Nonlinear system identification based on modelling of restoring force behaviour

Nonlinear system identification based on modelling of restoring force behaviour Arturo O. Cifuentes

The MacNeal-Schwendler Corporation, Los Angeles, CA, USA Wilfred D. lwan

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, C A 91125, USA This paper introduces a system identification algorithm based upon modelling of the restoring force behaviour of the structure. This algorithm is more efficient than traditional algorithms based upon matching the time history of response of the structure, since error evaluation does not require the solution of any differential equations. The effectiveness of this system identification approach, in coordination with a model for deteriorating structures is demonstrated by an example using actual earthquake data from the Bank of California building which was damaged during the 1971 San Fernando earthquake. INTRODUCTION In recent years a fairly large database of the earthquake response of both damaged and nondamaged structures has become available. This has resulted in a significant effort to apply system identification techniques to analyse these data. This effort is becoming very important for a number of reasons. First, system identification provides a means of calibrating different models for structural behaviour. Second, it permits a critical evaluation of whether a structure performed as designed. Perhaps most important, the knowledge gained through system identification makes it possible to provide more reliable estimates of structural response for future seismic events. Initially, system identification studies concentrated on the application of linear models. However, this class of models has frequently produced unsatisfactory results ~-3, since structures subjected to severe ground shaking may experience deformations beyond their elastic limit. Such structures exhibit nonlinear hysteretic behaviour which is often also associated with permanent stiffness reduction. Several models have been proposed to describe the restoring force behaviour of such structures ~'2'4~15. In a previous paper 2, the authors presented a Deteriorating Distributed Element model for the overall restoring force behaviour concrete structures. This model was shown to be capable of describing the major features of the restoring force while also retaining the inherent simplicitly of a physically motivated model. The present paper describes a new system identification algorithm that is suitable for use with this or similar models. The algorithm was used in the previous paper to illustrate the usefulness of the distributed element model, but was not described in any detail. The present paper provides the theoretical basis necessary for implementation of the algorithm and a numerical validation of its accuracy.

The algorithm is unique in that it is based on a matching of the restoring force behaviour of the structure and the model rather than on the customary matching of the time history. This results in avery substantial saving in computational effort required to perform the identification. A SIMPLE MODEL FOR REINFORCED CONCRETE STRUCTURES The Deteriorating Distributed Element (DDE) model 1'2 consists of an ensemble of linear springs and Coulomb slip elements arranged as shown in Fig. 1. The model is comprised of three types of elements: (1) a linear element, which is represented by a linear spring with stiffness Ke; (2) an elasto-plastic element, which consists of a linear spring with stiffness Kep in series with a slip element that allows a maximum force equal to Ke.Y~p, where Yepis the yielding displacement (see Fig. 2);~and (3) a set of N deteriorating elements, which consist of a 'breaking' linear spring with stiffness Ki in series with a slip element having a yielding force of K~Yi. The behaviour of the deteriorating element is similar to that of an elasto-plastic element, except that it 'fails' (i.e., the element no longer contributes to the restoring force) when the absolute value of the displacements exceeds a value flY~. The general model includes N deteriorating elements organized in ascending order of YiThe coefficient fl may be any number greater than 1. However, based on results obtained from several damaged structures, a value offl = 2 has been found to give satisfactory results for system identification purposes 1. It has also been found that it is acceptable to assume that each deteriorating element absorbs the same maximum energy. This leads to the relationship 1,

Accepted April 1987. Discussion closes July 1989. 9 Computational Mechanics Publications 1989

2

Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1

1 rT

K,=DT;-.-~.,

i=I ..... N

(1)

Nonlinear system identification based on modellin 9 of restoring force behaviour: A. O. Cifuentes and W. D. lwan

LINEAR ELEMENT

~p

~Yep ELASTOPLASTIC ELEMENT

KI

b Y~ K2

J 1
N DETERIORATING ELEMENTS

KN

b YN

Y1 < Y2 < " " " < YN

Fig. 1." The Deteriorating Distributed Element (DDE) model

force and p is the participation factor for the mode. In expressing the response by equation (2), it is assumed that the response can be separated into 'mode-like' contributions even though modes do not strictly exist for a nonlinear system. Although not rigorous mathematically, this approach is nevertheless useful. Considerable insight into the nature of a nonlinear system may be obtained from a knowledge of its generalized restoring force. By graphing f(x,k) as a function of x, one obtains the so-called restoring force diagram, which may be used to quantify the nonlinear behaviour exhibited by the system. Fig. 3 displays a typical modal restoring force diagram for the case of a reinforced concrete structure that has experienced strong shaking during an earthquake. This diagram corresponds to the roof response of the Imperial County Services Building during the 1979 Imperial Valley earthquake. Stiffness reduction with consecutive cycles of response is clearly apparent from this diagram and is associated with a progressive rotation of the hysteresis loops in a clockwise manner. In a nondeteriorating hysteretic structure, the 'orientation' of the hysteresis loops does not change and the end points of consecutive loops all lie along the same curve; the backbone curve. The 'effective' cyclic stiffness ofa hysteretic system for a given amplitude of oscillation may be defined as

f(xl, O)

Ke.r/--- - -

(3)

xi

where f(x~, 0) is the restoring force at a turning point xi which is either a local maximum or a local minimum of

I/%o', /

330.00

220.00

03 110.00

Fig. 2. Restoring force diagram of a typical elastoplastic element

~E

0.00

(..) rr

Normally, the Yi's may be chosen equally spaced over the displacement range of interest. Therefore, there are four free parameters to be identified; D, Ke, Kep and Ye~. SYSTEM RESPONSE

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03 LU rr'.

-220.00

It will be assumed that the structure in question responds primarily in a single mode and that the equation of motion may be expressed in the form -330.00 .

~(t) + f ( x , 5r = -- pE(t)

(2)

where x(t) is the relative displacement at some point in the structure measured from equilibrium, ;~(t) is the base excitation, f(x, 5c) is a mass normalized modal restoring

-24.00

J

-12.00

0.00

12.(~0

24.00

RELATIVE DISPLACEMENT (CM)

Fig. 3. Restoring force diagram for the Imperial County Services Building response

Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1 3

Nonlinear system identification based on modelling of restoring force behaviour: A. O. Cifuentes and IV. D. lwan the base is

x(t)=y(t)-pz(t)

In principle, the time history of the relative response of the structure can be determined by simple double integration and subtraction of the appropriate accelerograms. However, due to low frequency noise that is usually present in the data, this will result in unrealistic response behaviour. Since only a single mode model is sought in the present application, most of the undesirable noise can be eliminated by band-pass filtering the data 1. This allows isolation of the first mode response from the overall response of the structure. Consistent with the single-mode assumption, the equation of motion of the structure may be transformed to give the modal restoring force as

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<

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lO-

0.0 0.0

(4)

I

I

I

6.0

12.0

18.0

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Fig. 4. Effective stiffness diagram, hnperial County Services Building the response x (i.e., :~---0). By graphing Kefj- as a function of xi, it is possible to gain new insight into the nature of the dynamic system behaviour. In a nondeteriorating yielding system, the effective stiffness will return along the same backbone curve as the amplitude of oscillation decreases. Fig. 4 shows a typical effective stiffness diagram corresponding to the restoring force pattern depicted in Fig. 3. It will be noticed that the effective stiffness decreases monotonically as the amplitude of oscillation increases, corresponding to a form of 'softening" behaviour which is typical of yielding systems in which the slope of the backbone curve is a decreasing function of amplitude. However, in the example shown, when the response amplitude decreases, the effective stiffness does not follow the initial stiffness curve. This failure to recover stiffness is characteristic of deteriorating hysteretic systems. The difference between the initial and final stiffness, K o - K f , is a measure of the stiffness lost and is thereby related to the amount of deterioration suffered by the structure. IDENTIFICATION ALGORITHM Earthquake response data commonly consist of records corresponding to the absolute acceleration at two or more locations within the structure. Typically, at least one record is from the basement or ground level of the structure. Let y(t) be the absolute displacement at an upper level of the structure and let z(t) be the absolute displacement at the base of the structure. Then, the relative displacement x(t) of the structure with respect to

(5)

The accelerogram data provide j:(t) for a discrete set of points, q. Using these data, it is possible to generate a discrete representation of f(x, 50 at the times ti and thus obtain a discretized effective modal restoring force diagram for the structure. The idea of determining this diagram directly from response data was first suggested by Iemura and Jennings t6. Once the restoring force diagram has been obtained, the determination of the associated effective stiffness diagram may be carried out in a straightforward manner as indicated above. The identification algorithm presented herein is based upon matching the restoring force behaviour observed during the earthquake rather than the thne history of the response. It may be argued that if a model is able to reproduce the features of the restoring force, it should also provide a good estimate of the time history of the response. With this motivation, the modelling error is defined in terms of the restoring force diagram rather than the time history. The identification algorithm is described in detail below.

Selection of the parameters Yi Let x.~x be the maximum relative displacement experienced by the structure. Then, for a given value of N, a simple choice for the Y~is to assume equal spacing in the range zero to Xr,~x.From the assumption that fl = 2, it then follows that those deteriorating elements for which Xm~>2Y~ must fail during the response. Since the deteriorating elements in the DDE model have been arranged in ascending order of Y, this implies that the first J deteriorating elements must fail where J is the largest integer such that 2Yj
Determination of parameter D The net effective stiffness loss during the duration of structural response, K o - K f , can be estimated directly from the effective stiffness diagram. Invoking equation (1) yields s 1 K o - K f = 2=, D Y--~t (6) This allows-D to be estimated as

D = (K o - K f)

l/r~) i

4 Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1

(7)

Nonlillear s3,stem identification based on modelling of restoring force behaviour: A. O. Cifuentes and IV. D. Iwan function of the parameters K e, K~v and Yev" Thus, the following problem can be formulated to estimate the value of these parameters Mine = e(K~,K~v, Yev)

(lOa)

subject to

Ke>~O

+1

hj + 1

Fig. 5.

Typical hysteresis loop

Determillation of parameters I~, K~p and Y~ These parameters are chosen so that the restoring force diagram generated by the model matches the restoring force diagram obtained from the earthquake records with minimum error. Let the restoring force diagram corresponding to the actual structure be given by the discrete set of points (xi, f~) i = l , . . . , n . A typical hysteresis loop, for one cycle of oscillation will appear as shown in Fig. 5. The intersections of each hysteresis loop with the vertical axis of the restoring force diagram may be determined from the set (xi, fi) using linear interpolation. Let g~ and gi+ ~ denote the values of the restoring force at these points. Similarly, let hj and hj+ ~be the values of the restoring force for the maximum positive and negative displacements (turn around points) of the loop. Each hysteresis loop will then be characterized by the four points g~, gi+x, hi and h~+x. Let the restoring force diagram of the model for the same time history x(t) be characterized by the similar set of points g j,* g j+*1, h~' and h*+ i. Then, the error between the restoring force diagrams of the actual structure and the model will be defined in terms of the difference between the values hj and 9j, and h* and 9~. Thus, the error e is defined as

lhj-hyl+n j=l

I J-gYl

(8)

.i=1

where m is the number of points necessary to define all of the hysteresis loops of the restoring force diagram considered. The factor B is employed to homogenize the variance of both populations (h's and O'S). Hence, B is taken to be

h

B= j=l

g~

(9)

Ij=l

Note that the error as defined by equation (8) is only a

Kev>/O

Yev>/O

(lOb)

Equations (10a) and (10b) represent a standard nonlinear optimization problem, in which the optimum must be found by means of numerical techniques. Numerous approaches are available to solve such a problem. A detailed discussion of possible techniques which might be used is beyond the scope of this paper. For this purpose one can refer to the available literature 17-2~ In this study, the optimization problem is solved by means of a sequence of one dimensional minimizations, i.e., minimizing the error with respect to one variable at a time while keeping the other variables fixed. The procedure is continued until the error no longer decreases according to some established criterion. It is !mportant to notice that for the identification scheme proposed, the evaluation of the error does not reqllire the solution of ally differential eqtlation. It is only necessary to determine the set of points (xl, f * ) i= 1..... n i.e., the values of the restoring force fI' at each point xi. This provides a very marked computational advantage over traditional approaches which require that a new differential equation to be solved for each evaluation of the error. Accordingly, the proposed identification algorithm is much more efficient than its time history counterpart. VISCOUS DAMPING AND PARTICIPATION FACTOR The model for the restoring force presented herein accounts for energy dissipated through hysteretic behaviour only. It does not account for energy dissipated at very low amplitudes of oscillation, i.e., when x < Min { Yev, )'1}. Therefore, a viscous damping term Of the form e2 is also included. The viscous damping coefficient, c, is expected to be small; corresponds to 1 7o or 2 7o of critical damping based on the initial system stiffness. This is due to the fact that the large amplitude oscillations of the system are controlled primarily by the energy dissipated through hysteresis. Thus, the viscous damping coefficient only becomes important for small amplitude oscillations corresponding to the final portion of the record. The coefficient c is determined after all the other parameters of the model have been identified. For this purpose, two or three tentative values ofc are chosen, and the time history response of the model is computed. The value ofe selected is the one that gives the best fit for the low amplitude portion of the response. Since the system is nonlinear, the concept of normal modes as they are understood in the linear case is not strictly valid. However, one can define the participation factor b y analogy to the linear case. Thus, one may assume for the participation factor, p, the value corresponding to tile participation factor of the fundamental mode of a purely linear model having the initial stiffness of the structure. This value may be

Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1 5

Nonlinear system identification based on modelling of restoring force behaviour: A. O. Cifuentes and W. D. Iwan values of the optimal parameters of the model. The initial effective linear stiffness ofthe model is 21.95 s -2, while the final stiffness is 8.15 s -2. This difference between the initial and final effective stiffness corresponds to a stiffness decrease of approximately 60%. These values are in agreement with the value of K o - K s that can be

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Fig. 6. Bank of California Building, N I l E component. Restoring force diagram obtained from earthquake records

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0 w

LL U.

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w

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available from the structural analysis performed during the design of the building. Another possibility, if the mass distribution of the structure is known, is to use the relationship ~brM1 p = ~bTM~b

EXAMPLES O F A P P L I C A T I O N To illustrate the usefulness of the proposed identification algorithm, it is applied to the analysis of the response of the Bank of California building during the San Fernando earthquake of 1971. This building is a twelve storey reinforced concrete moment-resisting frame structure that suffered both structural and nonstructural damage during the earthquake 21'22. Previous investigations 3 have indicated that it is not possible to match the response of this building in either direction using linear models. Only by fitting different linear models to short time intervals, was McVerry 3 able to estimate the variation of the natural period of the structure during the earthquake. In so doing, it was found that the period in both directions 0N11E a n d N79W) increased by almost 60%. For the present study, the earthquake records were low-passed filtered using a cutofffrequency of 1 Hz. Figs 6 and 7 show the restoring force diagram and effective stiffness diagram obtained from the recorded data for the N11E component of response. A model consisting of nine deteriorating elements was considered adequate i n modelling the response. Table 1 shows the numerical

6

~

_

-

0.00

I

0.00

(11)

where M is the mass matrix of the structure and q~ is an assumed mode shape of the first mode of vibration.

i i

8.00

I

16.00

24.00

RELATIVE DISPLACEMENT

(CM)

Fig. 7. Bank of California Building, N I l E component. Effective stiffness diagram obtained from earthquake records Table 1. Numericalvalue of the parametersof the model Yl =

5cm

Ke4"75

1

}2= 8cm Y3=ll cm

Kep = 1.3

1 sec 2

Y4= 14cm Ys=17cm

Yet= 9.5 cm

Y6=20cm Y7= 23 cm

p=l.5

Ys=26cm

cn2 D = 202.9- -

Yg=29cm

Soil Dynamics and Earthquake Engineering, 1989, ~Vol. 8, No. 1

sec 2

1

viscous damping coeff.(c)=0.15-see

Nonlinear system identification based on modelling of restoring force behaviour: A. O. C~tentes and W. D. Iwan apparent. Fig. 9 compares the time history of the response recorded during the earthquake with that obtained using the DDE model. The agreement appears to be quite satisfactory. Fig. 10 shows the recorded absolute acceleration of the roof of the structure and the absolute acceleration predicted by the model. Although the overall approximation is good, the model fails to accurately reproduce the high frequency components that characterize the early part of the record. This is due to the fact that only a one-mode model is used. Nevertheless, the approximation improves as time progresses and the influence of the first mode becomes dominant.

estimated from the effective stiffness diagram (Fig. 7). The viscous damping coefficient used was equivalent to 1 ~o of critical damping based on the initial system stiffness. The restoring force diagram generated by the model is given in Fig. 8. The similarity to the restoring force diagram obtained from the recorded data (Fig. 6) is

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-300.00 0.00 -300.00

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-40.00

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Fig. 8. Bank of California Building, N I l E component. Restoring force diagram, given by the optimal DDE model I

16.00

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24.00

TIME (SECONDS)

Fig. 10. Bank of California Building, N I l E component. Time history of the absolute acceleration of the roof. (a) Actual time history. (b) Approximation given by DDE model

RELATIVE DISPLACEMENT (CM)

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Fig. 9. Bank of California Building, N I l E component. Comparison between the actual response (solid line) and the response given by the optimal DDE model (dashed line)

Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1 7

Nonlinear system identification based on modellin9 o f restorh19 force behaviour: A. O. Cifuentes and IV. D. hvan For the relative displacement time history, the singlemode model captures most of the essential features of the response throughout the record. If one is interested in obtaining a better approximation for the acceleration time history, it is possible to decompose the response of the system into several 'mode-like contributions' that include the effects of higher 'modes' of response. By filtering the data with band-pass filters, it is possible to isolate the response of each 'mode' arid determine a corresponding model. The contributions of these models can be combined to model the overall response of the structure.

REFERENCES 1 2

3

4

5 C O N C L U D I N G REMARKS It is concluded that the system identification algorithm introduced herein is an effective means for determining the optimal values of the parameters o f a hysteretic model using actual earthquake data. This algorithm, which is based upon matching the restoring force behaviour of the structure rather than the time history of the response, has significant advantages from a computational viewpoint. This is due to the fact that no differential equations need be'solved to evaluate the modelling error. In this regard the algorithm differs substantially from traditional algorithms in which each new evaluation of the error requires a new integration of the differential equation of motion. Furthermore, the Deteriorating Distributed Element model appears to be capable of capturing many of the most important features of the response of reinforced concrete frame buildings subjected to severe ground shaking. When used with actual data, the predictions of the relative displacement time history of response made by the model match very well those recorded during the earthquake event. The model and the system identification algorithm presented herein have been used to obtain an approximation of the response of a structure based on a single mode only. However, the approach indicated is still valid if used to approximate the higher modes of response. The major difference will be that the band pass filter will be centred about a higher mode of response so as to isolate the contribution of that particular mode. In summary, it is believed that the system identification algorithm presented herein constitutes a useful tool for studying the performance of buildings during actual earthquakes. Using this tool one can assess the damage suffered by a structure and obtain direct information regarding its future performance should another earthquake occur. This information could be vital to the assessment of the safety of a structure following a major earthquake.

6 7 8 9 10 I1 12 13

14

15

16

17 18 19 20 21

ACKNOWLED GEMENTS This research was sponsored by a grant from the US National Science Foundation. The opinions expressed are those of the authors and do not necessarily reflect those of the Foundation.

8

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Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 1

Cifuentes, A. System Identification of Hysteretic Structures, PhD Thesis, California Institute of Technology, Pasadena, California, September 1984 Iwan, W. D. and Cifuentes, A. O. A Model for Sysiem Identification of Degrading Structures, International Journal of Earthquake Engineeringand Structural Dynamics, 1986, 14, 877890 McVerry,G. H. Frequency Domain Identificationof Structural Models from Earthquake Records, Earthquake Engineering Research Laboratory Report No. EERL 79-02, California Institute of Technology, Pasadena, California, October 1979 Clough, R. W. and Johnston, S. B. Effect of Stiffness Degradation on Earthquake Ductility Requirements, ProceedhTgs,Japan Earthquake Engineering S)~nposium, Tokyo, Japan, October 1966, pp. 227-232 Saiidi,M. Influenceof Hysteresis Models on Calculated Seismic Response of R/C Frames, Proceeding of the Fifth Iibrld Conference on Earthquake Enghwerhlg, Rome, Italy, 1973, 5, 423--430 Iwan,W. D. A Distributed-Element Modelfor Hysteresisand its Steady-State DynamicResponse,Journal of Applied Mechanics, Trans. ASME, Voi.88, Series E, 33(4), 893-900 Iwan,W. D. On a Class of Models for the Yielding Behaviorof Continuous and Composite Systems, Journal of Applied Mechanics, September 1967, 34(3), 612~517 Takeda, T., Sozen, M. A. and Nielsen, N. N. Reinforced Concrete Responseto Simulated Earthquake, ASCE, Journal of the Structural D&ision, December 1970, 96, 2557-2573 Toussi,S. and Yao, J. Identification of Hysteretic Behavior for Existing Structures, Report CE-STR-80-19, School of Civil Engineering, Purdue University, December 1980 Toussi,S. and Yao, J. Hysteretic Identification of Multi-Storey Buildings, Report CE-STR-81-15,School of Civil Engineering, Purdue University, May 1981 Masri,S. F. and Caughey,T. K. A Nonparametric Identification Technique for Nonlinear DynamicProblems,Journal of Applied Mechanics, June 1979, 46, 433-447 Wen, Y. K. Method for Random Vibration of Hysteretic Systems, ASCE, Journal of the Engineerin9Division, April 1976, 102, 249-263 Saiidi,M. and Sozen, M. A. A Naive Model for Nonlinear Response of Reinforced Concrete Buildings, Proceedingsof the 7th Iibr/d Cotference on Earthquake Enghleering, Istanbul, Turkey, 1980, 7, 8-14 Muguruma,M., Tominaga, M. and Watanabe, F. Response Analysis of Reinforced Concrete Structures Under Seismic Forces, Proceedingsof the 5th Iibrld Conferenceon Earthquake Enghwerin9, Rome, Italy, 1974, 1, 1389-1392 Aoyama,H. Simple Nonlinear Models for the Seismic Response of Reinforced Concrete Buildings, Proceedings of the Review Afeetin9 US-Japan CooperativeResearch Programin Earthquake Engineering, Tokyo, Japan, 1976, pp. 291-309 Iemura,H. and Jennings, P. C. Hysteretic Responseof a NineStory Reinforced Concrete Building, International Journal of Earthquake Engineering and Structural Dynamics, 1974, 3, 183201 Gallagher, R. H. and Zienkiewicz, O. C. (Ed.), Opthnum Structural Design, Theory and Applications, John Wiley and Sons, 1973 Rosen,J. B. (Ed.), Nonlinear Programming, Academic Press, 1970 Fletcher,R. Practical Methods of Opthnization, John Wileyand Sons, 1980 Luenberger, D. G. Introduction to Lhzear and Nonlinear Programming, Addison-Wesley, 1973 Blume,J. A. Bank of California, 15250Ventura Boulevard, Los Angeles in San Fernando, California, Earthquake of February 9, 1971, (Ed. L. M. Murphy), Vol. I, Part A, 327-357, US Dept. of Commerce, National Oceanicand AtmosphericAdministration (NOAA), Washington DC, 1973 Foutch,D. A., Housner, G. W. and Jennings, P. C. Dynamic Response of Six Multistorey Buildings During the San Fernando Earthquake, Earthquake Engineering Research Laboratory Report No. EERL 75-02, California Institute of Technology, Pasadena, California, October 1975