seismic design of secondary systems

REVIEW PAPER' Seismic design of secondary systems

M. P. Singh

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA

The ever increasing demand of providing accurate yet practical solution to the problem of seismic analysis of secondary system in industrial units has led to the development of some very novel analytical techniques. The paper describes the evolution of the methods which have been used to analyse the secondary systems in the past two decades. The developments starting with the direct generation of floor response spectra up to the recent introduction of the cross floor response spectra as the seismic inputs for the analysis of multiply supported secondary systems are discussed. Lately, significant research efforts have also been directed to incorporate the effect of the dynamic interaction between the primary and secondary systems in their analyses. The latest developments in this area, utilizing the component mode synthesis approaches, are also described.

Key Words: seismic response, secondary systems, piping analysis, vibrations, random vibrations.

INTRODUCTION

In industrial and power generation facilities, the term secondary system is commonly used to describe the nonstructural mechanical, electrical and auxiliary subsystems attached to the walls and floors of a main structure. As these systems are usually enclosed in a main building, they are not subjected to environmental loads like wind, snow, and ambient temperature loads. They, however, have to be designed for their own operating loads and other extreme accidental loads likely to be encountered in the facility. Besides these loads, they could also be subjected to the vibratory effects of an earthquake induced ground motion which are transmitted to these subsystems through their supporting primary structures. Obviously, the vibratory effect felt by a secondary system during an earthquake is greatly influenced by the characteristic of the supporting primary structure. For proper seismic design of the secondary systems it is, therefore, necessary to consider the input filtering effect of the primary structures. This paper addresses this problem.

The secondary systems can be classified into several categories. For the purpose of analysis, its common to divide them into systems with single or multiple supports. A single support system could be a simple equipment with one degree of freedom or a more complex system with several degrees of freedom. The multiply supported secondary systems, on the other hand, are usually characterized as multi-degree-of-freedom systems, with possibly different input motions at the supports. The piping systems in industrial and power generation systems are the systems of the latter type. A common

characteristic of the secondary system is that they are usually light compared to their supporting structure, though in some cases they could also be heavy. See Figs 1 and 2. This characteristic is often used in devising simple analytical techniques for calculating their seismic response, as discussed later.

The response analysis of secondary systems for earthquake induced ground motions is a very interesting and challenging problem both from the practical as well as analytical standpoints. This problem has, thus, attracted attention of many researchers in the last two decades. The fact that these systems are relatively light

Suppor ts ~ ~.f~ Piping S t r u c t u r e

..... .%:-.:i:;.... .... ~--

Fig. I. Primary and multiply supported light secondary systems

© Computational Mechanics Publications 1988 Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 3 151

Seismic design of secondary systems: M. P. Singh

primar Str cture : mSe °ndarY ................... rSo0po ts

<

Fig. 2. Primary and multiply supported heavy secondary systems

compared to their supporting primary structures makes it difficult to prepare a combine analytical model which can be analysed with reasonable accuracy to obtain the response. Furthermore, as the multiply supported secondary systems such as pipings generally have large degrees of freedom, their combination with the primary structure also increases the size of the problem significantly and thus making it unwieldly and cumbersome to solve. The challenge, therefore, lies in developing rational and accurate analytical procedures in which it is not necessary to prepare a combined analytical model of the primary and secondary systems.

As the secondary systems are light, it is commonly assumed that they do not affect the response of their supporting structure. This implies that they can be considered decoupled for analytical purposes; and they they are only assumed to add to the mass and not stiffness of the primary structure. It is then only necessary to analyse the primary structure without any regard to the presence of the secondary structure. The motion of the primary structure where the secondary system is attached then defines the input to the latter system. That is, the secondary and primary systems are assumed to be in cascade for transmission of the ground motion. Although there are some problems associated with the assumption of decoupled analysis in some special applications, it is widely accepted in practice. In fact, the most commonly used analytical approaches in the industry make this assumption now. In the author's opinion also, this assumption is quite justified and valid in a majority of applications.

FLOOR RESPONSE SPECTRA

In the early sixties, the problem of defining design input for secondary systems in nuclear power plants was of great practical and research interest. For a given ground motion time history at the base of the primary structure, it was no problem to define the time history of the motion of the primary structure at the support of the secondary system as accurate analytical procedures were then available to conduct a step-by-step time history analysis. A time history analysis is, however, cumbersome and computationally expensive. As one could not rely on the

results from a single time history and thus must consider an ensemble of ground motion time histories in the analysis, it was found impractical to use the time history analysis for design purposes.

For the design of primary structure, it has been a common practice to define the design motion by smoothed ground response spectra, such as those prescribed by Nuclear Regulatory Commission 1'2 which supposedly consider an ensemble of possible ground motions at a site in their development. It was, therefore, of interest to obtain the design input for the secondary systems also in the form of response spectrum curves. Such spectrum curves are called as the floor response spectra. It is obviously necessary that these floor spectra inputs to the secondary systems be consistent with the input spectra used for the supporting structure.

A common approach at that time was to obtain a single ground acceleration time history (often called as the spectrum consistent time history) whose spectra closely enveloped the prescribed ground response spectra, and then use this time history in the analysis of primary structure to obtain the time history at the support of the secondary system. Several methods have been developed to obtain the spectrum consistent time histories 3 13. The response spectra developed for the floor acceleration time history then provided the desired floor response spectra. It was, however, observed 14'1s that such an approach could give different floor response spectra if different spectrum consistent time histories were used in the analysis. Obviously, such floor spectra generated for a single time history could not be used with confidence for the design of a secondary system. Again, to obtain floor response spectra suitable for design, one must consider an ensemble of the spectrum-consistent time histories in the analysis. As this is a costly approach, research efforts were aimed to obtain the floor response spectra directly from the ground response spectra.

The approaches which obtain floor response spectra from the ground response spectra directly without generating any intermediary input in terms of time histories or other forms are called as the 'direct' approaches. In the author's opinion, the first such method for direct generation of floor response spectra was developed by Biggs and Roesset 16 and Biggs 17. Later, a similar approach was also presented by Kapur and Shao 18. These approaches were based on the deterministic principles of the amplification of motion as it passes through the primary structure to the secondary system. However, as the design ground response spectra supposedly represent the ensemble of the possible ground motions that can occur at a site, the development of floor response spectra also must recognize and incorporate the probabilistic concepts. This motivation led the author 19 to develop a direct floor spectra generation approach based on the random vibration analysis of the primary structure subjected to stochastically characterized ground motions. This approach was further extended and improved by the author 2° to cover the case of a tuned oscillator. The approach utilizes the dynamic characteristics of the primary structure such as mode shapes, natural frequencies and participation factors, the dynamic characterstics of the oscillator representing the secondary system and of course the ground response spectra directly without generating any intermediate input such as a spectrum consistent time history or

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

spectral density function. Other extensions to this approach have also been developed in Ref. 21 to apply it to the nonclassically damped primary structure 22, and in Ref. 23 and 24 for the structures with dominant high frequency modes where mode acceleration approach may be more effective. Several other direct approaches, Refs 25-30, have also been developed with similar or somewhat different ideas.

To remove the nonuniqueness associated with the results obtained by a single spectrum consistent time history, the use of a spectrum consistent spectral density function as the design input was also suggested by Singh et al. 14°15 for the calculation of the primary structure response and generation of floor response spectra. To obtain a spectral density function consistent with a prescribed spectrum, the following equation is required to be solved:

P O.(o)lHp(o0)l 2 do--- R~(o0) (1) 0o

where ~o(o~) is the imput motion spectral density function consistent with response spectrum Rp(oo),/-/p(~Oo) is the complex frequency response function for an oscillator with frequency o~ 0 and damping ratio/3o- Py is the peak factor by which the mean square response is multiplied to obtain the response spectrum value. Practical methods6,3~.32 have been developed to solve equation (1) for the spectral density function. This spectral density function can then be used to calculate the secondary system response 15'33 36.

In the calculation of spectral density function it is implicitly assumed that the ground motions are the functions of a stationary random process. Some simplifying assumptions are also made in calculating PI in equation (1), as a precise solution of the first passage problem 25'37 which is required to define the peak factor is not known yet. As a result of these approximations, as well as because of some internal inconsistency between the prescribed response spectra for various damping ratios, different spectral density functions are obtained for different damping spectra. Which of these several spectral density functions is then the right one to define the ground motions? Often this question is conservatively circumvented by choosing the envelope of all the spectral density functions for various damping ratios as the seismic design input.

As the assumptions made in calculating a spectrum consistent spectral density function from ground response spectra and using such a spectral density function in the calculation of primary and secondary system response are similar to those made in the development of direct floor response spectra generation approach, the writer sees no special advantage in adopting the spectrum consistent spectral density function approach except, maybe, for the case of generation of the tertiary floor spectra from the secondary floor response spectra.

STRUCTURE EQUIPMENT INTERACTION EFFECT ON FLOOR SPECTRA

The earlier floor spectra generation procedures developed by the author and others were based on a cascaded analysis of the primary and secondary systems and thus ignored the effect of the feed-back or dynamic interaction


between the oscillator and supporting structure. In some cases, however, this interaction may not be entirely negligible 39'4°. It may modify the response of the supporting structure which in turn may affect the response of the oscillator and thus the floor response spectrum value. It specially happens when the oscillator is tuned to a dominant frequency of the supporting structure. As in the case of a classical tuned-mass-damper system 38, here also a small mass oscillator can cause significant interaction effect to alter the combined system response.

The aforementioned interaction effect can, of course, be included in the analysis by considering a combined oscillator-structure model. This approach, however, becomes impractical when one has to generate floor response spectra at various structural locations for different oscillator parameters. A practical analytical approach was first proposed by Kelly and Sackman 4° 42 to solve this problem. The approach utilized the first order perturbation analysis to obtain the modified eigenproperties of the combined structure-equipment system. After the eigenproperties were known, the analysis followed the deterministic approach to obtain the maximum oscillator response in terms of the ground response spectra.

Sackman and Kelly's pioneering work initiated a vigorous research activity to solve this problem with probabilistic methods of response analysis. Since then a series of papers have been written by Sackman, Der Kiureghian, Noor-Omid and Igusa 43 45 on this subject. These investigations are also based on the first order perturbation methods to obtain the perturbed eigenproperties of the combined primary-secondary systems. The combined system eigenproperties can then be used in a suitable mode response combination rule 22'46 to obtain the oscillator response. The results obtained by the perturbation methods are claimed to be accurate up to the order of the perturbation parameter, which in the problem of generation of floor response spectra can be related to the ratio of the mass of the oscillator to that of the supporting structure. A similar approach has also been used by Gupta 47'48 to synthesize the modal properties of an oscillator-structure system. Based on the second order matrix perturbation analysis49 51, Suarez and Singh have also developed closed form expressions to calculate the eigenproperties and floor spectra for an oscillator-structure system with different damping conditions 52-s4.

EXACT EIGENVALUE ANALYSIS OF OSCILLATOR-STRUCTURE SYSTEM

To incorprate the effect of dynamic interaction more accurately than a perturbation-based approach, Suarez and Singh 55 58 have also developed methods whereby exact eigenproperties of the combined primary-secondary system can be obtained easily in terms of the modal properties of the primary structure and oscillator. The methods have been developed for the classically and nonclassically damped primary structures. For a classically damped structure, the method utilizes the real eigenproperties of the primary structure to obtain the real s5'56 or the complex eigneproperties 57 of the combined structure to account for the presence of the possible nonclassical damping effects in the combined

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 3 153


system. For a nonclassically damped primary structure, however, one must ultilize its complex eigenproperties to obtained the complex properties of the combined system 58. In the most common case of a classically damped primary structure, the combined system eigenvalues are obtained as the solution of the following characteristic equation

L meV p2 + 2fleeCe p + 82 t=l ~ f - . H 2fleojep3+coZp2 =0 (2)

where 6i=p2+2fl~p+~02; p=the eigenvalue of the combined system; ~%, fie and me, respectively, are the frequency, damping ratio and mass of the oscillator; ~oi and /3~, respectively, are the natural frequency and damping ratio for mode i of the primary structure; n = the degrees of freedom of the primary structure and vi is the ith modal displacement of the point where the oscillator is attached.

This characteristic equation can be solved by a simple Newton-Raphson iterative technique 57. The initial estimates of the roots can be obtained by the closed-form expressions developed by Suarez and Singh 53. For light oscillators, these pertubation-based expressions also provide excellent estimates of the combined system eigenvalues. For the purpose of solving the characteristic equation, good estimates of the roots can also obtained by the complex version of the Raleigh's inclusion principle 59.

A striking advantage of this approach lies in the calculation of the exact eigenvector for a known eigenvalue, without solving any simultaneous equation, a s 5 7 :

n + l

Oij = 4 vk j/6j (3) k = l

where 4~a is the ith element of the kth eigenvector of the primary structure with q~i,,+l =q~,+l,i =0 for i= 1 to n and q ~ , + l , , + l = l / x ~ . The normalizing factor fit is defined as:

[-2/3e .+~ p21 1/2 z j= - - +2 ~ (p~+flpa,) (4)

L(2fleP~ +O~e) ,=, ~J

wherein p j--the jth eigenvalue of the combined system. As no approximations or assumptions are made in the

development of these expressions, the calculated eigenproperties are exact up to the accuracy of the precision used in the computations. Furthermore, there is no restriction about the heaviness of the secondary system. Thus, these equations provide an efficient way of calculating the exact eigenproperties of the combined oscillator-structure system. These properties can then be used for calculating the response of the oscillator to define floor response spectra. For a response spectrum analysis of the system with the complex eigenproperties, the mode combination approach developed by the author 22 can be used.

Like ground response spectra, only pseudo acceleration or absolute acceleration floor response spectra have been utilized in the design of secondary systems in seismic analysis practice. However, for the

analysis of multi-degree-of-freedom secondary systems, even supported at a single point, it is necessary to have relative velocity floor response spectra as the input along with the pseudo acceleration floor response spectra. Also for generating tertiary system spectra from floor respectra, it is desirable to have relative velocity floor spectra available. Such floor response spectra can also be developed in the same way as the pseudo acceleration spectra. See Singh and Burdisso 6°.

MULTIPLY SUPPORTED SECONDARY SYSTEMS

It has been a common practice to consider the multiply supported secondary systems as decoupled from, and in cascade with, the primary structures, with no interaction or feed-back. This assumption enables one to define the seismic inputs at the supports of the secondary system by analysing the primary structure alone.

In a cascaded type of analysis, the response of the secondary system { Us} can be obtained as the sum of the dynamic {U d} and pseudo-static {U p} parts of the response as:

{u l + { u f l (5)

where the dynamic and pseudo-static parts of the responses are in turn calculated as the solutions of the following two equations

{Ms]{ (J~} +[CJ{0~} + [Ks]{U d} = [r]{U,} (6)

{u.,} = - [ / (J (7)

where [Ms], [CJ and [K J , respectively, are the mass, damping and stiffness matrices of the secondary system, [Ks, ] is the cross coupling stiffness matrix between the support points and the secondary system degrees of freedom; [r] is a dynamic influence coefficient matrix, each column of which represents the distribution of forces in the unattached degrees of freedom due to the unit acceleration of each supports; and {U,} is the vector of the absolute displacement of the secondary system supports.

This strategy of dividing the response into the dynamic and pseudo static parts for multiple support inputs is given in the text by Clough and Penzien 6~ and has been used by several investigators 35'62 73,92 95 for seismic analysis of multiply supported secondary systems. An analysis approach has also been developed by Asfura and Der Kiureghian 74'75 where this separation of the response into dynamic and pseudo static components has not been employed. In such formulations, however, it is not possible to calculate the support or anchor forces which sometimes are the most important quantities of interest in the seismic design of multiply supported piping systems.

Although the pseudo acceleration floor response spectra are very commonly used to define the design input for the analysis of such systems, it is obvious from equations (6) and (7) that this description of the input is grossly inadequate. Such floor spectra alone cannot account for the effect of the cross correlation between the motions of various supports, which sometimes can be important. The random vibration analysis of these

154 Probabilistic Engineerin9 Mechanics, 1988, Vol. 3, No. 3

systems has shown 7°'7~ that, to incorporate the effect of cross correlation between the support, it is necessary to define the cross floor response spectra along with the auto floor response spectra as the inputs. As a cross spectral density function, characterizing the cross correlation between two stationary random processes, possesses the real and imaginary parts, the cross floor response spectra also have two components: (1) coincidence spectra and (2) quadrature spectra. In terms of the cross spectral density function of the motions of two supports, these two cross floor spectra components are defined as:

(8)

R~(o~j)=F2 2 f_~ o~'(~o)/Hj(o~)[ 2 do~ (9)

where R¢(co.i ) and Re(co 1), respectively, are the coincident and quadrature components of the cross floor response spectra for frequency cot; F~ and FE are the peak response factors; (I)n(co) and ¢~(09), respectively, are the real and imaginary parts of the cross spectral density function; and Hr(co) is the frequency response function of the oscillator with frequency o9 r and damping ratio fir'

The methods to obtain these spectra directly in terms of ground response spectra rather than the cross spectral density function are defined by Singh and Burdisso v 1, and the procedures to utilize these in the calculation of the secondary system response are given by Burdisso and Singh 7°. If it is assumed that there is no feed-back to the primary system, then these auto and cross floor spectra can be obtained in terms of the primary structure properties only. It is mentioned that other definitions of the cross floor response spectra also have been used TM.

COUPLED ANALYSIS OF MULTIPLY SUPPORTED SECONDARY SYSTEMS

The above mentioned decoupled analysis approaches follow the same basic methodology as the current industrial practice. That is, first the primary structure is analysed to obtain the input at the supports which is then used in the calculation of the secondary system response. This approach is reasonable if the secondary system is light and thus the dynamic interaction effect are negligible. Although for the light secondary systems the floor inputs can also be modified to include the dynamic interactionTZ 78, the methods to do this are still approximate. Also in these methods, it is difficult to incorporate the effect of the variability in the structural parameters on the secondary system response. The effect of such uncertainties in generation of auto floor response spectra can be included by peak widening or through procedures developed by Singh et al. 79'8° and Igusa and Der Kiureghian 8~,82 but it is not clear as to how it can be incorporated in the definition of the cross floor response spectra and then in the calculation of the secondary system response.

The aforementioned problems associated with a decoupled analysis of secondary system can be conveniently resolved in a coupled analysis of the two systems. Again a straightforward coupled analysis in which a combined model of the two systems is analysed is unwieldly and impractical, as mentioned earlier. A better

Seismic desi9n of secondary systems: M. P. Singh

approach is to adopt the mode synthesis technique s3,s4 where the modal characterstic of the two systems are synthesized to obtain the similar characteristics of the combined system. Such procedures are: (1) usually more efficient computationally, (2) can incorporate the effect of dynamic interaction between the two systems more accurately, (3) do not require any intermediate inputs in terms of floor response spectra, (4) can use ground response spectra directly and (5) can be adopted to include the effect of structural parameter uncertainties in the calculation of response. These procedures are also as flexible as the decoupled procedures inasmuch as any change in the primary or secondary system will only necessitate the reanalysis of the changed system alone.

The first step in these procedures is to obtain the modal characteristics of the combined primary and secondary system. Once these characteristics are known, the response for inputs defined by response spectra or other forms of inputs can be easily obtained by any one of the established modal response combination procedures.

As the secondary systems are usually light compared to the primary structure, the first order perturbation methods have been utilized to calculate the modal properties of the combined system by Igusa and Der Kiureghian 85'86 and Gupta and Jaw 76'77, Villaverde and Newmark 87. These procedures are primarily based on the assumption that since a secondary system is usually light, it causes only a small perturbation in the modal properties of the individual systems when it is combined with the primary structure. A systematic second order matrix perturbation approach has also been used by Suarez and Singh s8 to obtain the combined system eigenproperties. It is shown that after some simple transformation, the combined system eigenvalue problem, in general, can be written in the following form:

[[~0] + ~[A,] + ~2[A2]]q, r = Pr[[B0] + ~[B,] + ~2182]]q,~

(10) where [Ao] and [Bo] are symmetric matrices of the unperturbed eigenvalue problem, defined in terms of the matrices of the decoupled individual systems. Matrices [A1] and [B1] are symmetric perturbation matrices whose elements are one order of magnitude smaller than those of [A0] and [B0], whereas matrices [A2] and [B2] are two orders smaller than the unperturbed matrices. The parameters e identifies the order of the matrices and is also used to keep track of the order of magnitude of the different quantities involved. Thus, if we assume that [Ao] and [B0] are O(e°), then [AI] and [B1] are Off.) while [Az] and [Bz] are 0(~2). These matrices are' symmetric nonpositive definite of dimension (2N × 2N) where N = the combined degrees of freedom of primary and secondary system. Therefore, the eigenvalue problem in general possesses N pairs of complex conjugate eigenvalues. The eigenvalue and eigenvectors are pj and

For such an eigenvalue problem, the second order estimate of the eigenvalue can be expressed 88 as:

Pj=Poj+Plr+p2j (11)

where Poj is the unperturbed eigenvalue associated with matrices A o and Bo, whereas the first and second order correction terms Plr and P2r can be defined in terms of the unperturbed eigenproperties as:



T P l ; = Ooj[ A1 - pojB1]Ooj (12)

2N

p2J=K~ ' P~J ~-~Oo~[A2-PojB2-PljB1]Oo j (13) = (Poj--Pok)

KS)

For the case of the tuned eigenvalues, the expressions are similar, except there are some further restrictions in the summation term in P~i (Ref. 88). Similar expressions can also be developed for the perturbed eigenvectors.

MULTIPLY SUPPORTED HEAVY SECONDARY SYSTEMS

In a majority of cases, the perturbation-based approaches will provide an accurate estimate of the eigenproperties of the combined system for the calculation of the combined system response. However, for heavier secondary systems such as steam generator and reactor coolant pump in nuclear reactors, Fig. 2, more accurate values of the combined eigenproperties may be desired. Such values can be obtained by mode synthesis techniques of structural dynamics.

The currently available mode synthesis techniques invariably require the solution of a second eigenvalue problem. We have, however, developed approaches for various damping cases whereby a conventional solution of the second eigenvalue problem can be completely avoided. Sequential coupling procedures are developed89 91 which provide the exact solution of the eigenvalue problem. The procedure can be used with classically or nonclassically damped subsystems. For example, for classically damped primary and secondary subsystems but nonclassically damped combined system, the procedure consists of representing the stiffness matrix of each finite element connecting the two subsystems as the product of two vectors:

d [K~]= ~ K~(~,aT--la~ T) (14)

i=l

where [Ktc] is the stiffness of the/th coupling element. The coefficients Ki and vectors ~2 and/~i are defined in terms of the elements of [K~]. For truss or beam elements, the elements of these vectors can be defined by inspection 91. For other finite element matrices, they can be obtained in terms of the eigenvectors of matrix [K~c]. The summation in equation (14) is over the number of the deformable degrees of freedom (that is, the degrees of freedom excluding the rigid body modes).

The procedure consists of successive coupling of the two subsystems (primary and secondary) by including increasing number of terms in equation (14) for each connecting element. At each coupling stage, the eigenvalues are obtained as the solution of a characteristic equation similar to equation (2). The eigenvector corresponding to an eigenvalue is then obtained by a simple closed form expression. The eigenproperties obtained at a coupling stage are used in the immediately following coupling stage. The process is repeated until the two substructures are connected by all coupling elements at all degrees of freedom. The eigenproperties obtained at the final coupling stage then provide the combined system

properties. These properties can then be used to obtain the response of the primary or secondary system for any form of input. For the inputs defined by ground respovse spectra, a suitable mode combination procedure such as the one in Ref. 22 can be used for response analysis.

SOME RECENT RESEARCH AND OBSERVATIONS

Since the presence of a secondary system can affect the response of the primary structure through dynamic interaction, which, in turn, may also affect its own response, it is also conceivable that addition of more secondary systems to the primary structure may further change the response of the existing secondary systems. Thus, the problem of evaluating the effect of dynamic interaction between two or more secondary systems is also of interest. Our own investigation 96 indicates that this effect, indeed, can be significant, and thus must be considered in the design of interacting secondary systems supported on a primary structure. In Ref. 96, a method is developed to evaluate the response of such interacting secondary systems.

A topic where little has been done in terms of research is in the area of response analysis of secondary systems supported on nonlinear yielding primary structures. The yielding of primary structure can greatly alter the input motion to the secondary systems and thus their response, as shown by Lin and Mahin 97 in the study of a cascaded simple yielding structure-oscillator system. Our own work 9s on a multi-degrees-of-freedom yielding primary structure supporting a secondary systems shows that the effect of yielding, as well as the dynamic interaction is very significant on the secondary system response. Also, our analytical work 99'100 on the isolation of equipment in building from seismic effect through coulomb damping clearly shows that response of equipment can be significantly reduced by frictional energy dissipation devices. These initial studies also warrant the need for further research work on these topics.

ACKNOWLEDGEMENTS

The writer's research on the topics discussed in this paper has been financially supported by the National Science Foundation through Grants Nos CEE-810910, CEE- 8208897, CEE-8412830 and CES-8619306 with Dr S. C. Liu and Dr M. P. Gaus as their Program Directors. This support is gratefully acknowledged.

Every attempt has been made to include the significant works reported in the open literature in the bibliography of this survey paper. Yet, however, it is quite conceivable that several equally important publications may also have been left out of this list inadvertantly; the writer regrets such unavoidable omissions.

REFERENCES

1 US Nuclear Regulatory Commission, Design Response Spectra for Nuclear Power Plants, Nuclear Regulatory Guide No. 1.60, Washington DC, 1975

2 Neemark, N. M., Blame, J. A. and Kapur, K. K. Seismic Design Spectra for Nuclear Power Plants, ASCE Journal ~f Power Division, November 1973, 99

3 Tsai, N. C. Spectrum-Compatible Motions for Design Purpose,

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

Journal of Engineering Mechanics Division, ASCE, EM2, April 1972

4 Scanlan, R. H. and Sachs, K. Earthquake Time Histories and Response Spectra, J. of Eng. Mech. Div., ASCE, EM4, August 1974, 635~555

5 Scanlan, R. H. and Sachs, K. Floor Response Spectra for Multi- Degree-of-Freedom Systems by Fourier Transforms, Proceedings of SMiRT-3 Conference, London, paper K5/5, September 1975

6 Gasparini, D. and Vanmarcke, E. H. Simulated Earthquake Motions Compatible with Prescribed Response Spectra, Pub. No. R76-4, Dept. of Civil Engr., M.I.T., January 1975

7 Levy, S. and Wilkinson, P. D. Generation of Artificial Time Histories, Rich in all Frequencies, from Given Response Spectra, Nuclear Engineering and Design, 1976, 38, 241-251

8 Kaul, M. K. Spectrum Consistent Time History Generation, J. Eng. Mech. Div., ASCE, 1978, 104, 781-788

9 Iyengar, R. N. and Rag, P. N. Generation of Spectrum Compatible Accelerograms, J. Earthquake Engr. and Str. Dyn., 1979, 7, 253 263

10 Spanos, P. D. Digital Synthesis of Response-Design Spectrum Compatible Earthquake Records for Dynamic Analyses, Shock and Vibration Digest, 1983, 15(3), 21-30

11 Spanos, P. D. and Vargas, Loli L. M. A Statistical Approach to Generation of Design Spectrum Compatible Earthquake Time Histories, Soil Dynamics and Earthquake Eng., 1985, 4, 1

12 Perumont, A. The Generation of Spectrum Compatible Accelerograms for the Design of Nuclear Power Plants, J. Earthquake Engineering and Str. Dyn., 1984, 12, 481~,97

13 Perumont, A. The Generation of Nonseparable Artificial Earthquake Accelerograms for the Design of Nuclear Power Plants, Nuclear Engineering and Design, 1985, 88, 59-66

14 Singh, M. P., Singh, S. and Chu, S. L. Stochastic Concepts in Seismic Design of Nuclear Power Plants, SMiRT-2 Conf., Berlin, Paper K 1/4, September 1973

15 Singh, M. P. and Chu, S. L. Stochastic Consideration in Seismic Analysis of Structures, J. Earthquake Eng and Str. Dyn., 1976, 4

16 Biggs, J. M. and Rosset, J. M. Seismic Analysis of Equipment Mounted on a Massive Structure, in Seismic Design for Nuclear Power Plants, (Ed. R. J. Hansen), MIT Press, Cambridge, Mass., 1970

17 Biggs, J. M. Seismic Response Spectra for Equipment Design in Nuclear Power Plants, Proceedings of SMiRT-I Conference, Berlin paper K4/7, September 1971

18 Kapur, K, K. and Shag, L. C. Generation of Seismic Floor Response Spectra for Equipment Design, Proceedings Speciality Conference on Structural Design of Nuclear Plant Facility, Chicago, IL, 1973

19 Singh, M. P. Generation of Seismic Floor Spectra, ASCE, Journal of the Engineering Mechanics Division, October 1975, 101(EM5), Proc. paper 11651, 593-607

20 Singh, M. P. Seismic Design Input for Secondary Systems, ASCE, Journal of the Structural Division, February 1980, 106, 505 517

21 Singh, M. P. and Sharma, A. M. Floor Spectra for Nonclassically Damped Structures, J. of Structural Eng., ASCE, November 1985, 111(11), 2446-2463

22 Singh, M. P. Seismic Response by SRSS for Nonproportional Damping, Journal of Engineering Mechanics, ASCE, December 1980, 106, 1405 1419

23 Singh, M. P. and Sharma, A. M. Seismic Floor Spectra by Mode Acceleration Approach, J. of Eng. Mech., ASCE, November 1985, 111(11), 1402-1419

24 Sharma, A. M. and Singh, M. P. Floor Spectra by Mode Acceleration-Based Response Spectrum Approach for Nonclassically Damped Structures, Nuclear Engineering and Design, April 1986, 92(2), 181-194

25 Vanmarcke, E. H. Seismic Structural Response, Chapter 8 in Seismic Risk and Engineering Decisions, (Eds C. Lomnitz and E. Rosenblueth), Elsevier Scientific Publishing Co., New York, 1976

26 Vanmarcke, E. H. A Simple Procedure for Predicting Amplified Response Spectra and Equipment Response, Proceedings of 6th World Conference on Earthquake Engineering, New Delhi, India, January 1977

27 Scanlan, R. H. and Sachs, K. Development of Compatible Secondary Spectra Without Time Histories, Proceedings of SMiRT-4 Conference, San Francisco, CA, paper K4/13, August 1977

28 Schmidt, D. and Peters, K. Direct Evaluation of Floor Response

Seismic design o f secondary systems: M. P, Singh

Spectra from a Given Ground Response Spectrum, Proceedings of SMiRT-4 Conference, San Francisco, CA, paper K4/10, August 1977

29 Peters, K. A., Schmitz, D. and Wagner, U. Determination of Floor Response Spectra on the Basis of the Response Spectrum Method, Nuclear Engineering and Design, 1977, 44, 255-262

30 Atalik, T. S. On Upperbound Instructure Response Spectra, Proceedings of SMiRT-5 Conference, Berlin, paper K9/3, August 1979

31 Sundararajan, C. An Iterative Method for the Generation of Seismic Power Spectral Density Functions, Nuclear Engineering and Design, 1980, 61, 13 23

32 Unruh, J. E. and Kana, D. An Iterative Procedure for the Generation of Consistent Power/Response Spectrum, Nuclear Engineering and Design, 1981, 66, 427~435

33 Singh, A. K. A Stochastic Model for Predicting Maximum Seismic Response of Light Secondary Systems, Thesis presented to Univ. of Illinois at Urbana-Champaign, in partial fulfillment of requirements for the degree of Doctor of Philosophy, 1972

34 Chakravorty, M. K. and Vanmarcke, E. H. Probabilistic Seismic Analysis of Light Equipment within Buildings, Proceedings of 5th World Conference on Earthquake Engineering, Rome, Italy, 1973, Vol. II

35 Lee, M. C. and Penzien, J. Stochastic Analyses of Structures and Piping Systems Subjected to Stationary Multiple Support Excitations, Earthquake Engineering and Structural Dynamics, 1983, 11, 91-110

36 Perumont, A. The Spectral Analysis of Linear Structures Under Stationary Random Excitation, J. Earthquake Engineering and Str. Dyn., (in press)

37 Lin, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967

38 Den Hartog, J. P. Mechanical Vibrations, Fourth Edition, McGraw-Hill, 1965

39 Penzien, J. and Chopra, A. K. Earthquake Response of Appendage on Multi-Storey Building, Proceedings of 3rd World Conference on Earthquake Engineering, New Zealand, 1965, Vol. II

40 Kelly, J. M. and Sackman, J. L. Response Spectra Design Methods for Tuned Equipment-Structure Systems, Journal of Sound and Vibration, 1978, 59(2), 171-179

41 Sackman, J. L. and Kelly, J. M. Seismic Analysis of Internal Equipment and Components in Structures, Engineering Structures, 1979, 1(4), 179 190

42 Sackman, J. L. and Kelly, J. M. Equipment Response Spectra for Nuclear Power Plant Systems, SMiRT-5 Conf., Berlin, Paper K9/1, August 1979

43 Sackman, J. L., Der Kiureghian, A. and Nour-Omid, B. Dynamic Analysis of Light Equipment in Structures: Modal Properties of the Combined System, Journal of Engineering Mechanics, ASCE, 1983, 109, 73-89

44 Der Kiureghian, A., Sackman, J. L. and Nour-Omid, B. Dynamic Analysis of Light Equipment in Structures: Response to Stochastic Input, Journal of Engineering Mechanics, ASCE, 1983, 109, 90-110

45 Igusa, T. and Der Kiureghian, A. Generation of Floor Response Spectra Including Oscillator-Structure Interaction, J. Earthquake Engineering and Structural Dynamics, 1985, 13, 661 676

46 lgusa, T., Der Kiureghian, A. and Sackman, J. L. Modal Decomposition Method for Stationary Response of Nonproportionally Damped Structures, J. Earthquake Engineering and Structural Dynamics, 1984, 12(1), 121-135

47 Gupta, A. K. and Jing-Wen, Jaw Reports on Seismic Response of Secondary Systems, Dept. of Civil Engineering, N.C. State University, Raleigh, 1985

48 Gupta, A. K. and Jing-Wen, Jaw Complex Modal Properties of Coupled Moderately Light Equipment-Structure Systems, Nuclear Engineering and Design, 1986, 91, 171 178

49 Franklin, J. N. Matrix Theory, Prentice Hall, Englewood Cliffs, NJ, 1969

50 Meirovitch, L. and Ryland II, G. Response of Slightly Damped Gyroscopic Systems, Journal of Sound and Vibration, 1979, 67, 1 19

51 Lancaster, P. Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford, 1966

52 Singh, M. P. and Suarez, L. E. Perturbation Analysis of Structure-Equipment System, Nuclear Engineering and Design, November 1986, 97(2), 167-186

53 Suarez, L. E. and Singh, M. P. Perturbed Complex


Seismic design o f secondary systems: M. P. Singh

Eigenproperties of Classically Damped Primary Structure and Equipment Systems, Journal of" Sound and Vibration, July 1987, 116(2)

54 Suarez, L. E. and Singh, M. P. Eigenproperties of Nonclassically Damped Primary Structure and Equipment System by a Perturbation Approach, Journal of Earthquake En,qineerin,q and Structural Dynamics, (to appear)

55 Suarez, L. E. and Singh, M. P. Floor Response Spectra with Structure-Equipment Interaction Effects by a Mode Synthesis Approach, Journal of Earthquake En,qineerin,q and Structural Dynamics, February 1987, 15(2), 141 158

56 Suarez, L. E. and Singh, M. P. Seismic Response of Equipment- Structure Systems, En,qineerin,q Mechanics, ASCE, January 1987, 113(1), 16 30

57 Singh, M. P. and Suarez, L. E. Seismic Response Analysis of Structure-Equipment System with Nonclassical Damping Effects, Journal of Earthquake En,qineerin,q and Structural Dynamics, (to appear)

58 Suarez, L. E. and Singh, M. P. Eigenproperties of Nonclassically Damped Primary Structure and Oscillator System, Journal o[ Applied Mechanics, ASME, (to appear)

59 Rayleigh, J. W. S. The Theory o[" Sound, Vol. 1, Dover Publications, Inc., New York, NY, 1945

60 Singh, M. P. and Burdisso, R. A. Relative Velocity and Relative Acceleration Floor Spectra, SMiRT-8 Conf., Paper No. KI2/10, Brussels, Belgium, August 1985

61 Clough, R. W. and Penzien, J. Dynamics o[" Structures, McGraw- Hill, 1975

62 Amin, M., Hall, W. J., Newmark, N. M. and Kasawara, R. P. Earthquake Response of Multiply Connected Light Secondary System by Spectrum Methods, Proceedings ASME 1st Congress on Pressure Vessel and Piping, San Francisco, May 1971

63 Kasawara, R. P. and Peck, D. A. Dynamic Analysis of Structural Systems Excited at Multiple Support Locations, ASCE 2nd Specialty Conference on Structural Design of Nuclear Plant Facilities, Chicago, IL, December 1973

64 Vashi, K. M. Seismic Spectral Analysis for Structures Subject to Nonuniform Excitation, 83-PVP-69, ASME Pressure Vessel and Piping Conf., Portland, 1960

65 Gasparini, D. A., Shah, A. and Tsiatas, G. Random Vibration of Cascaded Secondary Systems, Report 83-1, Case Western Reserve Univ., Cleveland, OH, January 1983

66 Wu, R. W., Hussain, F. A. and Liu, L. K. Seismic Response Analysis of Structural Systems Subjected to Multiple Support Excitations, SMiRT-4 Conf., San Francisco, August 1977

67 Schmidt, H. J. and Ludwig, J. Design of Piping with the Procedure for Multiple Support Excitation, SMiRT-4 Conf., San Francisco, Paper K6/15a, August 1977

68 Subudhi, M. and Bezler, P. Alternate Procedure for the Seismic Analysis of Multiply Supported Piping Systems, SMiRT-8 Conf., Brussels, Belgium, Paper No. K 16/3, August 1985

69 Liemback, K. R. and Sterkel, H. P. Comparison of Multiple Support Excitation Solution Techniques for Piping Systems, 5th SMiRT Conf., Berlin, Paper KI0/2, August 1979

70 Burdisso, R. A. and Singh, M. P. Multiply Supported Secondary Systems, Part I: Response Spectrum Analysis, Journal of Earthquake En`qineerin,q and Structural Dynamics, January February 1987, 15, 53 72

71 Singh, M. P. and Burdisso, R. A. Multiply Supported Secondary Systems, Part II: Seismic Inputs, Journal of Earthquake En,qineerin,q and Structural Dynamics, January 1987, 15, 73 90

72 Burdisso, R. A. and Singh, M. P. Seismic Analysis of Multiply Supported Secondary with Dynamic Interaction Effects, Journal o1" Earthquake En,qineerin,q and Structural Dynamics, (in press)

73 Gupta, A. K. Seismic Response of Multiply Connected MDOF Primary and MDOF Secondary Systems, Nuclear Engineering and Desi,qn, 1984, 81~ 359 373

74 Asfura, A. and Der Kiureghian, A. A New Flow Response Spectrum Method for Seismic Analysis of Multiply Supported Secondary Systems, Report No. UCB/EERC-84/04, Earthquake Engineering Research Center, Univ. of Cal., Berkeley, CA, 1984

75 Asfura, A. and Der Kiureghian, A. Floor Response Spectrum Method for Seismic Analysis of Multiply Supported Secondary Systems, Journal o1' Earthquake Engineering and Structural Dynamics, 1986, 14, 245 265

76 Gupta, A. K. and Jing-Wen, Jaw Coupled Response Spectrum Analysis of Secondary System Using Uncoupled Modal Properties, Nuclear En`qineerin,q and Design, 1986, 92, 61~8

77 Gupta, A. K. and Jing-Wen, Jaw A New Instructive Response

Spectrum ORS) Method for Multiply Connected Secondary Systems with Coupling Effects, Nuclear En,qineerin,q and Design, 1986, 96, 63 80

78 Ruzicka, G. C. and Robinson, A. R. Dynamic Response of Tuned Secondary Systems, Report No. UILU-ENG-80-2020, Dept. of Civil Engr., Univ. of UII., Urbana, IL, November 1980

79 Singh, M. P. Seismic Response of Structures with Random Parameters, Proceedings of 7th World Conference on Earthquake Engineering, Istanbul, Turkey, September 1980

80 Ghafory-Ashtiany, M. and Singh, M. P. Seismic Response of Structural Systems with Random Parameters, Report No. VPI- E-81-15, Virginia Polytechnic Institute and State Univ., September 1981

81 Igusa, T. and Der Kiureghian, A. Stochastic Response of Systems with Uncertain Properties, 8th SMiRT Conference, paper M14/5, Brussels, Belgium, August 19 23, 1985

82 Igusa, T. and Der Kiureghian, A. Reliability of Secondary Systems with Uncertain Tuning, Proceeding of the 4th Intl. Cong. on Structural Safety and Reliability, Kobe, Japan, May 27 29, 1985

83 Hurty, W. C. Vibration of Structural Systems by Component Mode Synthesis, Proc. ASCE J. of Eng. Mech. Div., August 1960, 86, 51 69

84 Hurry, W. C. Dynamic Analysis of Structural Systems Using Component Modes, AIAA Journal, April 1965, 3(4)

85 Igusa, T. and Der Kiureghian, A. Dynamic Analysis of Multiply Tuned and Arbitrarily Supported Secondary Systems, Report No. EERC-83/07, Earthquake Engineering Research Center, University of California, Berkeley, July 1983

86 Igusa, T. and Der Kiureghian, A. Dynamic Response of Multiply Supported Secondary Systems, Journal Of Engineering Mechanics, ASCE, 1985, 111, 2041

87 Villaverde, R. and Newmark, N. M. Seismic Response of Light Attachments to Buildings, Structural Research Series No. 469, UILU-ENG 80-2006, Dept. of Civil Engineering, University of Illinois, Urbana, IL, February 1980

88 Suarez, L. E. and Singh, M. P. Exact Modal Synthesis Methods for Seismic Analysis of Combined Primary and Multiply Supported Secondary Systems, Report No. VPI-E-86-22, Virginia Polytechnic Institute and State Univ., Blacksburg, VA, September 1986

89 Singh, M. P. and Suarez, L. E. A Method for Dynamic Coupling with Nonclassical Damping Effects, J. ~j Sound and Vib., February 1988, 21(1)

90 Suarez, L. E. and Singh, M. P. An Exact Component Mode Synthesis Approach, J. Earthquake Engineerin,q and Str. S yn., (in press)

91 Suarez, L. E. and Singh, M. P. Dynamic Synthesis of Nonclassically Damped Structures, J. c~[ Eng. Mech., ASCE, fin press)

92 Biswas, J. K. Seismic Analysis of Equipment Supported at Multiple Levels, Dynamic and Seismic Analysis of Systems and Components, PVP, Vol. 65, ASME, Pressure Vessel and Piping Conf., Orlando, FL, July 1982, 133 142

93 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code Section III, Appendix N, Dynamic Analysis Methods, July 1981

94 Liembach, K. R. and Schmid, H. Automated Analysis of Multiple-Support Excitation Piping Problems, Nuclear En,qineerin,q and Design, 1979, 51,245 252

95 Lin, C. W. and Loeeff, F. A New Approach to Compute System Response with Multiple Support Response Spectra Input, Nuclear En,qineerin,q and Design, 1980, 60, 347 352

96 Suarez, L. E. and Singh, M. P. Floor Spectra with Equipment- Structure-Equipment Interaction Effects, J. qf En,qineerin,q Mechanics, ASCE, (in press)

97 Lin, J. and Mahin, S. A. Seismic Response of Light Subsystems on Inelastic Structures, J. of Structural En,qineerin,q, ASCE, February 1985, 111(2), 400~71

98 Singh, M. P., Maldonado, G. and Suarez, L. E. Seismic Response of Equipment on Hysteretic Structures, for publication in J. of Structural En,qineerin,q, ASCE, 1987

99 Malushte, S. R. and Singh, M. P. Seismic Response of Simple Structures with Coulomb Damping, ASCE Structural Congress. Orlando, FL, August 1987, Vol. 4

I00 Singh, M. P. and Malushte, S. R. Seismic Response of Structures with Sliding Interfaces, Report No. VPI-E-88-24, College of Engineering, Virginia Polytechnic Institute and State Univ., Blacksburg, VA, September 1988

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

seismic design of secondary systems

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