction Pm.x
i .80 I
,-, O
d U
•
•
. .
I
.
..
I
" 'i'-
~'"
"t"
/~max
1.20-
0.80-
11" 0.40-
0.00 0.00
a,." /
•
.
.
;
S2
-
0.03
.
.
;
•
.
.
~
'
. S3
,
0.06 0.09 FRICTION COEF.
O, 12
(a) I
0.20
I
.
.
.
I
p ,-, 0
d
S1
$2
$3
6th Floor
Base
VP Model 0-02 -40.4 0'04 - 38.7 0.06 -26"8 0'08 -14"3 0"10 -5"4 0.12 3.5
-47"8 - 54.8 -39"3 -15"4 -6-7 -0"9
-0"6 - 21-4 - 13.9 -8.3 --7.2 --2.5
-4"3 - 22.7 -24-5 -16"7 - 10-3 -3-1
9-2 7"9 0"3 -19"0 -38-9 -53.1
CS Model 0"02 - 33.3 0.04 -3'2 0'06 9.7 0.08 18'4 0'10 21-8 0"12 26.3
-47-9 -41-9 - 14.2 -3.8 1"1 6"9
- 16.7 - 14-2 - 1.7 7.0 8"7 11.1
- 21.7 - 15~9 -2.0 7"4 20"7 25"0
- 18"4 -31.5 -49"1 -57-1 -83.3 -93-7
0.15-
0 <
0.10-
n,
0.0§-
SIXTH FLOOR 0.00 0.00
.
.
;
0,03
•
.
.
;
•
.
,
~
•
.
.
1.12
0.06 0.09 FRICTION COEF.
(b) 0.80
I
I
.
I
.
.
£, 0 Z
97
0.60-
BASE
0.40O
01 0.20n,
0.00 0.00
"
"
;
0.03
"
"
'
I
"
"
"
;
0.06 0.09 FRICTION COEF.
"
"
L 12
(c) Fig. 2. Stationary RMS response versus friction for (a) secondary systems, (b) sixth floor and (c) base (-- VP model, - - CS ~ model, zxA Monte-Carlo simulation). All three secondary systems have a mass ratio M~/Mp = 0.5% and a viscous damping ratio ~ = 2.0%, with S1 being tuned to ms = 3.88 Hz (the second fundamental frequency o f the frame under fixed-base conditions) and the remaining ones detuned with ~o~equal to 8-0 Hz and 16.0Hz for $2 and $3, respectively. In general, Fig. 2 shows that the VP model, with its ability to simulate stick-slip episodes, is more appropriate than the CS model for higher base isolator friction. It should be noted, however, that stochastic linearization on which
both VP and CS models are based is approximate and, as a result, even the more realistic VP model exhibits errors, as summarized in Table 3. As discussed in Ref. 25, the term E[2bz] contained in the expressions for the equivalent coefficients of eqns (7) and (9) is responsible for introducing errors when z(t) approaches zero. Figure 3 compares the nonstationary RMS response of the same components as before for the velocitydependent, intermediate friction (#max = 11"9%) sliding bearing component o f the base isolator given in Table 2. The VP and CS models give nearly the same RMS accelerations for the detuned secondary systems. The CS model predicts somewhat better results for the tuned secondary systems and its supporting floor and for times greater than four seconds. Finally, the VP and CS models, respectively, act as upper and lower bounds for the R M S displacements o f the base isolator. This model behavior was also observed in Ref. 25 in conjunction with the standard deviations o f the motions o f the base isolator. Next, Figs 4-7 examine the influence o f various base isolator and secondary system parameters on the RMS acceleration spectra of a typical secondary system, originally placed on the top floor, and with the same mass and damping ratios as before. The RMS acceleration spectra are constructed from 50 values of the secondary system natural frequency cos in the range 0.2-25 Hz. In all cases, the VP base isolator model is used. In particular, Fig. 4(a) compares the RMS accelerations o f the secondary systems for all three cases (low, intermediate and high) o f friction exhibited by the sliding bearing component o f the base isolator, as given in Table 2. The fixed-base (FB) case (#m,× ~ O0) serves as the reference point. It is observed that as the friction coefficient drops, the RMS accelerations are reduced and the spectrum progressively flattens. Two basic mechanisms are at work here: 33 (a) the slider limits the intensity of shear force transmitted from the base to the super-
G. Juhn, G.Do Menol~s, MoCo Constantinou
98
.!
7 ~. o.~o-+
/-./
&80-
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F~go 3. Nonstationary RMS response for velocity-dependent intermediate friction (/~,~ = 0.1 t 9) for (a) secondary systems, (b) sixth floor and (c) base (-- VP model, - - CS model, e - , - e Monte-Carlo simulation). structure; and (b) as friction is lowered, nonlinearities in the base isolator predominate and tuning between primary structure and secondary system is progressively destroyed. Figure 4(b), which plots the secondary system RMS accelerations normalized with respect to the fixedbase case, clearly shows the high degree of acceleration reduction offered by the low friction slider at all frequeno cies except the very low (below 0.5 Hz) range. It should be noted that although low friction in the sliding bearings
is very desirable for secondary system protection, it invariably ieads to large superstructure disptacemen:~c. 3j This means that the intermediate friction slider, wHch offers passive protection o~er the entire frequene',~ rang% may be the best choice. Next, Fig. 5 compares the secondary system RMS accelerations as obtained using the ve_Tocity-dependes.L friction model, which is appropriate for the s:iding bearings with Teflon interfaces that were used m the experimental studieso 7:s33 and the constam fr.',ction mode1, which is c o m m o n l y employed in numerica~ ssmulations of sliding base isolators. Both intermediate friction (#~,~ = 0.119} and iow friction ( # ~ = 0-0572"~ designs are examined Significant differerzces arc observed between the variable and constant friction models past a frequency o$ 0-7Hzo Ciear~y, veioc~tv.dependent sliders are more efficient m protecting the secondary system tha~ sliders that exhibit constant friction for the ewo levels of friction considered. This is especially true in the critical regions of resonance (c<< - o%~, co~ = c~m) where m a x i m u m reductions of the order of 40% and 30% are observed for the intermediate and tow variable friction cases, respectively. F~rthermore, as friction is !owered. the R M S acceleration spectrum flattens more quickly in. the vdocity-dependent
Stochastic response of secondary systems in base-isolated structures . . . . . . . .
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Fig. 5. RMS acceleration spectra for secondary system on 6th floor for (a) intermediate (#m~x = 0'119) friction and (b) low (#m~x = 0"0572) friction. (Variable versus constant (Coulomb friction.)
Fig. 6. RMS acceleration spectra for secondary system on 6th floor for variable low friction (#m.x = 0"0572) versus (a) base frequency COband (b) mass ratio M~/Mp.
slider, indicating a more desirable design. The reason that velocity-dependent sliders perform better than the constant friction ones basically has to do with the fact that repeated slip-stick cycles in the latter introduce frequency components in the superstructure motions that correspond to the base isolator's own natural frequencies. This phenomenon is ameliorated when the slider is capable of producing a smooth transition between the slipstick cycles, as is the case with velocity-dependent friction. Subsequently, Figs 6(a) and (b), respectively, illustrate the effect of base isolator frequency cob and of mass ratio Ms/Mp on the secondary system response for the velocitydependent low friction slider ~m,x = 0-0572), which has succeeded in keeping the RMS accelerations at below 0.40g. In particular, secondary systems are rather insensitive to low values of stiffness of the base isolator Kb = coZMT. However, as the stiffness increases, low frequency (0-3-1-0 Hz) secondary systems experience an almost two-fold increase in their response. This can be explained as follows: at low values of friction and for stiff restoring force springs, the latter component predominates. As a result, the base isolator behaves almost linearly and peaks corresponding to the natural fie-
quencies of the combined structural system appear in the RMS acceleration spectra. It should be noted that since the presence of springs in the base isolation mechanism basically acts as a restoring mechanism for limiting large residual base displacements, 33 there is no reason for choosing a stiff base isolator design unless one wants to minimize these residual displacements for strong input motions. As far as primary structure-secondary system interaction is concerned, Fig. 6(b) shows that it is negligible as long as the mass ratio stays below 0.01. Also, heavy secondary systems (Ms/M p = 0.1) experience less motion (0.20g or below) than their lighter counterparts in the mid-frequency range, indicating significant interaction between the secondary system and its supporting floor. Finally, Figs 7(a) and (b) contrast the effect of location, i.e. 6th versus 2nd floor, for the velocitydependent, low friction slider and for the fixed base situation. The only conclusion which can be drawn here is that high frequency secondary systems remain relatively unaffected as their location within the superstructure changes, irrespective of the presence or absence of base isolation. It is noted finally that the presence of base isolation still reduces the overall maximum secondary
G. Juhn, G.D. Manotis, M.C. Constaneinou
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The authors wish ~o acknowledge the suppor~ o~ the National Center for Earthquake Engineenng Research under grants N C E E R 88-2006 and 89-2003. The authors also wish to thank Mary Anne Lang for ~yping the manuscript and an anonymous reviewer for his va~.uab!e comments.
1
REFERENCES t.00-
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ACKNOWLEDGEMENTS
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limited to as much as one-fifth of those exper~encec under fixed-base conditionso The aforementioned deazgn is also insensitive to variatio,~s in the raass of the secormary system and produces the same maximum response, irrespecwve of the floor on which the secondary sys~;em zs !oeatedo
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(b) Fig. 7. RMS acceleration spectra for secondary system at two different locations under (a) base-isolation with variable iow friction (#~,~ 0.0572) and (b) fixed-base conditions.
system RMS accelerations in the mid-frequency range down to 0.408 from the corresponding value of 1-75g observed under fixed base conditions.
CONCLUSIONS in this work, spectra were developed for the R M S acceierations o f a secondary system attached to a six-story frame that was base-isolated using sliding bearings and restoring force springs. The ground motions were represented by a doubly filtered, white noise power spectrum. The present study demonstrated that the protection offered to secondary systems by the Teflon-coated sliding bearings that produce a velocity-dependent friction coefficient is superior to that offered by sliders that exhibit constant friction, and in the frequency range that matters the most, i.e. the one encompassing the first few natural frequencies of the frame under fixed-base conditions. Atso, it was shown that a good design is one that employs an intermediate to low friction coefficient for the Tefloncoated sliding bearings and a soft restoring spring. This combination produces a secondary system spectrum that is relatively flat, with the maximum RMS accelerations
L Chert, Y.Q. & Soong, T.T., State-of-theoart review: Seisrmc response of secondary systems~ Engng StrucL~ ~ (I988) 218-28. 2. Igusa, T. & Der KiuregbAan, A. Dynamac response of muttiply supported secondary systems. A Engng M'eC~o (ASCE), !111(1) (t985) 20-4t 3. Suarez, L.E. & Singh, M.P., Floor response spectra ~,~-t}: structure-equipment interaction effects by a mode synthesis approach. Earthquake Engng StrucL Dyn., 15 (!987) i4i-58 4. Suarez, L.E. & Singh, M.Po, Eigenproperties of ~o~classically damped primary structure and eqmpmen*: systems by a perturbation approach. Earthquake Engng Struct. Dyno, 15 (1987) 565-83 5. Kelly. 7.M. & Chitty, D.E., Controi of seismdc response of piping systems and components in power pla~-ts by base isolation. Engng Struct., 2 (i980) 187-98. 6. Kelty, J.M., The influence of base isolation on the seismic response of light secondary equipment Report No. UCB/ EERC 81/ !7, Earthquake Engineering Research Center. University of California, Berkdey, California, I982 7o Manotis, G.D., Juhn, G., Constandnou, M.C. & Rdneo>q, A.M., Secondary systems in base-isolated structures: Experimental investigation, stochastic response and s~:o chastic sensitivity. NCEER Rept. No. 90=0013. Nat. Cen~e_~ for Earthquake Engng Res., Buffalo, New York, !990. 8. Kd!y, J.M., Aseismic base isoNtion: Review and b~bho graphy. Soil Dyn. Earthquake Engng, 5(3) (!986) 202-f& 9. Buckle, I.G., New Zealand seismic base isolation con~pts and their application to n~clear engineering. A%~.e;ear Engng and Design, 84 (1985) 3!3-26. 10. Kelly, LM. & Hodder. S.B, Experimental study of tend and elastomeric campers for base isolation systems in ~amihated neoprene bearings. Null of _grewZealand N~eto Soc Earthquake Engng, 15(2) (!982) 53-67. 11. Caspe, M.S.. The earthquake barrier. In Dynamic *~,es'ae~se of Structures, Proc. 3rd Conf. ,)f Engng ~gCech. Div (ASCEj, ed. Hart, G.C. & Ne}son, R.B. ASCE Pubhcation, New York, 1986. pp. 467-72. 12. [konomou, A.S., Afexisismon isolation engineering ~o~ nuclear power plants. Nuclear Engng and Desig& 85 (t985) 201-16. i3~ MosmgheL N. & Khodaverdian, Mo, Dynamics of resiliera> friction base isoiator (R-FBI). Earthquake Engng Struck. Dym.o ~5 (1987) 379-90
101
Stochastic response of secondary systems in base-isolated structures
14. Constantinou, M.C., Mokha, A.S. & Reinhorn, A.M., Teflon bearings in base isolation. II: Modeling. J. Struct. Engng (ASCE), 116(2) (1990) 455-74. 15. Mokha, A., Constantinou, M.C. & Reinhorn, A.M., Teflon bearings in base isolation. I: Testing. J. Struct. Engng (ASCE), 116(2) (1990) 438-54. 16. Mokha, A., Constantinou, M.C., Reinhorn, A.M. & Zayas, V.A., Experimental study of the friction pendulum isolation system. J. Struct. Engng (ASCE) (submitted). 17. Su, L., Ahmadi, G. & Tadjbakhsh, I.G., Comparative study of base isolation systems. J. Engng. Mech. (ASCE), 115(9) (1989) 1976-92. 18. Tsai, H.C. & Kelly, J.M., Non-classical damping in dynamic analysis of base-isolated structures with internal equipment. Earthquake Engng Struct. Dyn., 16 (1988) 29-43. 19. Tsai, H.C. & Kelly, J.M., Seismic response of the superstructure and attached equipment in a base isolated building. Earthquake Engng Struct. Dyn., 18 (1989) 551-64. 20. Chen, Y.Q. & Soong, T.T., Seismic behavior and response sensitivity of secondary structural systems. NCEER Rept. No. 89-0030, Nat. Center for Earthquake Engng Research, Buffalo, New York, 1989. 21. Singh, M.P. & Wen, Y.K., Nonstationary seismic response of light equipment. J. Engng. Mech. Div. (ASCE), 103(EM6) (1977) 1035-48. 22. Ahmadi, G., Stochastic earthquake response of structures on sliding foundation. Int. J. Engng Sci., 121 (1983) 93-102. 23. Constantinou, M.C. & Tadjbakhsh, I.G., The optimum design of a base isolation system with frictional dements. Earthquake Engng Struct. Dyn., 12 (1984) 203-14. 24. Constantinou, M.C. & Tadjbakhsh, I.G., Hysteretic dampers in base isolation: Random approach. J. Struct. Engng (ASCE) 111(4) (1985) 705-21. 25. Constantinou, M.C. & Papageorgiou, A.S., Stochastic
26.
27. 28. 29.
30. 31. 32. 33. 34. 35.
response of practical sliding isolation systems. Prob. Engng Mech., 5(1) (1990) 27-34. Lin, B.C., Tadjbakhsh, I.G., Papageorgiou, A.S. & Ahmadi, G., Response of base-isolated buildings to random excitations described by the Clough-Penzien spectral model. Earthquake Engng Struct. Dyn., 18 (1989) 49-62. Lin, B.C., Tadjbakhsh, I.G., Papageorgiou, A.S. & Ahmadi, G., Performance of earthquake isolation systems. J. Engng Mech. (ASCE), 116(2) (1990) 446-61. Igusa, T., Response characteristics of inelastic 2-DOF primary-secondary system. J. Engng Mech. (ASCE), 116(5) (1990) 1160-74. Pires, J.A., Analysis of nonlinear primary-secondary systems under random seismic loading. Proceedings of Fourth U.S. National Conference on Earthquake Engineering, May 20-24, 1990, Palm Springs, California. 1990, Vol. 3, pp. 51-60. Iwan, W.D. & Paparizos, L.G., The stochastic response of strongly yielding systems. Prob. Engng Mech., 3(2) (1988) 75-82. Suarez, L.E. & Singh, M.P., Seismic response of SDF equipment-structure system. J. Engng Mech. (ASCE), 113(1) (1987) 16-30. Wen, Y.K., Equivalent linearization for hysteretic systems under random excitation. J. Appl. Mech., 47(1) (1980) 150-4. Constantinou, M.C., Mokha, A.S. & Reinhorn, A.M., Study of sliding bearing and helical-steel-spring isolation system. J. Struet. Engng (ASCE), 117(4) (1991) 1257-75. Fang, T. & Wang, Z.N., Complex modal analysis of random vibrations. AIAA Journal, 24(2) (1986) 342--4. Shinozuka, M. & Sato, Y., Simulation of nonstationary random processes. J. Engng Mech. Div. (ASCE), 93(EM1) (1967) 11-40.
APPENDIX 1
Equations (1) and (2) can be combined to read
M, {Mp} k{Ms}
+
[Ms]
[o1
{o} {,
{O}
[
Kb
{M.} • {<}
{M,} )
{O}" 3 {O} [Kp] + Z Kq{ey}{ej} T j=l
{o }
-
{ej }f
{O}T -
-
[Ksj{ej}]
t xb}
[K I
l(xs}
+ &)
{x.}
(Al.1)
{Ms}
In the above, the DOF of the primary structure are pl-p6, while those of three secondary systems attached to the sixth floor are sl-s3. All symbols in eqn (AI.1) have been previously explained with the exception of the connectivities {ei}T = [0, 0, 0, 0, 0, 1]. It should be noted that since the damping terms behave in the same way as the stiffness terms, they have been omitted from eqn (AI.1) in the interest of brevity. Equation (AI.1) is
supplemented by eqn (11) to form an 11 × 11 system for the viscoplastic model. For the constant sliding model, variable z and coefficient/4 are ignored.
APPENDIX 2
By following the modal transformation described in
G. Juhn, G.D. ManNis, M.C. Consganginou
102
eqn (3), the expanded form of eqn (4) is 1
[(~s]T {Ms } (]~b
[t1
[o1
i
[o1
[z]
J
{o}~
{o}~
qb)
3
+ ]{o} [%1~ + Z <,[4,S{ej} {~,}'[+d ]=1
L{O}
- [ ~ ] T [K v{ ej}IT[~J
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f ~bMT#~g
+ ,{ {o}
(
{o}
z :
-
[~.]~{M.}}x~
;A2o i,
E
In the above, cob, [cop]and [cos]are the natural frequencies of the base ~solator, of the fixed-base frame and of the individual fixed-base secondary systems, respectively~ Furthermore, [I] is the identity matrix. Despite the fact that {F*} contains the actual model participation factors of the individual structural subsystems, matrices [M*I and [K*] are not diagonal so there is coupling of the
generalized co-ordinate equations° Assurmng ma~ the fixed-base frame is proportionally damped~ the~ eqn (A2.1) can be augmented by the t e , ~ ~[C*] {@ where [C*] has the same form as [K*] with cob, [a)~], ie~] a~d K~ replaced by 2~bco b ~-Ce~ [2~pcop], [2~s~s] a~d ;si~ respectivelyo
Stochastic response of secondary systems in base-isolated structures G. Juhn, G.D. Manolis & M.C. Constantinou Department of Civil Engineering, State University of New York, Buffalo, New York 14260, USA
(Received September 1990; accepted September 1991) In this work, the stochastic response of secondary systems attached to a baseisolated structure undergoing random ground motions is examined. It is assumed that the properties of this combined structural system are deterministic, while the ground motions are described by a filtered white noise model. The only nonlinear component of this structural system is its base isolation mechanism, which is linearized by using equivalent linearization. Also, a substructuring algorithm is developed which requires the dynamic properties of the individual, fixed-base components of the structural system. Both stationary as well as nonstationary cases are considered and comparisons are made with the results of Monte-Carlo simulations to ascertain the validity of this methodology. The example studied herein is a six-storey steel building frame with a base isolation system consisting of sliding bearings and restoring force springs. For this example, spectra are constructed that account for primary-secondary system interaction and depict the effect of variations in the base isolator's structural parameters and in the mass and location of the secondary system on the latter's root-mean-square (RMS) accelerations.
Only a limited number of experiments have been performed on shaking tables because of the high costs associated with such work. In this respect, the pioneering work of Kelly and his associates 5'6 with scaled steel frames containing secondary systems and under fixed base or base-isolated conditions must be mentioned. Some full-scale tests have also been conducted for nuclear power plant equipment and electrical equipment. Finally, experiments involving secondary systems alone have also been conducted using pseudodynamic testing. A detailed review of experimental methods for secondary systems can be found in Manolis et al. 7 Understanding the dynamic behavior of secondary systems is the first step in devising ways to protect them. This is very important in view of the fact that most secondary systems perform critical functions. A novel way that has recently emerged is base isolation, which is an aseismic design strategy with its objective being reduction of the transmission of motions from the ground to the structure. This is achieved by lowering the fundamental frequency of the structure to values that are below the dominant frequencies of the ground excitations and/or by dissipation of energy through inelastic action. A properly designed base isolation system protects not just the superstructure, but also its contents. A comprehensive review of base isolation systems can be found in Kelly. 8 A cursory classification of base
INTRODUCTION The behavior of secondary systems and critical equipment in conventional structures and in industrial facilities was reviewed by Chen & Soogn. 1 The two basic approaches for analyzing structures containing secondary systems are the floor response spectrum approach and the combined system approach. The former method, in its original form, ignores primary-secondary system interaction and consequently gives results that are valid under certain conditions only. The combined system approach, which includes perturbation methods, complex modal analyses, substructing concepts, etc., is more general but requires lengthier computations. What makes this problem difficult to analyze, besides its topological complexity, is the mismatch between the mass of the primary structure and that of the secondary systems, the existence of non-classical damping, and the possibility of inelastic deformations under intense ground shaking. Recent advances 2-4 have addressed the aforementioned problems, with attempts to include interaction effects in the floor response spectra as well as to improve numerical techniques for combined system analyses. Probabilistic Engineering Mechanics 0266-8920/92/$05.00
© 1992 Elsevier Science Publishers Ltd. 91
92
G. Yuhn, G.D. Mano[is, M.Co Constantinou
isolators includes rubber beatings with a lead core, 9 laminated rubber bearings, ~° the earthquake barrier system] ~ and various systems based on friction such as the alexisismon system, ~2 the resilient-friction base isolator, ~ Teflon sliding bearings, ~<~s and the friction pendulum system] 6 The basic components of a base isolator are: (a) A restoring element, such as a spring or a rubber bearing, that provides centering force capabilities. The restoring element may exhibit hysteretic material behavior resulting in energy absorption capabilities. (b) A sliding element, such as a Teflon bearing, that alters the transmision of motions across its ends. Sliding elements may exhibit either a constant (Coulomb) or a velocity-dependent friction coefficient. (c) Additional energy absorption capabilities provided by viscous dampers, yielding beams or similar devices. Base isolation has the advantage that all nonlinearities exhibited by the combined structural system are concentrated at the base level. A comparative study of six representative base isolation systems with respect to the 1940 El Centre and the t985 Mexico City earthquakes appears in Suet d. ~7Other recent analytical work includes examining the distortion in the high frequency content of the secondary system response caused by assuming ctassicaI instead of complex modes for the base-isolated structure, TM identifying the primary mode of the baseisoIated structure as controlling the response of high frequency secondary systems, ~ and studying the effect of yielding in a fixed base structure as a means of protecting the secondary systems9 All the work previously mentioned has been for a deterministic input. Among the earlier work to ~ook at the case of stochastic input is that of Singh & Wen, 2'~who employed the Markov vector approach to examine the nonstationary seismic response of a two degree-of-freedom (DOF) primary-secondary structure model. As far as base isolated structures are concerned, Ahmadi 2~- coe.sidered the response of a sliding structure to stationary as well as filtered white noise. Constantinou & Tadjbakhsh23 also used stochastic concepts in studying the optimum design of a friction-type base isolator supporting shear buildings. ~In later work, Constantinou & Tadjbakhsh ~4 employed equivalent linearization in conjunction with damper-type isolators whose hysteretic behavior was modeled by a nonlinear differential equation. This equiw alent tinearization technique was subsequently extended to cover friction-type elements, such as sliding bearings. ~ Also recently, L i n e t M.26"2v considered the stationary respor~se of base-isolated buildings subject to randorn earthquake excitations whose spectral content is modelled by a Kanai-Tajimi and a high-pass fiker. For the idealized case of a single DOF secondary system
attached to a single DOF primary str~ct~re where t"~eti~ components exhibit hysteretic ~.onlineari:ies~ igusa 2~ developed analytical expressior~s asiv~g the perturbation method in conjunction with equivalent 12nearizatien tha!-: showed a significant reduction, of the system response when it was subjected to a wide-band input. For the same problem, the n°americat results of Pires29 indicate a s:~5o. stantiai reduction of the secondary system accetera~oss in the vicinity of resonance due to energy dissipation, b ~ the forces transmitted to the secondary system increased ir~ the presence of inelasticity a~ higher freqaer~cies~ Finally, ~wan & Paparizos 3° studied the response e~ a nonlinear single DOF system ~.n temas of ~ts sewer spectral density fianction and concluded that equivalent ~i~aearization works well %r a system where friction: is supplemented by. a rest©ring spring. The present work focuses on the dynam~.c response o! secondary systems in a base-isolated st,~cmre azbjectec to random ground motions. ~n particular, ~he eq"oAvale~t lineariza~iola procedure for a base isolation system ses~. sisting of a sliding bearing component ~hat exhibits velocity-dependent friction plus a series of res~o£ng helical springs is discussed, aIor~g with modeiing aspec-~s concerning the combined structural system and the inp~a~ motions. The response statistics of 'the combined s~ruc~ tare augmented by the stochastic ground motion modei are obtained by solving a Liapunov matrix equatiOrio These results~ along with Monte-Carlo simuIations, for both stationary and nonstationary cases~ are ~he~ presented for a six-story s~cructura] system Fiu~all?~ a series of parametric studies are performed ;o ascer~xdn the effect of changes in the base isolator ann secondary system structural parameters on ~he RMS accderations of secondary systems attached to the structarai system and to estaNish the degree of passive pro~ection e~?re4 to the secondary systems by the base isolator
NUMERICAL MODELING ConnNeed sg~uct~are The combined structure includes the primary str~ctare.~ the secondary systems, and the base isolation mechanism° We focus on a mukioDOF frame with a number ef single D©F secondary systems attached to rt at va~cus locations, plus a single D©F base isolator. This discrete parameter, combined structure is shown in Fig 1 a~adits equations of motion are given beIow as
where
{x}
}
-
{xj
i
Stochastic response of secondary systems in base-isolatedstructures
former model requires the additional variable z(t) for its proper description that will also be subsequently discussed. As usual, in eqns (1) and (2), dots indicate time derivatives, g is the acceleration of gravity, superscript T denotes transposition while summation E extends over all superstructure DOF. It is obvious that a lumped mass representation is assumed, with MT being the total mass of the system. Finally, Appendix I gives explicit expressions for the case of a six-story frame containing three secondary systems attached to the top floor. Introduction of modal transformation at this point is advantageous because it requires the dynamic properties of the individual components rather than those of the combined structure. This advantage is important since most experimental identification procedures for structures are done at the component level. 7 In particular, the generalized co-ordinates {q} of the combined structure are given in terms of the physical co-ordinates {x} as 31
Z,Z f,¢ t",.
XXZ J,,
93
48 "
{X}: {Xb {Xp}I : [~b [•p] ]tqb t = [~b]{q}
'l
(a)
{qp}
{xs} )
[¢j J {, {q~} J
(3)
FRAME
'1
SLIDING BEARINGS 5HAK ING TABLE
(b)
Fig. 1. (a) Six-story frame with attached secondary systems and (b) base isolation mechanism.
MT~b + Cbkb + KbXb + #oMTgZ
Cb = (2~bOgb + ce)MT and
Kb
[M*]{~} + [C*]{~t} + [K*]{q} + {e*} =
-{F*}5~g
(4)
Expressions for the above system matrices, as well as for the modal participation factors {F*} and vector {e*} can be found in Appendix 2 for the six-story structure example.
for the superstructure and
= --MT~g -- E M)~, i where
where ~bb = (MT) -1/2, [q~p] are the eigenvectors of the fixed-base primary structure normalized with respect to the mass matrix [Mp], and [q~s] is a diagonal matrix containing the secondary system mass terms (Ms1)-1/2, (MJ-1/2, and (M~3)-1/2. By substituting eqn (3) in eqns (1) and (2) and pre-multiplying by [~b]T, the following modal equation is obtained
= oMT} (2)
for the base isolator. In the above, all displacements x are relative to the base except for xb that is relative to the ground, and subscripts p, s, b and g stand for primary structure, secondary system, base isolator and ground, respectively. Furthermore, M, C, and K stand for mass, damping and stiffness, respectively. Square brackets denote matrices, curly brackets denote vectors and absence of this symbolism denotes scalars. Also, Ogband {b respectively are the natural frequency and damping ratio, while ce and #o are damping and friction coefficients resulting from equivalent linearization of the base isolator that will be subsequently discussed. We will distinguish between a viscoplastic (VP) model and a constant sliding (CS) model for the base isolator. The
Base isolation The base isolation system consists of Teflon sliding bearings with restoring force capability provided by linear springs. The equation of motion of the base isolator, once it has been placed inside a multi-story structure, is eqn (2). The following two models have been proposed 25 for base isolation systems of the frictional type: (a) a model based on the theory of viscoplasticity which is capable of accounting for stick-slip conditions, and (b) a model based on the assumption that sliding is continuous across the Teflon sliding bearing interfaces. It should be noted that both models are capable of accounting for a velocity-dependent friction coefficient #(-tb), a fact which has been experimentally observed for Teflon sliding bearings. 15 In particular, the following expression has been found 14to best describe the friction
94
G. guhn, G.D. Mano[is, M . C . Constantinou
coefficient: #(2b) =
by
# ~ , - Dr exp ( - ~12~1)
(5)
tn the above, #~,~ is the coefficient of friction at high velocity, Dr is the difference between # ~ and the coefficient of friction at very tow velocity, and ~ is a constant dependent on the bearing pressure and on the conditions that exist at the sliding surfaces. Coulomb friction, where # is a constant, is recovered as a special case by setting Dr to zero in eqn (5). in the viscoplastic (VP) model case, the dimensionless variable z(t) o f e q n (2), whose range is [ - 1, 1], obeys the following constitutive equation proposed by Wen ~ for modeling hysteretic behavior =
- - ( V l , b l Z M ~-~ + ~ & t z ! ° -
A&)/y
(6)
where I3, ?, and A are dimensionless constants characteizing the shape of the sliding isolator's hysteresis toop, y is the yield displacement of Teflon and n controls the mode of transition of the model into the inelastic range. Constantinou e t a [ . ~4 found that eqn (6) satisfactorily reproduces the behavior of Teflon sliding bearings provided that A = t and 13 + 7 = 1. For this combb nation of parameters, the model described by eqns (5) and (6) collapses into a viscoplastic model. Finally, plots of the variation of the friction coefficient with vdocity and of the loops produced by friction force versus sliding displacement, obtaining from experiments as well as from numerical simulations, can be found in Refs ]4, 15 and 25. An important property of Teflon sliding bearings is reducton of the friction force mobilized across the sliding interfaces as the sliding velocity decreases. This means that tess force is needed to maintain relative motion as the superstructure's sfiding slows down. As a resu!L relative motion continues with almost complete absence of stick-slip episodes, a fact that has been confirmed in shake table tests of model structureSo 3~ Accordingty, the continuous sliding (CS) model for friction is well justified and can be recovered from the VP model by simply replacing z(t) in eqn (2) by sgn (kb), where sgn is the sign function. The CS model has Been found 2s to produce good results when compared with Monte-Carlo simu!ations, except for sliding bearings which exhibit high friction. Furthermore, both VP and CS models should be examined because they provide upper and lower bounds to the simulation results for the base isolator responseY As can be seen by reference to eqn (2), the method of stochastic equivalent linearizatiort 32 has been used in order to obtain approximate statistics for the response of the base isolation system and consequently of the combined structural system. The equivalent damping (G) and friction (be~) coefficients appearing in eqn (2) are obtained by minimizing the mean square difference between the original nonlinear equation of motion of the base isolator with the restoring force due to friction given
for the VP model or by f(kb)
-- ff(3~b)Jl~ygsgn txw
•* ic for the CS model and the linearized equanor. ~sem. :~e. eqn (2)° The resulting expressions for the VP model are gben by
c% =
E[f(J%)kb]/E{22] "
and
~
{9)
1 - h[f,Xb)Zl/E[z-k i g
#e
with E being the expectauon opera,or. By assuming ~ha; 2b and z of the linearized system are jointly Gaussian. zero mean processes, eqns (9) resuk in /
x exp (cdE[22]/2
xi/2
o er~c@E[xb!' /~, 2 ) / ~ ! z ~
>
and #~" =
Vm~. -- Dr exp @2E[2~]/2) erl% G[~'Ex~]:"a,,, Z;~, /
where erfc is the complementary error eqn (6) is Hnearized as
funcuon
C~2b - Kez
=-
&]so,,
~~~'~
where the equivalent damping and stiffness coe~iciems C~ and Ke, respectively, are given by the ~%Jlowing expressions for n = 3: (3~2[z j E l x b z j E t x b j
-
=
I
-
E[&z]~/s[mg]~/:)+ 29 k ~ )
<
and
;
(ECz2]E[k2U2
~ " "~--kXb: ~
'd
4- 6fl
E[kbz]g[z2] L'; j (i~2!
The CS model can be recovered from me above o } % disregarding eqn (1 i) and se~t~ '~ n g #o - 0 plus re~iaci~.g .. by
x
2 ~..g exp(0-5~ 2 or,) ~nc(~%)~!
/
:~
'in eqn (2), where £, is the standard deviatior~ of c~
13'
95
Stochastic response of secondary systems in base-isolated structures
recast as a first-order matrix differential equation in the form
Ground motions
In order to represent strong ground motions as a stochastic process, the usual Kanai-Tajimi (K-T) filter is supplemented with a high-pass (H-P) filter that removes singularities at zero frequency. This approach has been recently used in Lin et al. 26 and results in the following form for the power spectral density function of the
{I~} = [A]{Y} + {e}qJ(t)f(t)
(16)
In the above
{y}T = [{0}T, {q}V, z, 2g, Xg, 2f, xr] {ey = [{O} T, {O} T, 0, 0, 0, - 1, 0] and
[A] =
-- [M*]- 1[C@],
- [M*I-'[K*],
- [ M * ] - ' {e*},
[M*]-' {F*} (2{rcor,
(or,
-- 2~g cog,
-CO~)
[I]
[o1
{o}
0
0
0
0
{--Ce(/)b, {0} T}
{oF
-Ko
0
0
0
0
{o} T
{O} T
0
{O} T
o 0
1
0
0
0
(off
{oy {o} T
0
0
-- 2~fcof
-CO~
{O} T
{o} T
o
0
0
1
0
ground accelerations
&(co) = I1 - (±~212 + 4¢2g(~gg) co2
2~fOgf
-CO~ 2~%
[S] = [A][S] + [S][A]T + [B]
\cog/J
.so \~fl J (14) In the above, So is the power spectrum of white noise. What eqn (14) implies is that white noise is being filtered through the K-T filter (subscript g) and the H-P filter (subscript f) as 2~gcog2r + (D~Xf)
and
-
co2g
(17)
The covariance matrix of the response {Y}, Sij = E[Y~Yj], can be obtained by solving the following Liapunov differential equation of order N x N (N = 25 for our example)
1 + <~ ~
2 = 2~g "{- 2~gcog2g + (.OgXg
-
I
(15)
2r + 2(rcor2f + co~xf = ~ ( t ) f ( t ) respectively, where f(t) is a white noise process with intensity 2rcS0 and ~(t) is a deterministic time envelope function. Values chosen for the above parameters are So = 71"0cm2/s 3, COg = 3"47Hz, ~g = 0"59, COr= 0.50Hz, and ~r = 1.0, which correspond to a magnitude six ground motion o n r o c k . 26
where [B] is a matrix with all elements equal to zero except BN-~,N-~ = 2rcSoW2(t). Since [S] is symmetric, eqn (18) can be written as a system of N ( N + 1)/2 firstorder nonlinear differential equations that can be solved using standard techniques such as the Runge-Kutta or Adams-Moulton methods. The stationary case ([S] ~--0, BN_I, N 1 = 2zcS0) results in a system of N ( N + 1)/2 nonlinear algebraic equations that can also be solved using standard techniques such as Newton's iteration. All these techniques, however, require prior vectorization of [S]. There are other avenues open for solving eqn (18). It is possible, for instance, to work in the frequency domain as done by Lin et al. 26 The approach which was adopted here was to use complex modal analysis34 so as to diagonalize the covariance matrix. This first requires finding the eigenvalues 2i, the right eigenvectors [@R], and the left eigenvectors [q)e] of eqn (16). By introducing generalized co-ordinates {Q} and using the complex modal transformation { Y} = [@"] {Q}, eqn (16) can be decoupled to read as {{)} - [2]{Q} =
NUMERICAL SOLUTION The system of linearized differential equations (4) is augmented by the ground motion equations (15) and
(18)
{F}~(t)f(t)
(19)
where {F} = [@L],{e}. Consider now stationary white noise excitation (~(t) = 1.0). The stationary solution of eqn (19) is {Q(t)} = fo [H(u)] { r } f ( t - u)du
(20)
G. Yuhn, G.D: Manolis, M.Co Constantinou
96
Table L Prh~ary structure proFerties under fixed-base conN~io1~s
Mode
!
% (Hz)
2
1o17 3.88 5-0 5.0 0"068 0-I 70 0.138 0-301 0"200 0-271 0"25t 0"098 0-289 - O-t27 0-314 -- 0-301 AlI diagonal elements equal to 56-6 x
~ (%)
[Mp] (kg)
3
4
5
o5
6.64 5-0 - 0-256 - 0-232 0:077 0-3i I O"t35 - 0.264 t03
9-52 5.0 - 0-357 0-037 0°305 - 0-102 - 0°230 0 o166
] 2°40 5.0 0°237 ,0.273 0"073 0"227 -- 0-32~ O-136
?4°46 5.0 - 0 ~t6i 0~264 --0-310 0:280 - 0-198 0~065
where [H(u)] is a diagonal matrix whose ith component is hi(u) = e '~'~, with u > 0o Since {Q(t)} is a complex random response, the correlation function matrix of
{Q(0} is [Re(z)] :
E[({Q(t)} {Q*(t + r)}T] 2rcS0{F} {F*} ~ fo [Hiu)][H*(u + ~)]T du
:
(20 where * denotes complex conjugate. An element of
[RQ(~)] is t~CO
R~(Z) :
2zSoGi] J0 e;~e;¢(~+*)du
-
2rcs0a~je
V~ /()o, + )~)
(22)
with G~j being an element of matrix [G] = {F} {F*} v. The covariance matrix of the generalized co-ordinates {Q(t)} is s~
=
R~(o)
:
-2~soc,/(< + g )
(23)
Finally, the covariance matrix of the co-ordinates { Y(0} can be obtained as
s ] = [o ~1[sq [¢~*F
(~)
To revert to the physical co-ordinates [SX], eqn (3) needs to be used in conjunction with the above equation for the modal part {q} of response vector {Y}. It should be noted that matrix [A] contains coefficients Ce, K~, #~ and C~ from the tinearizafion process that are functions o f the response covariance and are thus unknown. This necessitates the use of an iteration scheme where: (a) initial values of these coefficients are assumed; (b) [S x] is cap culated based on those initial guesses; (c) the coefficients are computed via eqns (i0) and (12) using the results of the second step; and (d) the iterations are terminated when the values o f the coefficients converge. The same procedure is also applicable to the nonstationary response. The starting point is to write {Q(0}
:
~ [H(~ -- ~)] {F}~F(z)f(r)dz
(25)
in lieu of eqn (20). Then, the procedure for calculating the correlation function matrix IRe(t, z)] follows along the lines previously described.
NUMERICAL RESULTS The dynamic response o f the secondary sysmms attached to a base-isolated, six-story frame (see Fig. I) in ~he fo~:~ of RMS acceleration spectra is now examined. The numerical values assigned to the dynamic properties of the six-story frame under fixed-base condkions arm ~:o the base isolation mechanism are ~isted in TuNes ! and 2~ respectively° These properties were extrapolated from ar~ experimental program, 7n4~:sn3 where a base i~otation system using Teflon sliding bearings and restoring hekicat springs, originally detached from and subsequently attached to a quarter-scale six-story stee! frame contain. ing secondary systems, was tested on a shake m b b . The RMS response spectra were obtained by s o b i n g eqn (16~ via the complex mode approach previously discussed° The Monte-Carlo simuiations, 3s which are based on the so!ution of eqns (4) and (5), were obtained by generating 200 realizations of the input process and using ensemNe averages. Since an increase from 200 to 400 sampbs did not appreciably change the resuRs, the Monte-Carlo simulations are considered to be the exact solution. At firs% ~ Is necessary
Table 2. Base isolator #repert[es Nmax
(%)
Df
~
rd)b
~b
~
":
/{!
$2
(%~ (s/cm) (Hz) (%)
High f , iction case !7"0 10"0 3.22
0-5
0-0
0-t
0'9
!..0
3
0 25
Intermediate friction case 1.9 9-27 0"24 0"5
0~0
0,J
0-9
i-S
J
0.25
Low friction case 5.72 4 - 8 5 0'20
0"0
0"i
0~9
-g
b
0~25
0-5
Note: Total mass MT = 3774 x 103kg~
Stochastic response of secondary systems in base-&olated structures
Table 3. Error (as %) in the viscoplastic and constant sliding models versus base friction Pm.x
i .80 I
,-, O
d U
•
•
. .
I
.
..
I
" 'i'-
~'"
"t"
/~max
1.20-
0.80-
11" 0.40-
0.00 0.00
a,." /
•
.
.
;
S2
-
0.03
.
.
;
•
.
.
~
'
. S3
,
0.06 0.09 FRICTION COEF.
O, 12
(a) I
0.20
I
.
.
.
I
p ,-, 0
d
S1
$2
$3
6th Floor
Base
VP Model 0-02 -40.4 0'04 - 38.7 0.06 -26"8 0'08 -14"3 0"10 -5"4 0.12 3.5
-47"8 - 54.8 -39"3 -15"4 -6-7 -0"9
-0"6 - 21-4 - 13.9 -8.3 --7.2 --2.5
-4"3 - 22.7 -24-5 -16"7 - 10-3 -3-1
9-2 7"9 0"3 -19"0 -38-9 -53.1
CS Model 0"02 - 33.3 0.04 -3'2 0'06 9.7 0.08 18'4 0'10 21-8 0"12 26.3
-47-9 -41-9 - 14.2 -3.8 1"1 6"9
- 16.7 - 14-2 - 1.7 7.0 8"7 11.1
- 21.7 - 15~9 -2.0 7"4 20"7 25"0
- 18"4 -31.5 -49"1 -57-1 -83.3 -93-7
0.15-
0 <
0.10-
n,
0.0§-
SIXTH FLOOR 0.00 0.00
.
.
;
0,03
•
.
.
;
•
.
,
~
•
.
.
1.12
0.06 0.09 FRICTION COEF.
(b) 0.80
I
I
.
I
.
.
£, 0 Z
97
0.60-
BASE
0.40O
01 0.20n,
0.00 0.00
"
"
;
0.03
"
"
'
I
"
"
"
;
0.06 0.09 FRICTION COEF.
"
"
L 12
(c) Fig. 2. Stationary RMS response versus friction for (a) secondary systems, (b) sixth floor and (c) base (-- VP model, - - CS ~ model, zxA Monte-Carlo simulation). All three secondary systems have a mass ratio M~/Mp = 0.5% and a viscous damping ratio ~ = 2.0%, with S1 being tuned to ms = 3.88 Hz (the second fundamental frequency o f the frame under fixed-base conditions) and the remaining ones detuned with ~o~equal to 8-0 Hz and 16.0Hz for $2 and $3, respectively. In general, Fig. 2 shows that the VP model, with its ability to simulate stick-slip episodes, is more appropriate than the CS model for higher base isolator friction. It should be noted, however, that stochastic linearization on which
both VP and CS models are based is approximate and, as a result, even the more realistic VP model exhibits errors, as summarized in Table 3. As discussed in Ref. 25, the term E[2bz] contained in the expressions for the equivalent coefficients of eqns (7) and (9) is responsible for introducing errors when z(t) approaches zero. Figure 3 compares the nonstationary RMS response of the same components as before for the velocitydependent, intermediate friction (#max = 11"9%) sliding bearing component o f the base isolator given in Table 2. The VP and CS models give nearly the same RMS accelerations for the detuned secondary systems. The CS model predicts somewhat better results for the tuned secondary systems and its supporting floor and for times greater than four seconds. Finally, the VP and CS models, respectively, act as upper and lower bounds for the R M S displacements o f the base isolator. This model behavior was also observed in Ref. 25 in conjunction with the standard deviations o f the motions o f the base isolator. Next, Figs 4-7 examine the influence o f various base isolator and secondary system parameters on the RMS acceleration spectra of a typical secondary system, originally placed on the top floor, and with the same mass and damping ratios as before. The RMS acceleration spectra are constructed from 50 values of the secondary system natural frequency cos in the range 0.2-25 Hz. In all cases, the VP base isolator model is used. In particular, Fig. 4(a) compares the RMS accelerations o f the secondary systems for all three cases (low, intermediate and high) o f friction exhibited by the sliding bearing component o f the base isolator, as given in Table 2. The fixed-base (FB) case (#m,× ~ O0) serves as the reference point. It is observed that as the friction coefficient drops, the RMS accelerations are reduced and the spectrum progressively flattens. Two basic mechanisms are at work here: 33 (a) the slider limits the intensity of shear force transmitted from the base to the super-
G. Juhn, G.Do Menol~s, MoCo Constantinou
98
.!
7 ~. o.~o-+
/-./
&80-
°"..o.-~so . sz
~-
'
P
r
h
e
~.
.j
/
cJ
[
~
;,
'
ti
/i
'..
e~l
s'
{~O
.....
FB
........
~
=
9o-'7
O57
"°°
4 _L
U3
0 30-
....
c.~o4
)
o.oo4--
0 00 0.00
~~0.0
5.00 T~ME
k,--. %
~ , / ,
.
10 ~
~5.0
.
.
.
#
'17.,.At
.
.
10 =
20 0
-- . . . . C "
10
FREQ.
(Mz)
(SEC)
(a) 0~20-[
,
'
'
2-00 7 cg o
0 15 j-
~
F
1
.... ~'o =
l_
F-
,00-[
=
0
=
fA, 0 5 7
~]9
n
/
C
L
5 0.504 <
g.00--f--'
'
o.oo
~
-
'
s.oo
;
. . . .
~o.o TIME
;
'
~s.0
' --'--+-
O
.
.
(SEC)
.
L :
f~b~
F~g. 4. Secondary system on sixth floor: (a) RMS acceleration spectra for high, intermediate and 1ow s~iding friction° and (b) amplification factor for these spectra no~.,~.~a!ized with respec~ to the fixed=base case.
4~--~.
0.15-
_,g
i
i0
iO ~
20.0
(b) 0.20 .
O.O0 f ~0
0.100.05-
BASE
,' 0.OO0.00
5.0{)
I O.0 TIME
15.0
20.0
(SEC) (C)
F~go 3. Nonstationary RMS response for velocity-dependent intermediate friction (/~,~ = 0.1 t 9) for (a) secondary systems, (b) sixth floor and (c) base (-- VP model, - - CS model, e - , - e Monte-Carlo simulation). structure; and (b) as friction is lowered, nonlinearities in the base isolator predominate and tuning between primary structure and secondary system is progressively destroyed. Figure 4(b), which plots the secondary system RMS accelerations normalized with respect to the fixedbase case, clearly shows the high degree of acceleration reduction offered by the low friction slider at all frequeno cies except the very low (below 0.5 Hz) range. It should be noted that although low friction in the sliding bearings
is very desirable for secondary system protection, it invariably ieads to large superstructure disptacemen:~c. 3j This means that the intermediate friction slider, wHch offers passive protection o~er the entire frequene',~ rang% may be the best choice. Next, Fig. 5 compares the secondary system RMS accelerations as obtained using the ve_Tocity-dependes.L friction model, which is appropriate for the s:iding bearings with Teflon interfaces that were used m the experimental studieso 7:s33 and the constam fr.',ction mode1, which is c o m m o n l y employed in numerica~ ssmulations of sliding base isolators. Both intermediate friction (#~,~ = 0.119} and iow friction ( # ~ = 0-0572"~ designs are examined Significant differerzces arc observed between the variable and constant friction models past a frequency o$ 0-7Hzo Ciear~y, veioc~tv.dependent sliders are more efficient m protecting the secondary system tha~ sliders that exhibit constant friction for the ewo levels of friction considered. This is especially true in the critical regions of resonance (c<< - o%~, co~ = c~m) where m a x i m u m reductions of the order of 40% and 30% are observed for the intermediate and tow variable friction cases, respectively. F~rthermore, as friction is !owered. the R M S acceleration spectrum flattens more quickly in. the vdocity-dependent
Stochastic response of secondary systems in base-isolated structures . . . . . . . .
1.00
"~"
I .
.
,
.....
I
~
'~
- -
variable
FRICTION
'
I',
---
constant
t
0.75-
0.60
. . . . . . .
INTE~EDIATE
I
. . . . . . . .
LOW FRICTION
U
! I
'~
b U <
. . . . . . . .
99
--~
0.45-
I
. . . . . . . .
~_ D
...
b
':'\
~J
= =
0.25 0.50
Hz HZ
=
1.00
Hz
/'".
0.30-
0.50-
%%
Ij i
t%/|l
i! /
~
"
< i
03
0.15-
0.25-
0.0O
0.00
I0
1 0 -'
I0
*
FREQ.
10
'
10
10
o
(Hz)
10
FREQ.
(a) 0.60
. . . . . . . .
I
.
.
.
.
.
.
(a)
.
.
I
LOW FRICTION 0.45-
10
'
(Hz)
.
.
.
.
.
.
0.60
.
--
variable
p
---
constant
p
. . . . . . . .
I
. . . . . . . .
LOW FRICTION
--
0.45-
t ~
l I
[9
v
I
Ms/M P
. . . . . . . =
0.
---
=
0. 005
...
=
0.o1
---
=o.1
I
0.30-
,- " .
03
t
>
n-"
03~d~0.3o-
"
~
0.15-
0.00
Q~
. . . . . 10 "'
, , ' ;
. . . . . . . .
10 o
~ 10
FREQ,
. . . . . . . '
0.15-
0.00
10
(Hz)
(b)
. . . . . . . .
10
I
. . . . . . . .
10 o
;
10 ' FREQ.
. . . . . . . .
I0
(Hz)
(b)
Fig. 5. RMS acceleration spectra for secondary system on 6th floor for (a) intermediate (#m~x = 0'119) friction and (b) low (#m~x = 0"0572) friction. (Variable versus constant (Coulomb friction.)
Fig. 6. RMS acceleration spectra for secondary system on 6th floor for variable low friction (#m.x = 0"0572) versus (a) base frequency COband (b) mass ratio M~/Mp.
slider, indicating a more desirable design. The reason that velocity-dependent sliders perform better than the constant friction ones basically has to do with the fact that repeated slip-stick cycles in the latter introduce frequency components in the superstructure motions that correspond to the base isolator's own natural frequencies. This phenomenon is ameliorated when the slider is capable of producing a smooth transition between the slipstick cycles, as is the case with velocity-dependent friction. Subsequently, Figs 6(a) and (b), respectively, illustrate the effect of base isolator frequency cob and of mass ratio Ms/Mp on the secondary system response for the velocitydependent low friction slider ~m,x = 0-0572), which has succeeded in keeping the RMS accelerations at below 0.40g. In particular, secondary systems are rather insensitive to low values of stiffness of the base isolator Kb = coZMT. However, as the stiffness increases, low frequency (0-3-1-0 Hz) secondary systems experience an almost two-fold increase in their response. This can be explained as follows: at low values of friction and for stiff restoring force springs, the latter component predominates. As a result, the base isolator behaves almost linearly and peaks corresponding to the natural fie-
quencies of the combined structural system appear in the RMS acceleration spectra. It should be noted that since the presence of springs in the base isolation mechanism basically acts as a restoring mechanism for limiting large residual base displacements, 33 there is no reason for choosing a stiff base isolator design unless one wants to minimize these residual displacements for strong input motions. As far as primary structure-secondary system interaction is concerned, Fig. 6(b) shows that it is negligible as long as the mass ratio stays below 0.01. Also, heavy secondary systems (Ms/M p = 0.1) experience less motion (0.20g or below) than their lighter counterparts in the mid-frequency range, indicating significant interaction between the secondary system and its supporting floor. Finally, Figs 7(a) and (b) contrast the effect of location, i.e. 6th versus 2nd floor, for the velocitydependent, low friction slider and for the fixed base situation. The only conclusion which can be drawn here is that high frequency secondary systems remain relatively unaffected as their location within the superstructure changes, irrespective of the presence or absence of base isolation. It is noted finally that the presence of base isolation still reduces the overall maximum secondary
G. Juhn, G.D. Manotis, M.C. Constaneinou
100 0.80-
LOW FRICTION
~ -~-
6th 2nd
floor
]
floor
]
"~" 0 . 4 5 -i
O
0.50-
441
m Of
0.15-1
0.0010 e
10
10 ~
10 ~
(Hz)
VREO.
(a) 2.00-
- -
J FIXED BASE
t~
---
6th 2nd
floor floor
!
The authors wish ~o acknowledge the suppor~ o~ the National Center for Earthquake Engineenng Research under grants N C E E R 88-2006 and 89-2003. The authors also wish to thank Mary Anne Lang for ~yping the manuscript and an anonymous reviewer for his va~.uab!e comments.
1
REFERENCES t.00-
4--
in
a2
ACKNOWLEDGEMENTS
1.50-
b £J {3 <
limited to as much as one-fifth of those exper~encec under fixed-base conditionso The aforementioned deazgn is also insensitive to variatio,~s in the raass of the secormary system and produces the same maximum response, irrespecwve of the floor on which the secondary sys~;em zs !oeatedo
0.50{-
4
/
4--
?
0.0010 -'
10
10 '
°
~-REO.
!0 :
(Hz)
(b) Fig. 7. RMS acceleration spectra for secondary system at two different locations under (a) base-isolation with variable iow friction (#~,~ 0.0572) and (b) fixed-base conditions.
system RMS accelerations in the mid-frequency range down to 0.408 from the corresponding value of 1-75g observed under fixed base conditions.
CONCLUSIONS in this work, spectra were developed for the R M S acceierations o f a secondary system attached to a six-story frame that was base-isolated using sliding bearings and restoring force springs. The ground motions were represented by a doubly filtered, white noise power spectrum. The present study demonstrated that the protection offered to secondary systems by the Teflon-coated sliding bearings that produce a velocity-dependent friction coefficient is superior to that offered by sliders that exhibit constant friction, and in the frequency range that matters the most, i.e. the one encompassing the first few natural frequencies of the frame under fixed-base conditions. Atso, it was shown that a good design is one that employs an intermediate to low friction coefficient for the Tefloncoated sliding bearings and a soft restoring spring. This combination produces a secondary system spectrum that is relatively flat, with the maximum RMS accelerations
L Chert, Y.Q. & Soong, T.T., State-of-theoart review: Seisrmc response of secondary systems~ Engng StrucL~ ~ (I988) 218-28. 2. Igusa, T. & Der KiuregbAan, A. Dynamac response of muttiply supported secondary systems. A Engng M'eC~o (ASCE), !111(1) (t985) 20-4t 3. Suarez, L.E. & Singh, M.P., Floor response spectra ~,~-t}: structure-equipment interaction effects by a mode synthesis approach. Earthquake Engng StrucL Dyn., 15 (!987) i4i-58 4. Suarez, L.E. & Singh, M.Po, Eigenproperties of ~o~classically damped primary structure and eqmpmen*: systems by a perturbation approach. Earthquake Engng Struct. Dyno, 15 (1987) 565-83 5. Kelly. 7.M. & Chitty, D.E., Controi of seismdc response of piping systems and components in power pla~-ts by base isolation. Engng Struct., 2 (i980) 187-98. 6. Kelty, J.M., The influence of base isolation on the seismic response of light secondary equipment Report No. UCB/ EERC 81/ !7, Earthquake Engineering Research Center. University of California, Berkdey, California, I982 7o Manotis, G.D., Juhn, G., Constandnou, M.C. & Rdneo>q, A.M., Secondary systems in base-isolated structures: Experimental investigation, stochastic response and s~:o chastic sensitivity. NCEER Rept. No. 90=0013. Nat. Cen~e_~ for Earthquake Engng Res., Buffalo, New York, !990. 8. Kd!y, J.M., Aseismic base isoNtion: Review and b~bho graphy. Soil Dyn. Earthquake Engng, 5(3) (!986) 202-f& 9. Buckle, I.G., New Zealand seismic base isolation con~pts and their application to n~clear engineering. A%~.e;ear Engng and Design, 84 (1985) 3!3-26. 10. Kelly, LM. & Hodder. S.B, Experimental study of tend and elastomeric campers for base isolation systems in ~amihated neoprene bearings. Null of _grewZealand N~eto Soc Earthquake Engng, 15(2) (!982) 53-67. 11. Caspe, M.S.. The earthquake barrier. In Dynamic *~,es'ae~se of Structures, Proc. 3rd Conf. ,)f Engng ~gCech. Div (ASCEj, ed. Hart, G.C. & Ne}son, R.B. ASCE Pubhcation, New York, 1986. pp. 467-72. 12. [konomou, A.S., Afexisismon isolation engineering ~o~ nuclear power plants. Nuclear Engng and Desig& 85 (t985) 201-16. i3~ MosmgheL N. & Khodaverdian, Mo, Dynamics of resiliera> friction base isoiator (R-FBI). Earthquake Engng Struck. Dym.o ~5 (1987) 379-90
101
Stochastic response of secondary systems in base-isolated structures
14. Constantinou, M.C., Mokha, A.S. & Reinhorn, A.M., Teflon bearings in base isolation. II: Modeling. J. Struct. Engng (ASCE), 116(2) (1990) 455-74. 15. Mokha, A., Constantinou, M.C. & Reinhorn, A.M., Teflon bearings in base isolation. I: Testing. J. Struct. Engng (ASCE), 116(2) (1990) 438-54. 16. Mokha, A., Constantinou, M.C., Reinhorn, A.M. & Zayas, V.A., Experimental study of the friction pendulum isolation system. J. Struct. Engng (ASCE) (submitted). 17. Su, L., Ahmadi, G. & Tadjbakhsh, I.G., Comparative study of base isolation systems. J. Engng. Mech. (ASCE), 115(9) (1989) 1976-92. 18. Tsai, H.C. & Kelly, J.M., Non-classical damping in dynamic analysis of base-isolated structures with internal equipment. Earthquake Engng Struct. Dyn., 16 (1988) 29-43. 19. Tsai, H.C. & Kelly, J.M., Seismic response of the superstructure and attached equipment in a base isolated building. Earthquake Engng Struct. Dyn., 18 (1989) 551-64. 20. Chen, Y.Q. & Soong, T.T., Seismic behavior and response sensitivity of secondary structural systems. NCEER Rept. No. 89-0030, Nat. Center for Earthquake Engng Research, Buffalo, New York, 1989. 21. Singh, M.P. & Wen, Y.K., Nonstationary seismic response of light equipment. J. Engng. Mech. Div. (ASCE), 103(EM6) (1977) 1035-48. 22. Ahmadi, G., Stochastic earthquake response of structures on sliding foundation. Int. J. Engng Sci., 121 (1983) 93-102. 23. Constantinou, M.C. & Tadjbakhsh, I.G., The optimum design of a base isolation system with frictional dements. Earthquake Engng Struct. Dyn., 12 (1984) 203-14. 24. Constantinou, M.C. & Tadjbakhsh, I.G., Hysteretic dampers in base isolation: Random approach. J. Struct. Engng (ASCE) 111(4) (1985) 705-21. 25. Constantinou, M.C. & Papageorgiou, A.S., Stochastic
26.
27. 28. 29.
30. 31. 32. 33. 34. 35.
response of practical sliding isolation systems. Prob. Engng Mech., 5(1) (1990) 27-34. Lin, B.C., Tadjbakhsh, I.G., Papageorgiou, A.S. & Ahmadi, G., Response of base-isolated buildings to random excitations described by the Clough-Penzien spectral model. Earthquake Engng Struct. Dyn., 18 (1989) 49-62. Lin, B.C., Tadjbakhsh, I.G., Papageorgiou, A.S. & Ahmadi, G., Performance of earthquake isolation systems. J. Engng Mech. (ASCE), 116(2) (1990) 446-61. Igusa, T., Response characteristics of inelastic 2-DOF primary-secondary system. J. Engng Mech. (ASCE), 116(5) (1990) 1160-74. Pires, J.A., Analysis of nonlinear primary-secondary systems under random seismic loading. Proceedings of Fourth U.S. National Conference on Earthquake Engineering, May 20-24, 1990, Palm Springs, California. 1990, Vol. 3, pp. 51-60. Iwan, W.D. & Paparizos, L.G., The stochastic response of strongly yielding systems. Prob. Engng Mech., 3(2) (1988) 75-82. Suarez, L.E. & Singh, M.P., Seismic response of SDF equipment-structure system. J. Engng Mech. (ASCE), 113(1) (1987) 16-30. Wen, Y.K., Equivalent linearization for hysteretic systems under random excitation. J. Appl. Mech., 47(1) (1980) 150-4. Constantinou, M.C., Mokha, A.S. & Reinhorn, A.M., Study of sliding bearing and helical-steel-spring isolation system. J. Struet. Engng (ASCE), 117(4) (1991) 1257-75. Fang, T. & Wang, Z.N., Complex modal analysis of random vibrations. AIAA Journal, 24(2) (1986) 342--4. Shinozuka, M. & Sato, Y., Simulation of nonstationary random processes. J. Engng Mech. Div. (ASCE), 93(EM1) (1967) 11-40.
APPENDIX 1
Equations (1) and (2) can be combined to read
M, {Mp} k{Ms}
+
[Ms]
[o1
{o} {,
{O}
[
Kb
{M.} • {<}
{M,} )
{O}" 3 {O} [Kp] + Z Kq{ey}{ej} T j=l
{o }
-
{ej }f
{O}T -
-
[Ksj{ej}]
t xb}
[K I
l(xs}
+ &)
{x.}
(Al.1)
{Ms}
In the above, the DOF of the primary structure are pl-p6, while those of three secondary systems attached to the sixth floor are sl-s3. All symbols in eqn (AI.1) have been previously explained with the exception of the connectivities {ei}T = [0, 0, 0, 0, 0, 1]. It should be noted that since the damping terms behave in the same way as the stiffness terms, they have been omitted from eqn (AI.1) in the interest of brevity. Equation (AI.1) is
supplemented by eqn (11) to form an 11 × 11 system for the viscoplastic model. For the constant sliding model, variable z and coefficient/4 are ignored.
APPENDIX 2
By following the modal transformation described in
G. Juhn, G.D. ManNis, M.C. Consganginou
102
eqn (3), the expanded form of eqn (4) is 1
[(~s]T {Ms } (]~b
[t1
[o1
i
[o1
[z]
J
{o}~
{o}~
qb)
3
+ ]{o} [%1~ + Z <,[4,S{ej} {~,}'[+d ]=1
L{O}
- [ ~ ] T [K v{ ej}IT[~J
-IeSI<;{~,}Jm~ i {qP} I {q~} ;! H
f ~bMT#~g
+ ,{ {o}
(
{o}
z :
-
[~.]~{M.}}x~
;A2o i,
E
In the above, cob, [cop]and [cos]are the natural frequencies of the base ~solator, of the fixed-base frame and of the individual fixed-base secondary systems, respectively~ Furthermore, [I] is the identity matrix. Despite the fact that {F*} contains the actual model participation factors of the individual structural subsystems, matrices [M*I and [K*] are not diagonal so there is coupling of the
generalized co-ordinate equations° Assurmng ma~ the fixed-base frame is proportionally damped~ the~ eqn (A2.1) can be augmented by the t e , ~ ~[C*] {@ where [C*] has the same form as [K*] with cob, [a)~], ie~] a~d K~ replaced by 2~bco b ~-Ce~ [2~pcop], [2~s~s] a~d ;si~ respectivelyo