Response of frictional base isolation systems to horizontal-vertical

Response of frictional base isolation systems to horizontal-vertical random earthquake excitations Lin Su and Goodarz Ahmadi

Department of Mechanical and Industrial Enyineering, Clarkson University, Potsdam, N Y 13676, USA Responses of a rigid structure with a frictional base isolation system subjected to random horizontal-vertical earthquake excitations are studied. The ground accelerations are modelled by segments of stationary and nonstationary Gaussian white noise and filtered white noise processes. The technique of nonstationary equivalent linearization is used for the response analysis. The differential equation governing the covariance matrix is solved and for a range of values of parameters, the root-mean-square displacement and velocity responses of the structure are calculated. The results are compared with those obtained by a series of Monte-Carlo digital simulations and reasonable agreement is observed. For several values of parameters, responses of the structure to the accelerograms of the North-South and vertical components of El Centro 1940 earthquake are studied. Particular attentions are given to the effects of vertical excitation and the elastic stiffness of the isolation system. It is observed that vertical excitation amplifies the horizontal slip of the structure to an extent. The amount of amplification under various conditions are evaluated and discussed. It is also shown that the slip displacement decreases sharply as the elastic restoring force of the isolator increases.

1. I N T R O D U C T I O N Isolating the structure from the destructive earthquake ground motion as a mean of aseismic design has attracted considerable interest in recent years. A number of isolator designs ranging from rubber bearings, roller bearings and frictional types have been developed. Excellent reviews on the subject were provided by Kelly 1'2. In particular, using frictional foundation as a possible earthquake isolation systems was described in Refs 1 8. As a simple model for frictional base isolation systems, the slip of a rigid mass driven by the Coulomb friction force from a randomly moving foundation was studied by Crandall et al. 9, Ahmad? °, Constantinou and Tadjbakhsh ~1, S u e t al. 12, Lin and Tadjbakhsh 13 and Su and Ahmadi TM. In Refs 911, the stationary white noise model of Housner 15 and Bycroft 16 and the filtered white noise model of Kanai ~7 and Tajimi TM were used for seismic excitations. Earthquake ground motions, however, are by nature nonstationary 19'2° and therefore cannot be realistically represented by stationary models. In Refs 12 and 14, nonstationary random models for earthquake excitation were considered, the moments of the Fokker-Planck equation were used and the nonstationary performances of certain frictional isolation system were statistically analysed. One shortcoming of a purely frictional isolation system is that the slip of structure on its foundation could become unacceptably large. Recently, Mostaghe121 23 developed a frictional isolation scheme which is composed of a stack of several sliding plates with a central rubber core. The plates offer frictional resistance to motion while the rubber introduces certain elastic restoring force. This new base isolator is expected to © 1988 Computational Mechanics Publications 12 Probabilistic Enyineering Mechanics, 1988, Vol. 3, No. 1

reduce the structural slip to an acceptable level while effectively limiting the transmission of the ground acceleration to the superstructure. The effects of vertical ground acceleration in amplifying the horizontal deflection of the structures were studied by Lin and Shih 2426, Ahmadi and Mostaghe127, Ahmadi 2s'29 and Orabi and Ahmadi 3°. The conclusions of these studies were that except in certain extreme cases the vertical motion effects are small and at most secondary. The effects of vertical ground excitation on the performance of base isolators, however, have not been well understood and only a few preliminary studies in this direction appeared in the literature 13.14. In this work, the dynamic responses of a rigid structure with a frictional base isolation system subjected to horizontal-vertical random earthquake excitations are studied. Both the Coulomb frictional foundation and Mostaghel's resilient -friction base isolator are considered. Stationary and nonstationary, white, as well as, filtered white noise models of earthquake ground acceleration are used. The nonstationary equivalent linearization method 1°'31 33 are utilized for response analysis. For a filtered white noise excitation, the filter is treated as an additional governing equation as suggested in Ref. 31. Thus, the augmented response state vector becomes a vector Markov process and the moments of the associated Fokker-Planck equation may be conveniently formulated. The resulting set of differential equations for various moments are numerically solved and the rootmean-square displacement and velocity responses are evaluated. For several values of friction coefficient and elastic restoring force, the transient and nonstationary responses are obtained and discussed.

Response of frictional base isolation systems: Lin Su and G. Ahmadi

Monte-Carlo digital simulations are also performed. Ensembles of five hundred samples of earthquake excitations are generated and the response time histories of the structure are evaluated. Ensemble averaging is used and the response statistics of interest are determined. The simulation results are compared with those obtained from the statistical linearization scheme. The agreement is generally reasonable, although some discrepancies for the purely frictional system are observed. For several values of friction coefficient and elastic restoring force, responses of the structure to the accelerograms of the NS and vertical components of El Centro 1940 earthquake are analysed. Attentions are given to the effects of the elasticity of the isolation system and the vertical excitation. The results obtained from the statistical and deterministic analyses show that the presence of elastic restoring force of the rubber in the resilient-friction base isolator would reduce the peak slip of the structure to a significant extent. It is also observed that the vertical excitation somewhat amplifies the horizontal slip of the structure. The amount of amplification under various conditions are evaluated and discussed. 2. F O R M U L A T I O N Fig. 1 shows a sketch of a rigid structure of mass m with a resilient-friction base isolator subjected to horizontal and vertical earthquake excitations. Assuming that the structure is always in contact with its foundation, the equation of motion in terms of the horizontal relative displacement x is given as 3~+/~(g +//o) sgn(k) + co2x = - iia

(1)

Here, ii9 and i59 are the horizontal and vertical ground accelerations, k is the slip velocity, ~ is the friction coefficient, and g is the acceleration of gravity, o0 is the natural frequency of the isolation system and is directly proportional to its elastic restoring force 21 23. When ~ o = 0 , the isolator reduces to a simple Coulomb frictional foundation. Equation (1) is valid whenever the structure is sliding with respect to its foundation. When the structure is sticking to its foundation, no slip occurs as long as the ground acceleration ]//9] remains less than /~(g+/59). As soon as I//ol becomes greater than la(g+b'9), slip will occur and equation (1) applies. Note also that an additional linear viscous damping due to the internal friction of the rubber may be presented in equation (1). However, in order to better understand the effects of friction force at small /~, this additional damping is neglected in the present analysis. 5~1

_1

Equation (1) may be replaced by its equivalent linear system given as 5~+ P(9 +/~0)7(t)5c+ coZx= -//0

(2)

where 7 is proportional to the equivalent damping coefficient. The error in replacing equation (1) by equation (2) is given by error = 7(t):~- sgn(~)

(3)

Minimizing the mean-square error with respect to 7, it follows that , . (~ sgn(~)} 7(tl= ~

(4)

where the angular bracket, ( }, stands for the expected value (ensemble average). Using a Gaussian statistics, it follows that

•/

2

(5)

where it is assumed that (~} = 0. Equation (5) shows that 7(0 is time dependent and involves the yet to be determined (~2). When excitation is a Gaussian process, it is well known that the response of a nonlinear system is, generally, a non-Gaussian process. However, when the nonlinear system is replaced by an equivalent linear system the response becomes Gaussian. This latter results, in part, justifies the use of Gaussian closure techniques in estimating the equivalent parameters 31 33. Here, the equivalent linear system as given by equation (2) has a random parametric excitation which renders the response of the linearized system to be still non-Gaussian. However, in the absence of any other convenient, reliable and accurate technique the Gaussian closure method is used. The accuracy of this approximation will be assessed in the subsequent sections. Note also that this assumption was previously used by Shih and Lin 26 in response analysis of parametrically excited hysteretic structures.

3. RESPONSE ANALYSIS In this section, sliding of the rigid structure with a resilient-friction base isolator subjected to random horizontal and vertical earthquake excitations is analysed. The limit of a purely frictional base isolation system is also included in the analysis. The cases that the earthquake ground accelerations are modelled as shaped white noise and filtered white noise processes are discussed in the sequel. White noise excitations In this case, the horizontal and the vertical earthquake ground accelerations are modelled as segments of shaped Gaussian white noise (WN) processes, i.e., we assume that

zo

~

Foundation

Fig. 1. Model of the structure with a frictional base isolation system

iio=e(t)wl(t),

i/o=e(t)w2(t)

(6)

where e(t) is a deterministic envelope function and w l(t) and w2(t) are Gaussian white noise processes with the

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 1

13

Response of frictional base isolation systems: Lin Su and G. Ahmadi following statistics:

(w~ (t)) = (w,_(t)) = o

(7)

(W 1(t 1)W1(t2)) = 2rrSo(~(tI -- t2)

(8)

(w2(tl)wz(t2)) = 2rcS~f(t l - t2)

(9)

(wl(t1)w2(t2))=2uSlz6(t I - t 2 )

(I0)

Here, 6( ) is the Dirac delta function, S o, S~ and S~2 are the constant power spectra of the WN processes. Equation (2) may now be restated as ~'=Ay+BW

(ll)

with W=~w2(t) )

The explicit set of moment equations corresponding to equation (18) are listed in Appendix A. This deterministic system of equations may be solved by a variety of numerical techniques. The dependence of the equivalent parameter 7(t) on ~22~ =Q22 as given by equation (5) presents no difficulty in the numerical scheme of solution.

(12)

Filtered white noise excitations Now, consider the case when the horizontal earthquake ground acceleration is modelled as a filtered white noise (FWN) process. A FWN model is believed to provide a better representation of the ground motion. The vertical ground acceleration is still modelled by a WN process. That is, we assume iio= - (2~oo)o2f +~o2oxf)e(t),

b'o=e(t)Wz(t )

(20)

with and

-~I + 2~0~°02~+ (°2xl = w l (t)

o A=(_o3(~ _lug),) ' B=(_ o1

_~72) e(t)

(13)

Note that, in this particular case, the so-called Wong and Zakai 34 correction terms to the drift vector are identically zero. Response of system (11) is a Markov process. The associated Fokker-Planck equation for the transition probability density function f(y, t ]Yo, to) is thus given as

~- =/~1 ~2 -

j=l

?~Yi(-AijyJ)+~(Cijf) ~)icyj

(14)

where matrix C is given by S12

where x I is the response of the filter and wl(t) and wz(t) are white noise processes with their statistics given by equations (77(10), o9o is the predominant ground frequency, and -~ois the ground damping. Equations (20) and (21) imply that when e(t)= 1,//q will have a KanaiTajimi 1v.18 power spectrum given as 1 + 4~2(~o/'o90) 2 Sac(co)= So (1 - ((o/oog)2)2 + 4~0c or2 2/cos2

X y=

and the only nonzero element of C is (16)

A,,

+

0

Oy l )

-~y2e(t)

(23)

2f 0 A=

= j=l

0)

2 xf

The general moment equation associated with equation (14) is given as

dt

(22)

where S o is the power spectrum of the bedrock white excitation. Equations (2), (20) and (21) may be restated as the system given by equation (11) with

7~

C22=l~(So-)t- 2]gy2S12-+-]~1272228v)e2(t )

(21 )

1

--O)2o -ktg7 0 0

0 0

0

0

~oZe(t) 2~o~%e(t) 0 _ ~,~2. y

1

(24)

_

(17) where h is an arbitrary but twice differentiable function of y;. To obtain the equations governing the second-order moments let h=y~yj in equation (17) and the result expressed in a matrix form becomes O~= AQ + (AQ) r + G

(18)

where Q is the covariance matrix with Qu= ( y y j ) , and G is a square matrix whose all elements are zero except G22 which is given by G22 = 2(~So + 2pzs~)eZ(t)

(19)

Here, the fact that ( x ) = ( 2 ) = 0 is used in arriving at equation (19). The precedure for formulating the moment equations given by equation (18) is similar to that used in Refs 24, 30, 33 and 35. It is interesting to note that the cross-spectrum constant S~2 does not appear in the resulting second moment equations.

14

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 1

and W given by equation (12). Therefore, the augmented response vector y is a Markov process. The transition probability density function f now satisfies the FokkerPlanck equation given by equation (14) with A given by equation (24). The only nonzero elements of matrix C are C22 =/l:72pZe2

(t)22Sv,

C24 = C,2 = - rcYpe(t)2S12 C44 = 7tSo

(25)

The 4 x 4 response covariance matrix Q then satisfies equation (18) with G being a square matrix whose only nonzero elements are G22 = 41~2e2(t)Sv,

G,4 = 2~So

(26)

Note that in deriving equation (26) the fact that ( x ) = ( 2 ) = 0 was used. As a result, $12 does not appear in equation (18). To determine the transient and nonstationary second-order response statistics of the structure, it is sufficient to solve the moment equations

Response of frictional base isolation systems: Lin Su and G. Ahmadi given by equation (18) with zero initial conditions. The explicit system of second moment equations to be solved are given in Appendix B. It should be pointed out that the assumption that the vertical acceleration here is a white noise process is necessary for obtaining the closed system of moment equations given by equation (18) (and also in Appendix B). Should /Sg be a nonwhite process, an unclosed hierarchy of moment equations would results whose solution requires additional approximation.

given by (x2(t)) =

l Orc3S 3t

/~4g4 ,

as t---* ~

(27)

In Ref. 9, a (non-Gaussian) linearization scheme using exact stationary probability density function of the slip velocity was employed and the following asymptotic expression for the mean-square response was obtained:

8n3S3t (x2(t)) = p494 ,

as t---~ oo

(28)

4. STATISTICAL RESULTS The system of moment equations given by equation (18) is solved by using a second order Runge-Kutta scheme. Stationary, nonstationary, white and filtered white noise models of earthquake excitations are considered. For the white noise models So=55.44cmZ/sec 3 and Sv=~S o corresponding to the intensities of the NS and vertical components of E1 Centro 1940 earthquake are used in the analysis. Different values of Sv are also considered in some examples to study the effects of variations of the intensity of the vertical acceleration. For the filtered white noise horizontal earthquake excitation as given by equations (20) and (21), a ground frequency of coo= 15.5 rad/sec, a ground damping of (g = 0.42 and the same power spectral intensities are used. These are the recommended values for the site conditions of E1 Centro 1940 earthquake. To assess the accuracy of the results, several MonteCarlo digital simulations are also performed. Segments of stationary and shaped Gaussian white noise processes are generated and the response displacement and velocity time histories are calculated. Ensembles of five hundred samples are used to estimate the statistics of interest. In the digital simulation, /~g is generated as a white noise process which is independent from the horizontal acceleration. In other words, $12=0, is assumed. However, that is not at the expense of generality of the results, since St2 does not affect the second moment equations for zero-mean responses. The simulation results are compared with those obtained from the linearization method in the following section. Transient analysis When e(t) is a unit step function, the seismic excitations are modelled as stationary white and filtered white noise processes. A few sample results of transient root-meansquare (RMS) responses for WN and FWN earthquake excitations are shown in Figs 2-5. Several values of friction coefficient and too are used in the analysis. For comparisons, the transient RMS responses as obtained from the digital simulations are shown in Figs 2 and 5 by the dotted lines. From these figures it is observed that the simulation results are somewhat higher than the theoretical predictions and the observed differences increase as too decreases. In particular, Fig. 2 shows that the deviation is quite noticeably for a purely frictional system (i.e., too=0)- Similar discrepancies between the theoretical (Gaussian linearization) and simulation RMS responses were noticed in Refs 11 and 12. Based on the exact solution for the probability density function of the slip velocity obtained by Caughey and Dienes 36, Crandall et al. 9 provided an asymptotic exact solution for the mean-square slip displacement which is

These asymptotic RMS slip displacement are also shown by solid lines in Fig. 2 for comparison. It is observed that the exact asymptotic solution given by (27) is in excellent agreement with the digital simulation results. The agreement is surprisingly very good even for small t. The approximate solution given by (28), though superior to the Gaussian linearization result, underestimates the exact and simulation RMS responses by about ten percent. These results suggest that the deviations between the RMS responses obtained by the present Gaussian linearization scheme and digital simulation are due to the contributions of the non-Gaussian statistics which are lost in the present linearization scheme. In Refs 37-39 it was also found that the non-Gaussian statistics will increase the RMS responses. In this sense, the current Gaussian linearization scheme may provide a lower bound to the exact RMS values. For practical resilient-friction base isolator, too is in the range of n/2 to n rad/sec (i.e., a natural period of 2 to 4 sec). Fig. 5 shows that for these values the predictions of the Gaussian linearization scheme used in the present analysis are in reasonable agreement with the digital simulation results. The overall performance of the base isolator under various conditions may be observed from Figs ~ 5 . These results show that the RMS responses decrease rapidly as/~ increases. From Figs 2 and 3, it is observed that the vertical acceleration amplifies the horizontal slip of the structure to an extent. For a small friction coefficient, the vertical effect is rather small and increases as # increases. The amount of amplification is proportional to the intensity of vertical excitation. Fig. 3 shows when the intensity of vertical excitation is about three times that of ~=0.3 0.~-

1 I

°.2i ~