- Email: [email protected]
Stochastic response of practical sliding isolation systems
Stochastic response of practical sliding isolation systems M. C. Constantinou
Department of Civil Engineering, State University of New York, Buffalo, New York 14260, USA A. S. Papageorgiou
Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA Practical sliding isolation systems utilize Teflon sliding bearings in which the coefficient of friction exhibits a strong dependency on the velocity of sliding. Analytical solutions for the stochastic response of such practical sliding systems are presented in this paper and verified by extensive Monte Carlo simulations. INTRODUCTION Base isolation is an earthquake design concept receiving worldwide attention. In this approach an attempt is made to decouple a structure from the damaging components of earthquake motion by introducing flexibility and energyabsorption capacity through a system that is placed between the structure and its foundation 1. There is a variety of isolation systems but the focus of this paper is on sliding systems and, in particular, on sliding systems that utilize Teflon sliding bearings. In sliding isolation systems the entire weight of the structure is carried by Teflon sliding bearings which provide little resistance to lateral loading by virtue of their relatively low friction. Centering force capability is provided by separate mechanisms that may take the form of rubber springs, helical steel springs or even yielding mild steel dampers 2 7. A major advantage of these systems is their insensitivity to the frequency content of excitation, an apparent consequence of their frictional behaviour. Another property of sliding systems is the dependency on vertical load of the mobilized frictional forces. These forces tend to develop close to the centre of mass which in turn reduces adverse torsional motions. Applications of sliding isolation systems include a four-story structure in Japan s and a 50000 gallon fire water tank in California. In this paper we are concerned with the stochastic response of these practical sliding systems. Major concern in this study is the modelling of Teflon sliding bearings which exhibit a behaviour that is significantly different than that of Coulomb's theory. The coefficient of sliding friction of Teflon-steel interfaces depends strongly on velocity of sliding and bearing pressure 9-~2. Analytical solutions are presented for the following models of frictional behavior: (a) Based on a model of viscoplasticity theory which is capable of accounting for stick-slip conditions, and (b) based on the assumption of continuous sliding that appears to be a reasonable Paper accepted December 1989. Discussion closes August 1990.
© 1990Computational Mechanics Publications
assumption when considering the velocity dependence of the friction force. Both models account for the velocity dependence of friction coefficient. The derived analytical solutions are compared with results of Monte Carlo simulations and the degree of accuracy of each solution is established. Furthermore, the significant effect of velocity dependence is demonstrated by comparing results based on Coulomb's model of friction. MATHEMATICAL F O R M U L A T I O N The purpose of this study is to present an analytical method for the stochastic response analysis of practical sliding systems rather than to perform a parametric study. As such, the simplest structural model is used and excitation is limited to stationary and evolutionary white noise. Extension to more sophisticated models is simple and briefly discussed at the end. The system considered is a rigid mass, m, supported entirely by Teflon sliding bearings. Restoring force is provided by linear springs of total stiffness, K, and viscous damping constant, C. The equation of motion for the slip displacement x is:
Yc+ 2~oCOoJC+ogzx +F: = -5~o
(1)
F:= p(:~)mgZ
(2)
m
in which Oo is the natural frequency of the system and G0is the corresponding damping ratio (as computed from m, K and C) and F: is the mobilized frictional force./~(5:) is the coefficient of sliding friction which depends on the slip velocity, ~, and 5~o is the ground acceleration. Z is a variable taking values in the interval [ - 1 , 1]. This variable obeys the following constitutive equation which has been proposed by Wen 13 for modelling hysteretic behaviour: Y Z + TIYclZJZI"-1 + fl~IZl"- AYc=O
(3)
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 27
Stochastic response of practical s/iding isolation systems: M. C. Consmminou and A. S. Papageorgiou Mokha eta/. 9 - 1 2 have found that the model of frictional of equations (2) and (3) simulates well the frictional behaviour of sheet type Teflon bearings provided that A = 1 and r + °/= 1. For this combination of parameters the model of equations (2) and (3) collapses to a model of viscoplasticity!4o As such, this model will be referred to as the viscoplasticity model. Quantity Y represents the elastic deformation of Teflon prior to sliding which has been experimentally determined to be in the range of 0.005 to 0.02 in (0.12in to 0.5 ram) 9xx. Experimental results in Refs 9 to 12 suggest that the coefficient of stidlng friction of Teflon-steeI interfaces dose]y approximates the following equation: (4)
#(2)= fm~.- Df "exp(- al2})
in which f~,.. represents the maximum value of friction coefficient attained at large velocities of sliding arid ( f ~ , ~ - D f ) represents the value at essentially zero velocity. Table ! presents values of fr,~, Df and a for untitled sheet Teflon when sliding against stainless steel of surface roughness R~ = 0.03 gm ~''-~ under bearing pressure of 200 to 6500 psi (1.4 to 44.9 N/mmZ). Figure 1 shows the variation of the sliding coefficient of friction with velocity and pressure. The strong dependency on velocity and pressure is evident. The interested reader is referred to Ref. 9 for a physical explanation of this behaviour. A very important characteristic of Teflon-steel interfaces is the reduction of the mobilized frictional force as the velocity of sliding decreases (Fig. 1). Accordingly, as a structure sliding on these interfaces slows down less force is needed to maintain the motion. As a result of this behaviour motion continues with almost complete lack of stick-slip tendencies. This behaviour has been confirmed in shake table tests of model structure~ 5. Accordingiy, the assumption of continuous sliding in structures supported by Teflon bearings is well justified and is investigated in this study. In this case the equation of motion takes the simple form: ~" x-" + 2~0OoX + COoX+ # ( £ ) g sgn(2) = - 2~ •
2
(5)
where sgn(2) stands for the sign of 2.
VELOCITY
4
200 psi
0.15 -~
1000 psi ~U30.~O
"6,
o
Y Z + C 2 t k + K22Z=O in which 2¢ represents svolutionary white noise Coefficients C2~, C2~, K~2 and K22 are de~ermined b? minimizing the mean square difference between eq aafions {!) and (6) and {3) and (7). Wen ~ has reported expressions for coefficients C2: ova K22~ while C~ and K~ 2 take the following form: C ~~ =
2~o~ o + aDfgE[Z e x p ( - ai2 ) sgn(2)]
(8
in which E['] represents the expected value. For Gaussiar~ input the resoonse of the iinearized system {6, 7] is also Gaussian and coe~cients C~ and K~2 are evalua
~-/2~ ~-'=aDfg E[2Z] ®{B)
(10)
K I2=g[]<~.~- D) .~p(B )erfc(B)]
2 i/2 @(B) = 1 - 7r~"~Bexp(B2)edL'(B)
~i 3)
in which erfc is the complementary error funcJo~a and G stands for the standard deviation of the quantity in the subscript. The solution of the linearized stochastic system is obtained from the assoc,ated Lyapunov matrix differential equation for the zero time lag covaria~ce matrix S ~7. dS =PS+Spr+Q dt
{I~:~
in which P is given in the Appendix, S~-- El Y~Y~], Q u : 0 except Q3a =2~Oo I2(t} and YI =x, Y2O==Za~d Y3°=2- Go is the constant power spectra~ density (psd} of the two-sided white noise input and I(t) is the intensib function. The variance of the total acceleration ~ = 2 + x ~ is obtained from equation (1):
2000 psi 3000 psi 5500 psi
(5 Z 0.05
d~
1000 pmi= 6.9 N/rnm z 0.00
VELOCITY
in/see
Fig. 1. Sliding coefficient of friction of unfilled sheet Teflon in contact with stainless steel as fianction of pressure and sliding velocity
28
~s7
50
0.25 t U n f i l l e d Teflon Z ] Stainless Steel Ra=O.03 ~ m 0 i]2 Lt_
The method of stochastic tinearization '~ ~s empioyed m obtaining approximate statistics of the response of the slidi~g system. In the case of the viscoplasticity model equations (I) and (31 are replaced° respectiveiy, by:
cm/sec 410
0.20
STOCHASTIC LI[NEARiZAT~ON
Probabilistic Engineering Mechanics, t990, VoI. 5, No. J
9
2
2
2
in which the expected values of the bracketed expressions are given m the Appendix, [t should be noted that equation (15) is based on the nonlinear equation of motion rather than the linearized one. The method of equivalent linearization is Nso applied to equation (5) (continuous s~iding) resuking it: the
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. S. Papageorgiou following linearized equation: 2 + 2 ~e(_O0)C+ (_02X= -- 2 o
I ; =0,5 See ~, =0.05
(16)
Unfilled. 1 0 0 O p s i ( 6 .
where I
~,=¢o+
gEfmax--Df" ~(B)]
(17)
>-
"~ × ,o t3 .200psi(1.4
The second moments of x and 2 are obtained by solving the associated Lyapunov matrix differential equation. The variance of the total acceleration a = 2 + 2 o is obtained from the nonlinear equation (5):
o
tO
0.2 = 4~oO)oa; 2 2 2 + O)oa;, 4- 2 + 4~oe)gE[x2] + gZE[]22]
+ 4~oCoogE[p2 sgn(2)] + 2coZgE[#x sgn(2)]
(18)
I0 ~-
where the expected values of the bracketed expressions are given in the Appendix. C O M P A R I S O N TO M O N T E CARLO SOLUTIONS The accuracy of the derived approximate solutions is investigated by comparing to results of numerical simulation. Stationary and nonstationary responses are considered. Input is either stationary of psd GO or nonstationary of psd G O and intensity or modulating function l(t). Analytical stationary responses are obtained by solving the Lyapunov nonlinear matrix equation (14) with dS/dt set equal to zero. Numerical responses are obtained by creating a realization of stationary zero mean Gaussian process of psd G O(two-sided) and integrating numerically equations (1) through (4) up to 300 or 350 secs. Ergodicity is assumed and temporal averages are obtained over the last 2/3 of the duration of the response. Results are presented in terms of standard deviations of slip displacement, slip velocity and total acceleration, a~. a; and a., respectively, versus the normalized density D=(GoT3/rc3) 1/2 in which To is the natural period (To 2re/COo). Analytical nonstationary responses are obtained by solving equation (14). Numerical responses are obtained by generating 200 realizations of the input process and using ensemble averages. It should be noted that all numerical simulations (Monte Carlo solutions) are based on the solution of equations (1) to (44) which properly account for the stick-slip conditions. In these analyses a value of exponent t/=3 and Y=0.01 in (0.25 mm) were used. Both values are consistent with experimental observations9- ~2. For the numerical solution of equations (1) to (4) an adaptive integration procedure was used in which estimates of the local truncation error were kept within a prescribed tolerance. Figures 2 to 4 compare stationary responses produced by the two models (viscoplasticity and continuous sliding) to simulation results at various excitation levels. Data of friction of unfilled Teflon at a pressure of 200 and 1000 psi (1.4 and 6.9 N/mm 2) were used (see Table t) and To = 0.5, 1 and 2 sec while ~o = 0.05. The excitation is in the range of Go=3 to 100in2/sec 3 (19.4 to 645.2cm2/sec3). To get a perspective of the level of excitation we note that the Taft 1952 and E1 Centro 1940 motions could be represented by stationary white noise processes of Go=2.3 and =
N/ram z )
f:/ a
Simulation Viscoplosticity M o d e l Continuous Sliding
lo'
D/¥
;::°o:0%"°
/
Unfilled,1000psi(6,9
o
N/rnrnZ)
~
>b I
Unfilled.2OOpsi(l.4
N/ram ~ )
Simulation
0
Vlscoplosticlty
Model Continuous Sliding
10
10= D/Y
0.40 [o =0.5 S~C
to =0.05 0.30
///~/o
I 0,~_0 o
/
•
Unfitted,20 psi(l 4 N/ram a )
/ ~--~/"
0. I 0
.1000psi(6.9
......
0,00 I0
N/ram I
Simu]ahon Viscoplasticity Model Continuous Sliding I0 ~
O/Y
Fig. 2. Stationary response statistics of a sliding system with -I o = 0.5 sec Table 1. Constants f,..x, Df and a for unfilled sheet Teflon sliding against stainless steel o f R. = 0.03 pm Bearing pressure (N/mm2)
fmax
Df
a (sec/cm)
1.38 6.9 13.8 20.7 44.9
0.1900 0.I 193 0.0870 0.0703 0.0572
0.1200 0.0927 0.0695 0.0552 0.0485
0.236 0.236 0.236 0.315 0.197
7.5in2/sec 3 (14.8 and 48.4cm2/sec3), respectively is. A process with Go = 100 in2/sec 3 corresponds to a motion about three times stronger than E1 Centro •940. It is noted that the model of continuous sliding predicts
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 29
Stochastic response o f practical slidin9 isolation systems: M. C. Constandnou and A S. Papageorgiou 10
~o =0.05
period TO the resmnng force is weak and the system approaches a state of almost purely frictional behav~our. This is evident in the time needed for ~ to reach stationary in Figs 5 to 7. This dine increases as friction increases tsee differences between responses for 200 psi and 3000 psi). Deterministic evidence provided in Refs 9 and i2 suggests that sliding sys terns should be designed to exhibil this kind of behaviour. The resu!ts presented in Figs 5 to 7 further confirm the accuracy of the presented approximate solutions. Nonstationary responses for evdutionarv mpu,: are compared in Fig. 8. The input is modulated over an interva! of l0 sec by the intensity function shown in ~-he
o
To-1.0 Unfil~ed, l ~ec O00psi(6.9
~
)
~
. -
N/mrn~.
"
~/
>-
~× o ~D
o
&
.......
Simu{otion Viscoplosticlty Model Continuous Sliding ~
1
!0 ~
10
D/2Y T#--I.O
] r~ =2.0 sec $o =0.05
sec
/" //
r)
~ -0.05
Uefilled.lOO0psi(6.~
N/taro ~ )
-? >04 ,× b
~c o
.-
/
~
/'/£
~
,
~
(
~
.
S/m,~ ~)
~
/"
./-
Simulation
o
•
o
~
--
--
"
Simulation Vi~coplo~U¢itv Mo~el Continuous
~iiding
V~scoplastic~v Mode{
i
Slldin 9
Continuou~
......
!
0
10
o/sv
~0 ~ D/2Y [e ~
0.15
1
0.10
u . ~ i . ~ . 2 0 0 ~ i 0 4 N /~~
,
/"
] {o ~ 0 . 0 5
1.0 sec
.
o
r o ' 2 ' O sec
10:~-
0.20
~..
)
~
.2'
"//2
N/ram ~ )
Unliiled. l O O O u s t ( 5 . 9
.-~...~;...
ro
.....
]
_•
~X."" U n f i l t e d . 2 0 O u s i l ~ 4, N / r a m
o
~
0.05
_
~
o
/
}1
k'--- Unfilled- 1000us'( 6.9 N/mrnZ ~
o
m
......
Simulation Viscoplosticity Model Continuous Stidmg
\r~scoplQst;,ci~ Mo~e; Continuous
Simuia[ion
~idinq
10
0.00 ;0
o ~ - ......
!0 ;
~o '
D/6Y
D/2Y 0.20
-
Fig. 3. Stationary response statistics o f a sliding sysfem with T o = t sec
I o = 2 0 sec ~o :0.05
0 ~5 -, U~filled.200psq ? ~ N/ram z ) -
resonses in very good agreement with simulation results at a!l response tevels except in the low level tail and when friction is large (e.g., data for 1.4 N/ram z pressure}. For these conditions the viscoplasticity model provides accurate results (see standard deviation of velocity at low level excitation tail). This of course is a result of the ability of the viscoplasticity model to simulate stick-slip. It is very interesting to observe that the two models provide in all cases close upper and lower bounds to the simulation resMts. Figures 5 to 7 compare evolutionary responses for stationary input when the bearing pressure is 200, 1000, and 3000 psi (1.4, 6.9, 20.7 N/ram2), T o = 4 sec, 4o=0.05 and Go= 10 in2/sec 3 (64.5 cm2/sec3). For this value of
30
Probabilistic Engineering Mechanics, I990, VoL 5, No, ]
~
." -"
0[0
-
-
-
/
D
0 05
-
- - -o" - J
q
" - - -,~
/
~.;-"
..5~..-'~
/
. --o
%
.-J
dnh!le¢ 1000es~96.
0
~
I/
~,I/mma')
Sirnu!c~ior
V~scopl~s~iCi~y .......
300 --
Contir~uous -
,'Mode~ Siid~ng " ...... J
D/6Y
Fig. 4. Stationary response statistics o f a sliding system with T O = 2 sec
Stochastic response of practical sliding isolation systems." M. C. Constantinou and A. S. Papageorgiou 80 [ Te = 4 , 0 sec t o =0.05 G, =10 inys ) (64.5 c m ' / s ' )
approximation introduced in equation (22) of the Appendix. In this approximation Z was replaced by sgn(k) which is incorrect under conditions of free vibration in which the structure tends to stick to its foundation and Z attains values close to zero. In this case the approximation E[#xZ]~(fmax-Df)E[xZ ] for the free vibration part resulted in an acceleration response that closely followed the simulation results.
Unfilled Teflon, 200psi(l.4 N/ram a)
60
>~'~40 × b
COMPARISON TO COULOMB'S MODEL
20"
The presented solutions reduce to those of Coulomb friction with constant coefficient of friction p =fmax when
Simulation
o
10
o
TIME 1 O0
T0=4.0 sec . =0.05 c o =~o ~n%' ( 6 4 . 5
120
sec
T =4.0 sec Unfilled Teflon. 100Opsi(6.9 N/ram = ) ~** =0.05 0 ° =10 in~/s~ (64.5 cm~/s ' )
Unfilled Teflon, 200psi(1.4 N/ram =)
~m'IJ )
90
80 ¸
>.x b
60'
X b
60-
40
30
20-
Viscoplasticit 7 Model Continuous Sliding
. . . . . . . . . . . . .
-o
o-
..........
-~
Simulation Viscoplosticity Model Continuous Sliding
120-
~
to
T - 4 0 sec Unfilled Teflon. fO00psi(6.g N/ram = ) ~: =0.05 C 0 = I 0 inys ) (54.5 c m y s ~I)
sec
0.20 To =4.0 sec t o =0.05 c o = t o in~/,' ( 6 4 . 5 c m ~ s ' )
~
TIME s e c 10
TIME
;,
Unfilled Teflon. 200psi(1.4 N / m m =)
90
0.15
60 b
I 0.10
30 0.05
-o .o .
.
.
o.
.
.
.
.
.
Simulation Vlscoplasticlty Model Continuous Sliding
~
TIME
~
Viscopiosticity Model Continuous Sliding
0 10
0 TIME
0.00 0
o- Simulation
. . . . . . . . . . . . .
~
to
sec
O. lO To=4.0 sec ( , =0.05 ( ; , = 1 0 in?s ) (64~5 cm?s ) )
Fig. 5. Nonstationary response statistics of a system supported by Teflon bearings at 1.4N/mm 2 bearing pressure figure. The response is computed for an additional 5 secs of free vibration. Both approximate solutions provide accurate results in the range 0 to 10 sec. For the range beyond 10 secs (free vibration) the continuous sliding model fails as a result of numerical instabilities in the Lyapunov differential matrix equation (negative elements along the diagonal of matrix S in equation (14)). In contrast, the viscosity model provides accurate response predictions including the standard deviation of slip displacement during free vibration. However, the model failed to predict precisely zero standard deviation of accelerations after a long interval of free vibration (longer than that shown in Fig. 8). This is a result of the
sec
Unfilled Teflon.
t000psi(6.9 N/ram z
tffl I 0.05 0 b
"~' .........
0.00
TIME
~
Simulation Viscoplosticit Model Continuous ~liding
tO sec
Fig. 6. Nonstationary response statistics of a system supported by Teflon bearings at 6.9N/mm 2 bearing pressure
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 31
Stochastic response of practical sliding isolation systems. M. C. Conseandnou and A. S. Papageorgiou ~80TT
-4.o
=o.o~
l,d
sec
the standard deviations and in the power spectra~ density (which reveals the frequency content} of the response. Figaro 9 s~ows the standard deviations of sli# dispiacement, a~, and totai acceleration. %, of a syster~ with To=4Sec, ~o=0.05 and whe~ subiected to stationary input of psd Go equat to 2.3 and 7.5 ine 'sacs (14.8 and 48.4cm 2 sect). As discussed earlier, these processes could be used in the representation of the i952 Taft and 1940 Et Centro motions° respective13. The sliding bearings either obey the taw of friction el equation (4} for 1000psi ~6.9N/mmB~ pressare oz CoulomVs !aw with coefficient of friction #=a~.=:~. ."Fh~ .. responses shown were computed by the viscop!asticity
Unfilled Teflon. 3 O O 0 p s i ( 2 0 . 7 N / r a m a )
[
120
~-. 90
t//-
-'"
50
-t~
~° 1
o}
q>
............. 4
0
Co=Ie
Continuous
8 TiME
180" /T r = 4 , 0 see ~ =O.05 W/s' (s4.5 150-
Simulolion
Viscopiasticity Mode~ Sliding
~0
8
sec
Unfilled
180 --
Teflon, 3 0 0 0 p s i ( 2 0 . 7
N / t u r f ~a
)I )
cm'l~~)
~c =4,0 S~C
[ehon,
Unhtlec
{~ = o . o 5 Go =10 i ~ / s ~ 6 4 5
5000os~ 2 0 7
N/,-nm ~ "i]
Simu~atio~ *~
leod
cm~/s a )
~50 -
120-
-~. 9o I >".×- .
I
. . . . . . . .
90
60 )
~JO 4
30"L ............. 0
2
4
1
g
10
6 TIME
OAO
Simulotion Viscoplos~icity Mode~ Con|inuous Sliding
r I i'vtE 180~
3000psi(2017
<:,-----~-
I
sac
UnfilledTeflon.
go=4,0 ~eC
t:
To=4"O see ~o = 0 0 5 C° = I 0 *n~S~ (64,5
J
N/turn ~ )
sec
Unfilled Teflon, 3 0 0 0 0 ~ t { 2 0 . 7
~,W~',#r ~Q
crn/s ~ B)
50
( o =0.05
7 120>-~. 9 0 ~ -x u)
I 0.05
g
6oj ~0~ -o-------o-
$imulotion Viscoplosti¢ity
Mode~ ........... Continuous Sfiding o.oo
o
. . . . .
5
~
-
--~----;
T~ME
•
-,
'
i
3
0
5
12
9 "f~E
5
sac
--7o
sac OAO
T o = 4 . 0 .~ec Untgied Teflon, 3000Dsi{20.7 o =0.05 G o = t O inYs (64.5 c m T s •
fig. 7. Nonstationary response statistics of a system supported by Teflon bearings at 20.T N/mm 2 bearing pressure
either Df = 0 or when B ~ co. The latter case corresponds to a~--+ eo. This should be expected since when the velocity of sliding is large, the coefficient of friction in Teflon bearings attains its maximum value (see equation (4) and Fig. (1)). Deterministic studies presented in Refs 9 and t2 have shown that the use of Coulomb's model with #=fm~x coutd result in useful estimates of the peak displacement and acceleration responses. However, the details of the responses in the two models of Coulomb friction and velocity dependent friction were substantially different. [n stochastic analysis this difference shoutd appear both in
32
Simufotion V~scopmstic~ty Modet Continuous Sliding
..........
f
Probabilistic Engineering Mechanics, 1990, Vot. 5, No. /
o.o~- ~
0
~
5
6
9
i-(ME
Fig. 8. input
N/turf, x ) ~
~2
5
sec
Nonstationary response sta¢istics for epoiutionar)
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. S. Papageorgiou ~00
-
T, =4.0 s e c ~o =0.05
0.010
--Untitled Teflon, lO00pSi . . . . . Coulomb• [/.= fmox
80
Tn =0.5 ~ec ~G** "0"05
Coutomio. # -
fmGx
-10 in~/,~ (64.5 c~/, 3)
G° = 7 , 5 in~/s j
0.008
0.006 I
Z ~1.~--0.004
b
_.---20
......
2.3
~ 0.002 0
.--'''" 2.3
""
o"
0
2
6
4
TIME
0.12
Te =4,0
8
10
sec
sec
-
~o =0.05
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Teflon,
G 75 n'/'_._-__._j__r_
.
0.08
_
_
n
2.3 7.5
.................................................
o~ I
.
•
i 2
,
,
.
i 4
,
,
'
6 '
FREQUENCY
.J ~- 0.010 0 1I, 0 0.008 LO Q k-
lO00psi Coulomb. /.Z= fmox Unfilled
-
.....
.
•
0
C)
0.10 .
~__j0.000 DJ
le
~0.5
'
'
'
8
'
'
fO
Hz
Unfilled. lO00pni (6.9 N/ram z)
~ec
-0.05
~'. -lo i.'/,'(64.5 =.%'3
0.006
<[ r~ LtJ
0.06
r7 0.004
g
"3 0 h
2.3
0.04 0.002
0.02 0.000 0.00
o
~
~
~ TIME
~
0
2
4
FREQUENCY
io
6
8
10
Hz
sec
Fig. 9. Comparison of responses predicted by proposed law of friction and Coulomb's theory
Fig. 10. Frequency content in motion predicted by proposed law of friction and Coulomb's theory
model. As expected, the two models predict substantially different responses. Coulomb's model underestimates the displacement response by a factor of 2 in the strongest motion and by a factor of 3 in the weakest motion. This example demonstrates the significance of the velocity dependence of friction in practical sliding isolation systems. While the results of Fig. 9 show substantial differences in the response statistics, they do not reveal the differences in frequency content. Frequency content is very important in structures housing sensitive equipment. A comparison of frequency content of the acceleration responses is presented in Fig. 10. Portions of the stationary acceleration response of a system with To = 0.5 secs were Fourier transformed and compared in this figure. Clearly, Coulomb's model predicts higher frequency content than a system obeying the law of friction of equation (4).
solutions is good in a wide range of system parameters and response levels. It is, furthermore, demonstrated that the stochastic response of the studied systems is substantially different than that of systems obeying Coulomb's law of friction. The extension of the presented analytical method to the case of correlated excitation is simple and briefly discussed in the Appendix. It should be noted, however, that the presented method has been verified only for the case of white noise input. ACKNOWLEDGEMENT Financial support for this study has been provided by the National Science Foundation grant CES-8857080. REFERENCES 1
CONCLUSIONS A method of analysis of practical sliding isolation systems under random excitation has been presented. These systems obey a law of friction in which the coefficient of friction exhibits a strong and nonlinear dependency on velocity. Comparisons of results with M o n t e Carlo solutions indicate that the accuracy of the presented
2
3 4
Kelly,J. M• Aseismic base isolation: review and bibliography, Soil Dynamics and Earthquake Engineering, 1986, 5, 202-216 Caspe, M. S. and Reinhorn, A. M. The earthquake barrier. A solution for adding ductility to otherwise brittle buildings, Proceedings, ATC-17 Seminar on Base Isolation and Passive Energy Dissipations, 1986, 331-342 Ikonomou, A. S. Alexisismon isolation engineering of nuclear power plants, Nuclear Engineering andDesion, 1985, 85,201-216 Mostaghel, N. and Khodaverdian, M. Dynamics of resilientfriction base isolator (R-FBI), Earthquake Engineering and Structural Dynamics, 1987, 15, 379-390
Probabilistic En#ineering Mechanics, 1990, Vol. 5, No. 1 33
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. So Papageorgiov~ 5 6 7 8 9
10 1i 12 I3 14
i5
14 15 16
Nagashima, I , Hisano, M., Kitazawa. K. and Kawamura, S. A base isolation system utilizing sliding bearings, Tasei Technical Research Report, 1988, 20, 71-87 (in Japanese) Watson Bowman Acme Corporation, Wabo--Fyfe earthquake protection system, Amherst, New York, t981 Zayas, V., Low, S. and Mahin, S. TEe FPS earthquake resisting system: experimentai report, Report No. UCB/EERC-87107, University of California, Berkeley, 1987 Kelly, J. M. Base isolation in Japan, 1988, Report No. UCB/EERC-88/20, University of California, Berkeley, 1988 Mokha, A., Constantinou, M. C. and Reinhorn, A. M. Teflon bearings in aseismic base isolation: experimental studies and mathematical modeling, Technical Report NCEER-88-0038, 1988 Constantinou, M. C. and Mokha, A. A model of friction of Teflon sliding bearings, Report to NSF, Dept of Civil Engineering, SUNY/Buffalo, 1989 Mokha, A., Constantinou, M. C. and Reinhorn, A. M. Teflon bearings in base isolation. Part i: Testing, Journalof Structural Engineering, ASCE, 1990, t16(ST2), 438-454 Constantinou, M. C., Mokha, A. and Reinhorn, A. M. Teflon bearings in base isoiafion. Part 2: Modeling, Journal qf Structural Engineering, ASCE, 1990, 116(ST2), 455-474 Wen, Y.-K. Equivalent linearizationfor hysteretic systems under random excitation, Journal of Applied Mechanics, ASME, 1980, 47, 150-154 Constantinou, M. C. and Adnane, M. A. Dynamics of soil-baseisolated structure systems: Evaluation of two models for yielding systems, Report to NSF, Dept of Civil Engineering, Drexel University, September 1987 Constantinou, M. C, Caccese, J. and Harris H. G. Frictional characteristics of Teflon-steel interfaces under dynamic conditions, Earthquake Engineering and Structural Dynamics, i987, 15, 751-759 Atalik, T. S. and Utku, S. Stochastic linearization of multi-degree-of-freedom nonlinear systems, Earthquake Engineering and Structural Dynamics, 1976, 4, 411-420 Lin, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill Book Company, New York, New York, 1967 Clough, R. W. and Penzien, J. Dynamics of Structures, McGraw-Hill Book Company, New York, New York, 1975.
"2i "
E[/~xZ] ~ E[#x sgn(:t)] \re/
F(B}-- - 2 B + 2=l;2(B~-+O.5}exp(Ba)e~jc{~) p=
E[~z]
24
G~ z
Expected values of bracketed expresswns in equatioe ( i8~
Ell/2]
2 ( ~ , ;2 exp,4B ¢ 2 )~Oc~2B) o g t =fmZ~×+,D)
exp(B2)erfd g ~,
-- 2 ( D f ) f ~
~26)
E[#x sgn(2)] is given by e q u a t i o n (22). i t s h o u l d be n o t e d tha~ the E[ffxZ] has n o t been e v a i u a t e d {it r e q m r e s the e v a l u a t i o n of a triple interval}, but r a t h e r a p p r o x i m a t e d by E[~x sgn{2)]. This is b a s e d on the o b s e r v a t i o n tb~a~ whi~e in sliding m o d e Z represents a contiw_mes a p p r o x i m a t i o n to the sgn(2) function. A c c o r d i n g l y , the a p p r o x i m a t i o n in e q u a t i o n (22) is valid H o w e v e r . when s t i c k - s i i p c o n d i t i o n s p r e v a i l the value of Z is c!oser ~o zero rather t h a n to + 1. F o r these c o n d i t i o n s the approxio m o t i o n in e q u a t i o n (22} i n t r o d u c e s e r r o r .
F o r filtered white noise excitation a shot no~se is passed t h r o u g h a filter of frequency coo a n d d a m p i n g factor ~;g. T h e statistical r e s p o n s e of ~he ~inearized viscoplasticity m o d e l is given by e q u a t i o n ~14l with
Matrix P
P=
0 0
0 -K22/Y
-cog
-K~
t
t
--C21/Y
(~9)
-ci; J
g
i
Expected values of bracketed expressions in equation (15) P= ~3 = fm~J~ + (1 - pb(Df) ~
x exp(B2)erfc(B)+,~/e(t_p~)F(B)
34
(23"~
Extensio;~ m filtered wh#e noise
APPENDIX
E[/z
~.,
(20)
Probabilistic Engineering Mechanics, 1990, VN. 5, No. d
©
0
0
-K12
-co8
vI
~ -Ctl
0
0 cog
1
0
0
0
0
\
0
0
0
-eo~2
0
0
i ()D
~,~co~ ! ~" -zCdv)~/
a n d S u = E [ Y ~ Yj], Y1 = Z . Yz =x, Y3=2, Y4=:xz, Ys = 2 r where x r is the r e s p o n s e of the filter. F u r t h e r m o r e . Qu = 9 except Q s s = nGo I2 (t) where G Ois p s d at z e r o frequency of the one-sided filtered white noise input.
Department of Civil Engineering, State University of New York, Buffalo, New York 14260, USA A. S. Papageorgiou
Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, USA Practical sliding isolation systems utilize Teflon sliding bearings in which the coefficient of friction exhibits a strong dependency on the velocity of sliding. Analytical solutions for the stochastic response of such practical sliding systems are presented in this paper and verified by extensive Monte Carlo simulations. INTRODUCTION Base isolation is an earthquake design concept receiving worldwide attention. In this approach an attempt is made to decouple a structure from the damaging components of earthquake motion by introducing flexibility and energyabsorption capacity through a system that is placed between the structure and its foundation 1. There is a variety of isolation systems but the focus of this paper is on sliding systems and, in particular, on sliding systems that utilize Teflon sliding bearings. In sliding isolation systems the entire weight of the structure is carried by Teflon sliding bearings which provide little resistance to lateral loading by virtue of their relatively low friction. Centering force capability is provided by separate mechanisms that may take the form of rubber springs, helical steel springs or even yielding mild steel dampers 2 7. A major advantage of these systems is their insensitivity to the frequency content of excitation, an apparent consequence of their frictional behaviour. Another property of sliding systems is the dependency on vertical load of the mobilized frictional forces. These forces tend to develop close to the centre of mass which in turn reduces adverse torsional motions. Applications of sliding isolation systems include a four-story structure in Japan s and a 50000 gallon fire water tank in California. In this paper we are concerned with the stochastic response of these practical sliding systems. Major concern in this study is the modelling of Teflon sliding bearings which exhibit a behaviour that is significantly different than that of Coulomb's theory. The coefficient of sliding friction of Teflon-steel interfaces depends strongly on velocity of sliding and bearing pressure 9-~2. Analytical solutions are presented for the following models of frictional behavior: (a) Based on a model of viscoplasticity theory which is capable of accounting for stick-slip conditions, and (b) based on the assumption of continuous sliding that appears to be a reasonable Paper accepted December 1989. Discussion closes August 1990.
© 1990Computational Mechanics Publications
assumption when considering the velocity dependence of the friction force. Both models account for the velocity dependence of friction coefficient. The derived analytical solutions are compared with results of Monte Carlo simulations and the degree of accuracy of each solution is established. Furthermore, the significant effect of velocity dependence is demonstrated by comparing results based on Coulomb's model of friction. MATHEMATICAL F O R M U L A T I O N The purpose of this study is to present an analytical method for the stochastic response analysis of practical sliding systems rather than to perform a parametric study. As such, the simplest structural model is used and excitation is limited to stationary and evolutionary white noise. Extension to more sophisticated models is simple and briefly discussed at the end. The system considered is a rigid mass, m, supported entirely by Teflon sliding bearings. Restoring force is provided by linear springs of total stiffness, K, and viscous damping constant, C. The equation of motion for the slip displacement x is:
Yc+ 2~oCOoJC+ogzx +F: = -5~o
(1)
F:= p(:~)mgZ
(2)
m
in which Oo is the natural frequency of the system and G0is the corresponding damping ratio (as computed from m, K and C) and F: is the mobilized frictional force./~(5:) is the coefficient of sliding friction which depends on the slip velocity, ~, and 5~o is the ground acceleration. Z is a variable taking values in the interval [ - 1 , 1]. This variable obeys the following constitutive equation which has been proposed by Wen 13 for modelling hysteretic behaviour: Y Z + TIYclZJZI"-1 + fl~IZl"- AYc=O
(3)
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 27
Stochastic response of practical s/iding isolation systems: M. C. Consmminou and A. S. Papageorgiou Mokha eta/. 9 - 1 2 have found that the model of frictional of equations (2) and (3) simulates well the frictional behaviour of sheet type Teflon bearings provided that A = 1 and r + °/= 1. For this combination of parameters the model of equations (2) and (3) collapses to a model of viscoplasticity!4o As such, this model will be referred to as the viscoplasticity model. Quantity Y represents the elastic deformation of Teflon prior to sliding which has been experimentally determined to be in the range of 0.005 to 0.02 in (0.12in to 0.5 ram) 9xx. Experimental results in Refs 9 to 12 suggest that the coefficient of stidlng friction of Teflon-steeI interfaces dose]y approximates the following equation: (4)
#(2)= fm~.- Df "exp(- al2})
in which f~,.. represents the maximum value of friction coefficient attained at large velocities of sliding arid ( f ~ , ~ - D f ) represents the value at essentially zero velocity. Table ! presents values of fr,~, Df and a for untitled sheet Teflon when sliding against stainless steel of surface roughness R~ = 0.03 gm ~''-~ under bearing pressure of 200 to 6500 psi (1.4 to 44.9 N/mmZ). Figure 1 shows the variation of the sliding coefficient of friction with velocity and pressure. The strong dependency on velocity and pressure is evident. The interested reader is referred to Ref. 9 for a physical explanation of this behaviour. A very important characteristic of Teflon-steel interfaces is the reduction of the mobilized frictional force as the velocity of sliding decreases (Fig. 1). Accordingly, as a structure sliding on these interfaces slows down less force is needed to maintain the motion. As a result of this behaviour motion continues with almost complete lack of stick-slip tendencies. This behaviour has been confirmed in shake table tests of model structure~ 5. Accordingiy, the assumption of continuous sliding in structures supported by Teflon bearings is well justified and is investigated in this study. In this case the equation of motion takes the simple form: ~" x-" + 2~0OoX + COoX+ # ( £ ) g sgn(2) = - 2~ •
2
(5)
where sgn(2) stands for the sign of 2.
VELOCITY
4
200 psi
0.15 -~
1000 psi ~U30.~O
"6,
o
Y Z + C 2 t k + K22Z=O in which 2¢ represents svolutionary white noise Coefficients C2~, C2~, K~2 and K22 are de~ermined b? minimizing the mean square difference between eq aafions {!) and (6) and {3) and (7). Wen ~ has reported expressions for coefficients C2: ova K22~ while C~ and K~ 2 take the following form: C ~~ =
2~o~ o + aDfgE[Z e x p ( - ai2 ) sgn(2)]
(8
in which E['] represents the expected value. For Gaussiar~ input the resoonse of the iinearized system {6, 7] is also Gaussian and coe~cients C~ and K~2 are evalua
~-/2~ ~-'=aDfg E[2Z] ®{B)
(10)
K I2=g[]<~.~- D) .~p(B )erfc(B)]
2 i/2 @(B) = 1 - 7r~"~Bexp(B2)edL'(B)
~i 3)
in which erfc is the complementary error funcJo~a and G stands for the standard deviation of the quantity in the subscript. The solution of the linearized stochastic system is obtained from the assoc,ated Lyapunov matrix differential equation for the zero time lag covaria~ce matrix S ~7. dS =PS+Spr+Q dt
{I~:~
in which P is given in the Appendix, S~-- El Y~Y~], Q u : 0 except Q3a =2~Oo I2(t} and YI =x, Y2O==Za~d Y3°=2- Go is the constant power spectra~ density (psd} of the two-sided white noise input and I(t) is the intensib function. The variance of the total acceleration ~ = 2 + x ~ is obtained from equation (1):
2000 psi 3000 psi 5500 psi
(5 Z 0.05
d~
1000 pmi= 6.9 N/rnm z 0.00
VELOCITY
in/see
Fig. 1. Sliding coefficient of friction of unfilled sheet Teflon in contact with stainless steel as fianction of pressure and sliding velocity
28
~s7
50
0.25 t U n f i l l e d Teflon Z ] Stainless Steel Ra=O.03 ~ m 0 i]2 Lt_
The method of stochastic tinearization '~ ~s empioyed m obtaining approximate statistics of the response of the slidi~g system. In the case of the viscoplasticity model equations (I) and (31 are replaced° respectiveiy, by:
cm/sec 410
0.20
STOCHASTIC LI[NEARiZAT~ON
Probabilistic Engineering Mechanics, t990, VoI. 5, No. J
9
2
2
2
in which the expected values of the bracketed expressions are given m the Appendix, [t should be noted that equation (15) is based on the nonlinear equation of motion rather than the linearized one. The method of equivalent linearization is Nso applied to equation (5) (continuous s~iding) resuking it: the
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. S. Papageorgiou following linearized equation: 2 + 2 ~e(_O0)C+ (_02X= -- 2 o
I ; =0,5 See ~, =0.05
(16)
Unfilled. 1 0 0 O p s i ( 6 .
where I
~,=¢o+
gEfmax--Df" ~(B)]
(17)
>-
"~ × ,o t3 .200psi(1.4
The second moments of x and 2 are obtained by solving the associated Lyapunov matrix differential equation. The variance of the total acceleration a = 2 + 2 o is obtained from the nonlinear equation (5):
o
tO
0.2 = 4~oO)oa; 2 2 2 + O)oa;, 4- 2 + 4~oe)gE[x2] + gZE[]22]
+ 4~oCoogE[p2 sgn(2)] + 2coZgE[#x sgn(2)]
(18)
I0 ~-
where the expected values of the bracketed expressions are given in the Appendix. C O M P A R I S O N TO M O N T E CARLO SOLUTIONS The accuracy of the derived approximate solutions is investigated by comparing to results of numerical simulation. Stationary and nonstationary responses are considered. Input is either stationary of psd GO or nonstationary of psd G O and intensity or modulating function l(t). Analytical stationary responses are obtained by solving the Lyapunov nonlinear matrix equation (14) with dS/dt set equal to zero. Numerical responses are obtained by creating a realization of stationary zero mean Gaussian process of psd G O(two-sided) and integrating numerically equations (1) through (4) up to 300 or 350 secs. Ergodicity is assumed and temporal averages are obtained over the last 2/3 of the duration of the response. Results are presented in terms of standard deviations of slip displacement, slip velocity and total acceleration, a~. a; and a., respectively, versus the normalized density D=(GoT3/rc3) 1/2 in which To is the natural period (To 2re/COo). Analytical nonstationary responses are obtained by solving equation (14). Numerical responses are obtained by generating 200 realizations of the input process and using ensemble averages. It should be noted that all numerical simulations (Monte Carlo solutions) are based on the solution of equations (1) to (44) which properly account for the stick-slip conditions. In these analyses a value of exponent t/=3 and Y=0.01 in (0.25 mm) were used. Both values are consistent with experimental observations9- ~2. For the numerical solution of equations (1) to (4) an adaptive integration procedure was used in which estimates of the local truncation error were kept within a prescribed tolerance. Figures 2 to 4 compare stationary responses produced by the two models (viscoplasticity and continuous sliding) to simulation results at various excitation levels. Data of friction of unfilled Teflon at a pressure of 200 and 1000 psi (1.4 and 6.9 N/mm 2) were used (see Table t) and To = 0.5, 1 and 2 sec while ~o = 0.05. The excitation is in the range of Go=3 to 100in2/sec 3 (19.4 to 645.2cm2/sec3). To get a perspective of the level of excitation we note that the Taft 1952 and E1 Centro 1940 motions could be represented by stationary white noise processes of Go=2.3 and =
N/ram z )
f:/ a
Simulation Viscoplosticity M o d e l Continuous Sliding
lo'
D/¥
;::°o:0%"°
/
Unfilled,1000psi(6,9
o
N/rnrnZ)
~
>b I
Unfilled.2OOpsi(l.4
N/ram ~ )
Simulation
0
Vlscoplosticlty
Model Continuous Sliding
10
10= D/Y
0.40 [o =0.5 S~C
to =0.05 0.30
///~/o
I 0,~_0 o
/
•
Unfitted,20 psi(l 4 N/ram a )
/ ~--~/"
0. I 0
.1000psi(6.9
......
0,00 I0
N/ram I
Simu]ahon Viscoplasticity Model Continuous Sliding I0 ~
O/Y
Fig. 2. Stationary response statistics of a sliding system with -I o = 0.5 sec Table 1. Constants f,..x, Df and a for unfilled sheet Teflon sliding against stainless steel o f R. = 0.03 pm Bearing pressure (N/mm2)
fmax
Df
a (sec/cm)
1.38 6.9 13.8 20.7 44.9
0.1900 0.I 193 0.0870 0.0703 0.0572
0.1200 0.0927 0.0695 0.0552 0.0485
0.236 0.236 0.236 0.315 0.197
7.5in2/sec 3 (14.8 and 48.4cm2/sec3), respectively is. A process with Go = 100 in2/sec 3 corresponds to a motion about three times stronger than E1 Centro •940. It is noted that the model of continuous sliding predicts
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 29
Stochastic response o f practical slidin9 isolation systems: M. C. Constandnou and A S. Papageorgiou 10
~o =0.05
period TO the resmnng force is weak and the system approaches a state of almost purely frictional behav~our. This is evident in the time needed for ~ to reach stationary in Figs 5 to 7. This dine increases as friction increases tsee differences between responses for 200 psi and 3000 psi). Deterministic evidence provided in Refs 9 and i2 suggests that sliding sys terns should be designed to exhibil this kind of behaviour. The resu!ts presented in Figs 5 to 7 further confirm the accuracy of the presented approximate solutions. Nonstationary responses for evdutionarv mpu,: are compared in Fig. 8. The input is modulated over an interva! of l0 sec by the intensity function shown in ~-he
o
To-1.0 Unfil~ed, l ~ec O00psi(6.9
~
)
~
. -
N/mrn~.
"
~/
>-
~× o ~D
o
&
.......
Simu{otion Viscoplosticlty Model Continuous Sliding ~
1
!0 ~
10
D/2Y T#--I.O
] r~ =2.0 sec $o =0.05
sec
/" //
r)
~ -0.05
Uefilled.lOO0psi(6.~
N/taro ~ )
-? >04 ,× b
~c o
.-
/
~
/'/£
~
,
~
(
~
.
S/m,~ ~)
~
/"
./-
Simulation
o
•
o
~
--
--
"
Simulation Vi~coplo~U¢itv Mo~el Continuous
~iiding
V~scoplastic~v Mode{
i
Slldin 9
Continuou~
......
!
0
10
o/sv
~0 ~ D/2Y [e ~
0.15
1
0.10
u . ~ i . ~ . 2 0 0 ~ i 0 4 N /~~
,
/"
] {o ~ 0 . 0 5
1.0 sec
.
o
r o ' 2 ' O sec
10:~-
0.20
~..
)
~
.2'
"//2
N/ram ~ )
Unliiled. l O O O u s t ( 5 . 9
.-~...~;...
ro
.....
]
_•
~X."" U n f i l t e d . 2 0 O u s i l ~ 4, N / r a m
o
~
0.05
_
~
o
/
}1
k'--- Unfilled- 1000us'( 6.9 N/mrnZ ~
o
m
......
Simulation Viscoplosticity Model Continuous Stidmg
\r~scoplQst;,ci~ Mo~e; Continuous
Simuia[ion
~idinq
10
0.00 ;0
o ~ - ......
!0 ;
~o '
D/6Y
D/2Y 0.20
-
Fig. 3. Stationary response statistics o f a sliding sysfem with T o = t sec
I o = 2 0 sec ~o :0.05
0 ~5 -, U~filled.200psq ? ~ N/ram z ) -
resonses in very good agreement with simulation results at a!l response tevels except in the low level tail and when friction is large (e.g., data for 1.4 N/ram z pressure}. For these conditions the viscoplasticity model provides accurate results (see standard deviation of velocity at low level excitation tail). This of course is a result of the ability of the viscoplasticity model to simulate stick-slip. It is very interesting to observe that the two models provide in all cases close upper and lower bounds to the simulation resMts. Figures 5 to 7 compare evolutionary responses for stationary input when the bearing pressure is 200, 1000, and 3000 psi (1.4, 6.9, 20.7 N/ram2), T o = 4 sec, 4o=0.05 and Go= 10 in2/sec 3 (64.5 cm2/sec3). For this value of
30
Probabilistic Engineering Mechanics, I990, VoL 5, No, ]
~
." -"
0[0
-
-
-
/
D
0 05
-
- - -o" - J
q
" - - -,~
/
~.;-"
..5~..-'~
/
. --o
%
.-J
dnh!le¢ 1000es~96.
0
~
I/
~,I/mma')
Sirnu!c~ior
V~scopl~s~iCi~y .......
300 --
Contir~uous -
,'Mode~ Siid~ng " ...... J
D/6Y
Fig. 4. Stationary response statistics o f a sliding system with T O = 2 sec
Stochastic response of practical sliding isolation systems." M. C. Constantinou and A. S. Papageorgiou 80 [ Te = 4 , 0 sec t o =0.05 G, =10 inys ) (64.5 c m ' / s ' )
approximation introduced in equation (22) of the Appendix. In this approximation Z was replaced by sgn(k) which is incorrect under conditions of free vibration in which the structure tends to stick to its foundation and Z attains values close to zero. In this case the approximation E[#xZ]~(fmax-Df)E[xZ ] for the free vibration part resulted in an acceleration response that closely followed the simulation results.
Unfilled Teflon, 200psi(l.4 N/ram a)
60
>~'~40 × b
COMPARISON TO COULOMB'S MODEL
20"
The presented solutions reduce to those of Coulomb friction with constant coefficient of friction p =fmax when
Simulation
o
10
o
TIME 1 O0
T0=4.0 sec . =0.05 c o =~o ~n%' ( 6 4 . 5
120
sec
T =4.0 sec Unfilled Teflon. 100Opsi(6.9 N/ram = ) ~** =0.05 0 ° =10 in~/s~ (64.5 cm~/s ' )
Unfilled Teflon, 200psi(1.4 N/ram =)
~m'IJ )
90
80 ¸
>.x b
60'
X b
60-
40
30
20-
Viscoplasticit 7 Model Continuous Sliding
. . . . . . . . . . . . .
-o
o-
..........
-~
Simulation Viscoplosticity Model Continuous Sliding
120-
~
to
T - 4 0 sec Unfilled Teflon. fO00psi(6.g N/ram = ) ~: =0.05 C 0 = I 0 inys ) (54.5 c m y s ~I)
sec
0.20 To =4.0 sec t o =0.05 c o = t o in~/,' ( 6 4 . 5 c m ~ s ' )
~
TIME s e c 10
TIME
;,
Unfilled Teflon. 200psi(1.4 N / m m =)
90
0.15
60 b
I 0.10
30 0.05
-o .o .
.
.
o.
.
.
.
.
.
Simulation Vlscoplasticlty Model Continuous Sliding
~
TIME
~
Viscopiosticity Model Continuous Sliding
0 10
0 TIME
0.00 0
o- Simulation
. . . . . . . . . . . . .
~
to
sec
O. lO To=4.0 sec ( , =0.05 ( ; , = 1 0 in?s ) (64~5 cm?s ) )
Fig. 5. Nonstationary response statistics of a system supported by Teflon bearings at 1.4N/mm 2 bearing pressure figure. The response is computed for an additional 5 secs of free vibration. Both approximate solutions provide accurate results in the range 0 to 10 sec. For the range beyond 10 secs (free vibration) the continuous sliding model fails as a result of numerical instabilities in the Lyapunov differential matrix equation (negative elements along the diagonal of matrix S in equation (14)). In contrast, the viscosity model provides accurate response predictions including the standard deviation of slip displacement during free vibration. However, the model failed to predict precisely zero standard deviation of accelerations after a long interval of free vibration (longer than that shown in Fig. 8). This is a result of the
sec
Unfilled Teflon.
t000psi(6.9 N/ram z
tffl I 0.05 0 b
"~' .........
0.00
TIME
~
Simulation Viscoplosticit Model Continuous ~liding
tO sec
Fig. 6. Nonstationary response statistics of a system supported by Teflon bearings at 6.9N/mm 2 bearing pressure
Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 31
Stochastic response of practical sliding isolation systems. M. C. Conseandnou and A. S. Papageorgiou ~80TT
-4.o
=o.o~
l,d
sec
the standard deviations and in the power spectra~ density (which reveals the frequency content} of the response. Figaro 9 s~ows the standard deviations of sli# dispiacement, a~, and totai acceleration. %, of a syster~ with To=4Sec, ~o=0.05 and whe~ subiected to stationary input of psd Go equat to 2.3 and 7.5 ine 'sacs (14.8 and 48.4cm 2 sect). As discussed earlier, these processes could be used in the representation of the i952 Taft and 1940 Et Centro motions° respective13. The sliding bearings either obey the taw of friction el equation (4} for 1000psi ~6.9N/mmB~ pressare oz CoulomVs !aw with coefficient of friction #=a~.=:~. ."Fh~ .. responses shown were computed by the viscop!asticity
Unfilled Teflon. 3 O O 0 p s i ( 2 0 . 7 N / r a m a )
[
120
~-. 90
t//-
-'"
50
-t~
~° 1
o}
q>
............. 4
0
Co=Ie
Continuous
8 TiME
180" /T r = 4 , 0 see ~ =O.05 W/s' (s4.5 150-
Simulolion
Viscopiasticity Mode~ Sliding
~0
8
sec
Unfilled
180 --
Teflon, 3 0 0 0 p s i ( 2 0 . 7
N / t u r f ~a
)I )
cm'l~~)
~c =4,0 S~C
[ehon,
Unhtlec
{~ = o . o 5 Go =10 i ~ / s ~ 6 4 5
5000os~ 2 0 7
N/,-nm ~ "i]
Simu~atio~ *~
leod
cm~/s a )
~50 -
120-
-~. 9o I >".×- .
I
. . . . . . . .
90
60 )
~JO 4
30"L ............. 0
2
4
1
g
10
6 TIME
OAO
Simulotion Viscoplos~icity Mode~ Con|inuous Sliding
r I i'vtE 180~
3000psi(2017
<:,-----~-
I
sac
UnfilledTeflon.
go=4,0 ~eC
t:
To=4"O see ~o = 0 0 5 C° = I 0 *n~S~ (64,5
J
N/turn ~ )
sec
Unfilled Teflon, 3 0 0 0 0 ~ t { 2 0 . 7
~,W~',#r ~Q
crn/s ~ B)
50
( o =0.05
7 120>-~. 9 0 ~ -x u)
I 0.05
g
6oj ~0~ -o-------o-
$imulotion Viscoplosti¢ity
Mode~ ........... Continuous Sfiding o.oo
o
. . . . .
5
~
-
--~----;
T~ME
•
-,
'
i
3
0
5
12
9 "f~E
5
sac
--7o
sac OAO
T o = 4 . 0 .~ec Untgied Teflon, 3000Dsi{20.7 o =0.05 G o = t O inYs (64.5 c m T s •
fig. 7. Nonstationary response statistics of a system supported by Teflon bearings at 20.T N/mm 2 bearing pressure
either Df = 0 or when B ~ co. The latter case corresponds to a~--+ eo. This should be expected since when the velocity of sliding is large, the coefficient of friction in Teflon bearings attains its maximum value (see equation (4) and Fig. (1)). Deterministic studies presented in Refs 9 and t2 have shown that the use of Coulomb's model with #=fm~x coutd result in useful estimates of the peak displacement and acceleration responses. However, the details of the responses in the two models of Coulomb friction and velocity dependent friction were substantially different. [n stochastic analysis this difference shoutd appear both in
32
Simufotion V~scopmstic~ty Modet Continuous Sliding
..........
f
Probabilistic Engineering Mechanics, 1990, Vot. 5, No. /
o.o~- ~
0
~
5
6
9
i-(ME
Fig. 8. input
N/turf, x ) ~
~2
5
sec
Nonstationary response sta¢istics for epoiutionar)
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. S. Papageorgiou ~00
-
T, =4.0 s e c ~o =0.05
0.010
--Untitled Teflon, lO00pSi . . . . . Coulomb• [/.= fmox
80
Tn =0.5 ~ec ~G** "0"05
Coutomio. # -
fmGx
-10 in~/,~ (64.5 c~/, 3)
G° = 7 , 5 in~/s j
0.008
0.006 I
Z ~1.~--0.004
b
_.---20
......
2.3
~ 0.002 0
.--'''" 2.3
""
o"
0
2
6
4
TIME
0.12
Te =4,0
8
10
sec
sec
-
~o =0.05
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Teflon,
G 75 n'/'_._-__._j__r_
.
0.08
_
_
n
2.3 7.5
.................................................
o~ I
.
•
i 2
,
,
.
i 4
,
,
'
6 '
FREQUENCY
.J ~- 0.010 0 1I, 0 0.008 LO Q k-
lO00psi Coulomb. /.Z= fmox Unfilled
-
.....
.
•
0
C)
0.10 .
~__j0.000 DJ
le
~0.5
'
'
'
8
'
'
fO
Hz
Unfilled. lO00pni (6.9 N/ram z)
~ec
-0.05
~'. -lo i.'/,'(64.5 =.%'3
0.006
<[ r~ LtJ
0.06
r7 0.004
g
"3 0 h
2.3
0.04 0.002
0.02 0.000 0.00
o
~
~
~ TIME
~
0
2
4
FREQUENCY
io
6
8
10
Hz
sec
Fig. 9. Comparison of responses predicted by proposed law of friction and Coulomb's theory
Fig. 10. Frequency content in motion predicted by proposed law of friction and Coulomb's theory
model. As expected, the two models predict substantially different responses. Coulomb's model underestimates the displacement response by a factor of 2 in the strongest motion and by a factor of 3 in the weakest motion. This example demonstrates the significance of the velocity dependence of friction in practical sliding isolation systems. While the results of Fig. 9 show substantial differences in the response statistics, they do not reveal the differences in frequency content. Frequency content is very important in structures housing sensitive equipment. A comparison of frequency content of the acceleration responses is presented in Fig. 10. Portions of the stationary acceleration response of a system with To = 0.5 secs were Fourier transformed and compared in this figure. Clearly, Coulomb's model predicts higher frequency content than a system obeying the law of friction of equation (4).
solutions is good in a wide range of system parameters and response levels. It is, furthermore, demonstrated that the stochastic response of the studied systems is substantially different than that of systems obeying Coulomb's law of friction. The extension of the presented analytical method to the case of correlated excitation is simple and briefly discussed in the Appendix. It should be noted, however, that the presented method has been verified only for the case of white noise input. ACKNOWLEDGEMENT Financial support for this study has been provided by the National Science Foundation grant CES-8857080. REFERENCES 1
CONCLUSIONS A method of analysis of practical sliding isolation systems under random excitation has been presented. These systems obey a law of friction in which the coefficient of friction exhibits a strong and nonlinear dependency on velocity. Comparisons of results with M o n t e Carlo solutions indicate that the accuracy of the presented
2
3 4
Kelly,J. M• Aseismic base isolation: review and bibliography, Soil Dynamics and Earthquake Engineering, 1986, 5, 202-216 Caspe, M. S. and Reinhorn, A. M. The earthquake barrier. A solution for adding ductility to otherwise brittle buildings, Proceedings, ATC-17 Seminar on Base Isolation and Passive Energy Dissipations, 1986, 331-342 Ikonomou, A. S. Alexisismon isolation engineering of nuclear power plants, Nuclear Engineering andDesion, 1985, 85,201-216 Mostaghel, N. and Khodaverdian, M. Dynamics of resilientfriction base isolator (R-FBI), Earthquake Engineering and Structural Dynamics, 1987, 15, 379-390
Probabilistic En#ineering Mechanics, 1990, Vol. 5, No. 1 33
Stochastic response of practical sliding isolation systems: M. C. Constantinou and A. So Papageorgiov~ 5 6 7 8 9
10 1i 12 I3 14
i5
14 15 16
Nagashima, I , Hisano, M., Kitazawa. K. and Kawamura, S. A base isolation system utilizing sliding bearings, Tasei Technical Research Report, 1988, 20, 71-87 (in Japanese) Watson Bowman Acme Corporation, Wabo--Fyfe earthquake protection system, Amherst, New York, t981 Zayas, V., Low, S. and Mahin, S. TEe FPS earthquake resisting system: experimentai report, Report No. UCB/EERC-87107, University of California, Berkeley, 1987 Kelly, J. M. Base isolation in Japan, 1988, Report No. UCB/EERC-88/20, University of California, Berkeley, 1988 Mokha, A., Constantinou, M. C. and Reinhorn, A. M. Teflon bearings in aseismic base isolation: experimental studies and mathematical modeling, Technical Report NCEER-88-0038, 1988 Constantinou, M. C. and Mokha, A. A model of friction of Teflon sliding bearings, Report to NSF, Dept of Civil Engineering, SUNY/Buffalo, 1989 Mokha, A., Constantinou, M. C. and Reinhorn, A. M. Teflon bearings in base isolation. Part i: Testing, Journalof Structural Engineering, ASCE, 1990, t16(ST2), 438-454 Constantinou, M. C., Mokha, A. and Reinhorn, A. M. Teflon bearings in base isoiafion. Part 2: Modeling, Journal qf Structural Engineering, ASCE, 1990, 116(ST2), 455-474 Wen, Y.-K. Equivalent linearizationfor hysteretic systems under random excitation, Journal of Applied Mechanics, ASME, 1980, 47, 150-154 Constantinou, M. C. and Adnane, M. A. Dynamics of soil-baseisolated structure systems: Evaluation of two models for yielding systems, Report to NSF, Dept of Civil Engineering, Drexel University, September 1987 Constantinou, M. C, Caccese, J. and Harris H. G. Frictional characteristics of Teflon-steel interfaces under dynamic conditions, Earthquake Engineering and Structural Dynamics, i987, 15, 751-759 Atalik, T. S. and Utku, S. Stochastic linearization of multi-degree-of-freedom nonlinear systems, Earthquake Engineering and Structural Dynamics, 1976, 4, 411-420 Lin, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill Book Company, New York, New York, 1967 Clough, R. W. and Penzien, J. Dynamics of Structures, McGraw-Hill Book Company, New York, New York, 1975.
"2i "
E[/~xZ] ~ E[#x sgn(:t)] \re/
F(B}-- - 2 B + 2=l;2(B~-+O.5}exp(Ba)e~jc{~) p=
E[~z]
24
G~ z
Expected values of bracketed expresswns in equatioe ( i8~
Ell/2]
2 ( ~ , ;2 exp,4B ¢ 2 )~Oc~2B) o g t =fmZ~×+,D)
exp(B2)erfd g ~,
-- 2 ( D f ) f ~
~26)
E[#x sgn(2)] is given by e q u a t i o n (22). i t s h o u l d be n o t e d tha~ the E[ffxZ] has n o t been e v a i u a t e d {it r e q m r e s the e v a l u a t i o n of a triple interval}, but r a t h e r a p p r o x i m a t e d by E[~x sgn{2)]. This is b a s e d on the o b s e r v a t i o n tb~a~ whi~e in sliding m o d e Z represents a contiw_mes a p p r o x i m a t i o n to the sgn(2) function. A c c o r d i n g l y , the a p p r o x i m a t i o n in e q u a t i o n (22) is valid H o w e v e r . when s t i c k - s i i p c o n d i t i o n s p r e v a i l the value of Z is c!oser ~o zero rather t h a n to + 1. F o r these c o n d i t i o n s the approxio m o t i o n in e q u a t i o n (22} i n t r o d u c e s e r r o r .
F o r filtered white noise excitation a shot no~se is passed t h r o u g h a filter of frequency coo a n d d a m p i n g factor ~;g. T h e statistical r e s p o n s e of ~he ~inearized viscoplasticity m o d e l is given by e q u a t i o n ~14l with
Matrix P
P=
0 0
0 -K22/Y
-cog
-K~
t
t
--C21/Y
(~9)
-ci; J
g
i
Expected values of bracketed expressions in equation (15) P= ~3 = fm~J~ + (1 - pb(Df) ~
x exp(B2)erfc(B)+,~/e(t_p~)F(B)
34
(23"~
Extensio;~ m filtered wh#e noise
APPENDIX
E[/z
~.,
(20)
Probabilistic Engineering Mechanics, 1990, VN. 5, No. d
©
0
0
-K12
-co8
vI
~ -Ctl
0
0 cog
1
0
0
0
0
\
0
0
0
-eo~2
0
0
i ()D
~,~co~ ! ~" -zCdv)~/
a n d S u = E [ Y ~ Yj], Y1 = Z . Yz =x, Y3=2, Y4=:xz, Ys = 2 r where x r is the r e s p o n s e of the filter. F u r t h e r m o r e . Qu = 9 except Q s s = nGo I2 (t) where G Ois p s d at z e r o frequency of the one-sided filtered white noise input.