Response of sliding structures with restoring force to stochastic excitation

Response of sliding structures with restoring force to stochastic excitation A. S. Papageorgiou

Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, New York, 12180-3590, USA M. C. Constantinou

Department of Civil Engineering, State University of New York at Buffalo, Buffalo, New York, 14260, USA The stochastic response of a rigid structure with a restoring force resting on a frictional foundation and excited by stationary or evolutionary white noise is studied analytically and by Monte Carlo simulation. The degree of accuracy and range of applicability of the following three analytical methods is established: (1) Equivalent Linearization Method, (2) stationary solution of the Fokker-Planck equation and (3) direct closed form stochastic linearization of a viscoplasticity model. It is demonstrated that the Viscoplasticity Model is a very accurate and effective tool for studying sliding base isolation systems for all levels of excitation and any value of the coefficient of friction/~. However, the Equivalent Linearization Method also gives accurate predictions even for relatively low levels of excitation when the coefficient of friction is less than 0.1.

INTRODUCTION There are two approaches in confronting the earthquake problem. One is the traditional approach of strengthening the structure so that the structural system absorbs the earthquake forces through inelastic action. By allowing inelastic deformations - which are usually accompanied by some structural damage - the energy dissipation capacity of the structure increases and its fundamental period lengthens. The other approach, which has been developed in the last few years, is the use of isolation systems to uncouple the structure from the damaging horizontal components of ground motion by a mechanism that provides flexibility and energy absorption capacity. The basic feature of base isolation systems is that they restrict large deformations to special components of the system while the structure vibrates almost as a rigid body. Isolation systems that have either found application in modern structures or are at a stage of development that promises application in the future are classified as elastomeric bearing systems, sliding systems, sliding systems with restoring force and other systems (for a review see Kelly ~ and Mokha et al.2'3'4). Analytical studies with results that are pertinent to sliding systems have been made by Caughey and Dienes 5, Crandall et a/. 6'7, Ahmadi 8 and Constantinou and Tadjbakhsh 9 among others. In this paper we will be concerned with modelling dynamic response of sliding base isolation systems with restoring force. Six such systems have been proposed so far, all utilizing Teflon-steel interfaces for carrying the vertical load. These are the Barrier System, Alexsismon,

Taisei or TASS, Resilient-Friction Base Isolation System (R-FBI), Wabo-Fyfe Earthquake Protection System, and the Friction Pendulum System (Mokha et al.2'3). The system under study consists of a rigid mass which is attached to a spring and a dashpot and rests on an interface that exhibits Coulomb-type friction. The rigid mass represents the superstructure which to a first approximation responds to earthquake excitation as a rigid body. The system is subjected to support excitation which is represented by stationary or evolutionary white noise. We consider Coulomb-type friction despite experimental evidence that the coefficient of friction of Teflon-steel interfaces exhibits a significant dependence on the sliding velocity, bearing pressure and condition of interface 2-4'1°. The effect of such dependencies of the coefficient of friction on the dynamic response of a rigid structure are addressed in an accompanying paper. We compare the results of three methods of analysis: (1) Equivalent Linearization (E-L), (2) stationary solution of the associated Fokker-Planck (F-P) equation and (3) direct closed form stochastic linearization of a Viscoplasticity Model which was originally proposed by Boucl 1 and subsequently extended and used by Wen ~2,13. For the first two methods of analysis it is assumed that the time intervals over which sticking occurs are of such short duration that the probability of sticking is negligible. This assumption is not necessary for the third method of analysis. The degree of accuracy of each of the above three analytical methods is established by extensive Monte Carlo simulations which are based on the equation of motion with conditions of separation and reattachment. MATHEMATICAL M O D E L L I N G

Paper accepted December 1989. Discussion closes August 1990.

© 1990Computational Mechanics Publications

The system under study consists of a rigid structure of

Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 19

Response qf sliding structures whh reswring force to stochastic excitation: A. S. Papageorgiou and ML C mass m attached to a spring and dashpot and in contact with a moving base. The horizontal force Ff between the structure and the base exhibits Coulomb friction characteristics. When sliding occurs, the equation of motion in terms of the slip displacement x is 2

+

2~ocoo 2

+ coZox+ iF~ FI =

....

or~x~a~u~wz~

the following Lyapunov matrix differential equad:on for the zero time tag covariance matrix S, d --S=PS+SPr+~ dt

(7)

where

- 2~

where the frictional force Fz is given by the expression F f = # m g sgn(2)

(2)

The evaluation of the complete history of motion requires the use of conditions for separations and reattachment which take the following form. The structure remains stuck to its foundation as long as

When condition (3) fails, the structure slides and equations (1) and (2) apply. Reattachment occurs when the velocity, 2, becomes zero and condition (3) is satisfied. if condition (3) is not satisfied at a point of zero velocity, the structure remains in the sliding phase. In the above equation the dot (") stands for time derivative, ~o is the damping ratio and % is the natural circular frequency of the system. The above model is referred to as the Coulomb model. In order to solve the problem stated above for stationary or evolutionary white noise input 2~ we adopted the following three different approaches: StochasHc equivalent linearization For an approximate solution to the problem, we apiy the method of equivalent linearization (e.g. Lin14). We start by assuming that the intensity of the base acceleration is high enough so that the intervals in which sticking occurs are of such short duration that probability of sticking is negligibie and therefore equation (t) is always valid. Furthermore, we assume that an approximate solution of equation (1) can be obtained from the 'equivalent' linear equation

The elements of matrices S and B are given by which represents an evolutionary white noise exc~tadon with psd S~{co~=G o. Equation (7) is nonlinear since the matrix P ~s a function of ~ which in turn is a Nnction of the unknown standard deviation of the slip velocity a~ (see equation {6)}, Utilizing the symmetry of matrix S, equation (7', may be written as a system of three first-order nonlinear differential equations. Numerical solutions can be obtained by adaptive integration techniques utilizing the stiff methods of GeapS° For stationary white noise excitatiom the variances of the approximate response q aantities x and k are given by (e.g. Lin~). 2

7~

Z~C0o and ~r~= -2-~~o

{iOI

Go

From equations (6L t9), and {I0) we obtain

r2

~

-~

L~r#2gz + 8~°co°~zG°~_A ~-2-bL~ ! /;~ G:

4~ocoo

-

and

(4) By minimizing (with respect to {, and %) the mean-square error introduced in replacing equation (t) by equation (4), and observing that for a Gaussian, stationary zero-mean input process 2g the approximate displacement and velocity x and 2 respectively are also Gaussian zero-mean processes approaching stationarity for a stable system, we obtain co~ = coo

7rG O

~-

(!2~

For ~o=0 equations (1i ! and {12) reduce ~o

L2A

~.la!

(5)

and i

~

(6)

The tinearized governing equation (4) may be written as a system of two first-order differential equations and subsequently, application of standard methods of analysis of linear systems (e.g. Bryson and Ho 15, Lint4), leads to

20

Probabilistic Engineering Mechanics, 1990, VoL 5, No. i

From equation (ita) we observe that for {o=0 the stationary value of the standard deviation ~ of the slip velocity is independent of the natural circular frequency coo of the system. Stationary solution of the Fokker-Planck equano~ Assuming stationary response, the associated Eokker-

Response of sliding structures with restoringforce to stochastic excitation." A. S. Papageorgiou and M. C. Constantinou Planck equation takes the form (14)

2

0

[g(x, 2 ) f , . i ( x , 2)]

fx.k(x,

~2 -

f X,s,(x, 2)=0

(13)

where g(x, 2) = 2~oCOo2+ cogx+ ~ sgn(2) and fx.•(x, 2) is the joint probability density function of the displacement x and slip velocity k. Following Matsushima 17, an approximate solution of equation (13), when 4o=0, is given by

~292(D0 ( (D2#292X2 x~ fx,k(x, 2)-4w/~zc2G 2 exp

~

sticking (elastic behaviour), the absolute value of Z is less than ufiity: Y represents the elastic displacement prior to sliding (yield displacement), fl, 7 and A are dimensionless constants, and the parameter t/ is an integer which controls the smoothness of the transition from elastic to plastic response. The conditions of separation and reattachment are accounted for by equation (19). Constantinou and Adnane 19 have shown that when A = 1 and fl+ ? = 1, the model of equation (19) collapses to a model of viscoplasticity that was proposed by Ozdemir 2°. Constantinou and Mokha 21 have, furthermore, shown that under motion without reversal and provided that A = 1 and f l + ? = 1, the solution of equation (19) is given by

j V(2) (14)

Z = (-- 1)l/"y(t) (20)

with

f dy

fexp( )

y--C5+1= ( -

if 2 > 0 1

(15a, b)

if 2 < 0 J

1)l/.(x/y)

Explicit solution is possibly only for t/= 1 or 2 for which

Z= 1 - e x p ( - x/Y)

when r/= 1

Z = tan(x/Y)

when q = 2

(21) From equations (14) and (15a, b) we can easily obtain the standard deviations of slip x and slip velocity 2. a~ = x/2. rcG° /tgco0

(16)

a~ = x//2 •7cG° P~

(17)

The expression of or; given above agrees with the expression derived by Constantinou et al.18 who solved the Fokker-Planck equation for a system of sliding mass with no restoring force (i.e. co0=0 ). This agreement derives from the fact that o; is independent of c0o, as pointed out above.

Viscoplasticity model The solutions presented above are based on the assumption that the occurrence of sticking is negligible, that the mass slides continuously and therefore equation (1) is always valid. In order to investigate the range of validity of this assumption we solved the problem using a third method which makes use of a model that accounts for sticking and sliding by itself. This model is based on principles of the theory of viscoplasticity and it will be referred to as the Viscoplasticity Model. The governing equation of motion of the mass is again equation (1), but now the frictional force is given by

FI=#mgZ

(18)

where Z is a hysteretic dimensionless quantity that may be regarded as a continuous approximation to the unit step function and which obeys the following constitutive equation

Y2+yl2]ZIZI '-~ +fi2lZl'-A2=O

(19)

During sliding (yielding), Z takes values -t- 1 and during

whereas for t/ larger than 2 numerical evaluation of equation (20) results in solutions for Z with sharper variation than those given by equation (21). It is clear that a large value oft/and/or a small value of Ymay result in an almost rigid-plastic behaviour as required by the Coulomb model. The value of Y used in this study is 0.01inches (0.25mm) which is consistent with the experimental observations of Mokha et al.2"3"4 who studied the behaviour of Teflon sliding bearings. This value together with I/= 3 and A = 1, fl= 0.1 and 7 = 0.9 produces hysteresis loops for sliding bearings that are in fair agreement with experimental results 2'4. Furthermore, time histories of the response of sliding systems obtained by this model compare very well with results obtained by Coulomb's model 2. One such case illustrated by Fig. 1, is discussed further in the sequel. It should be noted that the selection of the parameters of the viscoplasticity model is based only on analytical considerations which are supported by experimental observations. They have not been selected to obtain a good fit with the simulation results. The exact solution of equations (1), (18) and (19) is very difficult to obtain. An approximate solution can be obtained by equivalent linearization. Equation (19) is replaced by

YZ + Ce±+ KeZ = 0

(22)

where C e and K e are given by Wen 13 for integer odd values of the parameter t/. From the linearized equations (1), (18), and (22) one can easily obtain the corresponding Lyapunov covariance matrix differential equation whose form is given by equation (7). However, in this case the matrix P takes the form

P=

1

o1

2flo~o

- COo 2

--Ce

--K~

(23)

Probabilistic Engineering Mechanics, 1990, Vol. 5, No. 1 21

Response of sliding structures with restoring force to stochastic exc~t~_tion: Ao S, Papageorgiou and M. Co ~" - ~ and y r = (x, 2, z) and F r = (0, - 2 0 , 0). Exploiting again the symmetry of the zero-lag covariance matrix S, the L y a p u n o v matrix equation can be written as a system of six first-order differential equations which are nonlinear because C~ and K , are functions of cr~ and c~=.Numerical solutions can be obtained by adaptive integration as described earlier. RESULTS We started by demonstrating that the Viscoplasticity Model (equations (1), (18), and (19)), compares welt with Coulomb's Model (equations (1) and (2)). F o r this, we considered a system with (~o = 0, ~o = 0 (Joe., pure friction, no restoring force) and p = 0 . 1 or 0.i5, and we integrated the governing equations of each model numerically. As input motion we considered a realization of stationary white noise with psd Go = 25 in2/sec 3. F o r the viscoplasticity model we considered Y=0.01 in and the values i and 3 for the parameter ~. The computed responses (slip) are shown in Fig. 1. We observe that the agreement of the two models is very good. The agreement could be further improved by reducing the value of the yield displacement I7. Also the differences between the responses corresponding to the two different values of the parameter ~ of the Viscoplasticity Model are insignificant. Therefore, for the rest of the study we used Y= 0.0i in and ~ = 3.

]

5

q

}4=0.10

Co=25in a / s ~

0

2

~

6

8

1'0

l ....... n=l:---n=3: Viscoptastieity Modeh }a=0.15 --~ Coulomb's Model

2-

-1-

-

t

2-

0

,

½

, - - r -

4

,

"-

6

,

8

[

i0

"

We nex~ proceeded to compare ~he three methods of solution for stationary input, in Figs 2So b. the normalized stavonary values of o'x and cr~ are plotted ~erscs normalized intensity D = ( G ~~d.;z~ ~ 3 ' a~, ,- of the input, for # = 0o t 5o ~o = 0.0 and natural period To = 0.5 or 2.0 see. t n the same figure we ptm a~so the resmts of the Monte Carlo simulations. The statistics of the simulations were obtained by integrating numerically equations (1!. ,?_g} and (191 up to 300 sec. and considering temporal averages (assuming ergodicity) over the last 2/3 of the durat!on of the response. It should be noted that in the simulation the conditions of separation and reattachment are accounted for properly by equauon (19L From Figs ha. b we observe that the results obtained from EoL Method underprejdict the results of the F - P Equation :~y a factor o~ 2 . . ~" as expected from equations ~11~, ~szj~ a~,,d ;, ~ , ~ ~ c predictions of the Viscoplasticity Model are cona~.,_zen~. ...... higher as compared to the results of both EoL method avid F - P Equation for all levels of excitationo The .~su~r~ ~',*~ o~;t~.~a~ simulations are bracketed by the above ;ave extremes. A similar comparison is presented in Figs 3a~ b but n o w for three different values of the coefficient of friction ~=0.05o 0.!0 amd 0.!51 and for nonzero {o From these stationary analyses the main conclusion is that ~he viscop!asticity model nerforms very well when friction ~s large, the natural period To of the system is large and tae excitation is weak. This is to be expected because under the physical conditions stated above, significant sticking ~s taking ptace which is accounted properly by the mode! On the other hand for strong motion and small fr'~ctio~. the E~L Method - which is based on the assumption of continuous sliding - gives good results It should be pointed out that for a wide range of values of input intensity, the differences in the values of ~ pr~d~c~.~c[ by the various methods of analysis are considerably less t~an a factor of 2. The same applies a!so for ff~ except for vet;; weak motions and high coefficient of friction ~ > 0 ° ! L i~ which case the E-L Method underpredicts results by ,_~ factor less than 3. We concluded our studies by considering the :.espouse statisucs of the system under study when it is subjected te stationary (Fig. 4) or evolutionary (Fig. 5) wh~_te ~.oise The Monte Carlo simutatio~ results are based on an ensemble size of 200 realizations. The ~ s p o n s e of the system approaches stadonar~ty unlike that of the system with oure friction and no restoring lbrce wn c:~a never becomes stationary 9. It ~s remarkable how fast ~; reaches ~ts stationary vaiue [much less than i sec) in Fig. 4..~n all cases, the results of the Viscoptasticity model compare better to the simulation results than those of the E-L model. Concluding, we notice that [he sma~ter ~he intensity of the base acceleration for a given value of #. the longer i~ takes to reach stationarity. This phenomenon is associated with the retative importance of the frictional force and the restoring force. F o r weak excitatin-c the ,orce reduces while the frictional force remains restoring ~ unchanged As such. the weaker the excitation ~s the closer the system resembles a purely frictional system.

-[!ME s e c

Fig. ]. Comparison of the Viscoplasticity Model (equations (1), (18) and (t9)) with Coulomb's Model (equations (1), (2) and (3)). The system considered has no restoring force (o%=0, ~o=0). The input moEon is a realization of stationary white noise with psd G o = 25 inZ/sec 3

22

Probabilistic Engineering Mechanics, ]990, VoL 5, No. 1

CONCLUSION Summarizing, we have established that the Viscoplasticity Mode1 is a very accurate and effective tool for studying sliding base isolation systems that obey Coutomb~s law of friction for at1 levels of excitation and any value of the coefficient of friction #. As a matter of fact.

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30 ¸

60

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60 ¸

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Fig. 4. Nonstationary response to stationary input when the system starts f r o m rest

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°X

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30 1 To=2.0 sec to =0,05 t~=0.15 GO=5 in~/s t

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Response o f sliding structures with restoring force ~o s~ochastic excitation: A. S, Papageorgiou and M, C. Co~.~s¢am&ou REFERENCES

I 0 ~ g=020 g o =20 in~s ~

4

1

)--\ 2

2

B- 10

3

c -05 ×

4 5

//" /¢¢" 0

~ -2

~imuJo,lon Anolytic(~l

! 6 6

4

8

~0

T~ME sec

2,

7

I ta=0.20 C,, =20 ~ny~~

8 9 10

f

-x t

I1

/

12 #

I

.....

0

2

Analytical "' ~ Simulation Lineorizo~ion Continuous Sfiding, Equiv.

4

6

8

10

TtME s e c

Fig. 5. Nonstationary response to nonstafionary inpu~ (intensity funcffon shown in figure)

~,3 i4 15 16 I7

t h e V i s c o p l a s t i c i t y M o d e l - w h i c h c a n b e r e g a r d e d as a continuous approximation to Coulomb's friction iaw provides a more realistic description of the experimentally observed frictional behaviour of Teflon-steel interfaces than Coulomb's law. The Equivalent Linearization Method gives accurate predictions even for relatively tow levels o f e x c i t a t i o n w h e n t h e c o e f f i c i e n t o f f r i c t i o n p is less t h a n 0.1. F i n a l l y , t h e m e t h o d of analysis of the Viscoplasticity Model and the E-L Method can easily be e x t e n d e d for t h e c a s e o f f i l t e r e d w h i t e n o i s e w i t h s p e c t r a l content fitted to represent earthquake sources '-''~.

Ministry of Construction. Japan. Research Repor~ No. ~,2, J~i~' 18

19

20 21 22

ACKNOWLEDGEMENT This work was partially supported by NSF grant CES 88-57080 and by the National Center for Earthquake Engineering Research under grant NCEER-88-1501.

26

Probabilistic Engineering Mechanics, 1990, Voi. 5, No. 1

Kelly, J. M. Aseismic base isoiation: review and bibhography Soil Dynamics and Earthquake Engineerino, 1986,. 5. 202--2i6 Mokha, A, Constantinou, M. C. and Reinhorn A. M Te~o~ bearings in aseismic base isolation: experimentai studies and mathematical modeling, Technical Report. NCEgR-88-C~938, 1988 Mokhm A- Constantinou. M. C. and Reinhorm A. M Tef]o~ bearings m base isolation. Part l : Testing, & Nfructura] E~?yng ~kSCE. 1990, it6{2/. 438-454 Constantinou. M. C.. Mokha, A. and Reinhorn. A. M "1ef~or_~ bearings in base isolation. Part 2: Modeling, Z S~rucluraLEi~qnF ASCE, 1990, 116{2L 455-474 Caughey, T. K. and Dienes_ J. K. Analysis of ~oniinear first-order system with a white noise input, Journal qf Applied Physics. 1961, 32{I 1L 2476-2~79 CrandatI, S. H., Lee. S. S. and Williams, J. H. Accumulated, slip of a friction-contro!!ed mass excited by earthquake m~v,~ns, 7ournaI of Applied Mechanics, ASME. ]974, 4L !094 1©98 CrandatL S. H. and Lee S. S. Biaxiat slip cf a mass o:~ foundation subject ~o eartMaake mo~ion° ]ngenwuroArehir, 1~76. 45 361-370 Ahmadi. G. Stochastic earthquake response of s~ruc~ures vn sliding foundatiom lmernationai Journal of Engineenny; 5cie~:ce, i983.2I(2t° 93-102 Constanti~ou. M. C. and Tadjbakhsh. I, G. Response oi a sliding structure to filtered random excitation. Jour~a[ o[ Str~cru~uf Mechanics~ 1984, 1213L 401-4!8 Constantinou. M. C., Caecese, J. and Harns~ H. O Fnc~;,o~al characteristics of Teflon-steel in~eLdhces unde,~ dynamic conditions~ Earthquake Engineering and S~ruem:,'of Dynamics I987 15, 75!-759 Bouc_ R. Modele Mathematique d'Hysteresiso Aeusti-:'~, !~971 24, i6-25 Wen. Y.-K. Method of random vibration of hysteretic systems, Journal of Engineering Mechanics Divisiom ASCF, 1976. [02{EM2L 249-263 Wen Y.°K. Equivalent linearization for hysteretic systems uneer random excitation° Journal oJ Applied Mechanics. ASME t980. 47. i50-154 Lm. Y. K. Probabilis~ic Theory oj S~ruc~urat Dve~um,,es. McGraw-Hill Book Co.. New York. N.Y., !967 Bryson. A. E. and Ha. Y.-C. Applied Opfimai Cc~,~ro~. Hemisphere Publishing Co,, Washington, !975 Gear. C. W. Numerical ~nitiat-Value Problems m Or4hvJr: Differential Equations, Prentice-Hallo Englewooa C!i~s. New Jersey, t971 Ma~sushima. Y. Random response of a single degree of fieedom system with general slip hysteresis, BuEdi~g Research ]~s,,dru~e

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i979 Constantinou, M. C., Gazetas, G. and Tadjbakhsh. i. Stochastic seismic sliding of rigid mass supported through non-symmetric friction. Earthquake Engineering and &ruc~uratDynamics. t984, 12. 777-793 Censtantinou, M. C. and Adnane, M. ~o Dynamics el soilobase-isoiated structure systems: evaluation of two models t~r yielding systems, Report go ,VSF, Dept of Civil Engineering, Drexel University, Sept. 1987 Ozdemir, H. Nonlinear transient oynamic ana~;,sis of yielding structures. Ph.D. Thesis, University of Caiifornia. Berkeiey, i976 Constantinou, M. C. and Mokha, A. A model of friction of Tenon sliding bearings, Repor'~ ~o ~SFo Dep~ of Civi~ Engineering, State University of New York, Buffalo, A~rit !999 Lin. B. C., Tadjbakhsh, I. G., Papageorgiou, A, S. and Ahmadi. G. Response of base-isolated buildings to random excitations described by the C!ough-Pentien spectral mode1, Earthquake En~i.neermg and Struc~ura! Dynamics, [989. ~g. 49-62 Lin. B. C., Tadjbakhsh. L G.. Papageorgiou. A, S. and Anmadi G Performance of earthquake isolation systems, Jaur;Tai q/ Engineering Mechanics. ASCE 1990. H6(2), 446-461