ABSTRACT. YANG, JIANFENG. Ritz Vector Approach for Evaluating the Dynamic Properties of Structural

Transcription

1 ABSTRACT YANG, JIANFENG. Ritz Vector Approach for Evaluating the Dnamic Properties of Structural Sstems and Incabinet Spectra. (Under the direction of Abhinav Gupta) The earthquake input needed in a shake table test for the seismic qualification of safet related electrical instruments, tpicall mounted on electrical cabinets or control panels, is defined in terms of incabinet response spectrum. This dissertation presents modifications to the originall proposed Ritz vector approach developed b Rustogi et al. (998) for evaluating the dnamic properties of the cabinets and the incabinet spectra. Modifications are needed to overcome the limitations encountered in the application to actual cabinets. The accurac of the modified formulations is evaluated b comparison of results for actual cabinets with the corresponding results obtained from detailed finite element analses. The modified Ritz vector approach can account for actual rotational constraints imparted b supporting structural members such as stiffeners. It can also be applied to bench board tpe cabinets in which instruments are mounted on plates or frames that are inclined to the global aes as well as to frames in which parallel frame members can vibrate in different vibration shapes. In this dissertation, detailed finite element analses are used to stud the rocking behavior of cabinets and to show that accurate representation of the boundar conditions at the cabinet base is essential in the evaluation of cabinet rocking mode. Simple formulations are developed for evaluating the rocking stiffness in three different tpes of cabinet mounting arrangements. These formulations enable incorporation of a cabinet rocking mode in the Ritz vector approach. The Ritz vector approach is also used to develop new formulations for evaluating static and dnamic characteristics of rectangular slabs with edge beams. The effect of elastic edge restraints is considered b including appropriate integrals for edge beams in the epressions for total kinetic and potential energies in a Raleigh-Ritz approach. The effect of various tpes of boundar conditions at the beam ends is accounted for b considering the corresponding Ritz vectors. The contribution of beam mass to the total kinetic energ is also considered in the proposed approach.

2 RITZ VECTOR APPROACH FOR EVAUATING THE DYNAMIC PROPERTIES OF STRUCTURA SYSTEMS AND INCABINET SPECTRA B JIANFENG YANG A dissertation submitted to the graduate facult of North Carolina State Universit in partial fulfillment of the requirements for the Degree of Doctor of Philosoph CIVI ENGINEERING Raleigh 00 APPROVED BY:

3 To m beloved parents, Yong-Cheng Yang and Hui-Ying Yang To m wonderful wife, Yue Wang ii

4 BIOGRAPHY Jianfeng Yang received his Bachelor of Engineering degree in Structural Engineering from Beijing Poltechnic Universit, Beijing, China, in 994. He received his Master of Science degree in Structures & Mechanics from the same universit in 997. He came to North Carolina State Universit, Raleigh, North Carolina, USA, in August 997 for his Ph.D. in Structures & Mechanics. During his stud, he worked as Research Assistant in the Center for Nuclear Power Plant Structures, Equipment and Piping at North Carolina State Universit. Jianfeng Yang s research interests include structural analsis and design, structural dnamics, space structures, finite element method, dnamic behavior of electrical instruments, and computer applications. iii

5 ACKNOWEDGEMENTS This research was partiall supported b the Center for Nuclear Power Plant Structures, Equipment and Piping at North Carolina State Universit. Resources for the Center come from the dues paid b member organizations and from the Department of Civil Engineering and College of Engineering in the Universit. The author wishes to epress his appreciation to Dr. Abhinav Gupta for his constant inspiration and guidance throughout the course of this research. Appreciation is also etended to Drs. A. K. Gupta, V. C. Matzen, J. M. Nau and C. C. Tung for their valuable suggestions. iv

6 TABE OF CONTENTS Page IST OF TABES.. IST OF FIGURES vi vii PART I INTRODUCTION... Introduction.. Objective Organization References... 8 PART II MODIFIED RITZ VECTOR APPROACH FOR DYNAMIC PROPERTIES OF EECTRICA CABINETS AND CONTRO PANES. 0 Abstract.... Introduction.. Ritz Vector Approach imitations of the Eisting Formulations Torsional Stiffness of Supporting Members Bench Board Cabinet Internal Frame Results Summar and Conclusions Acknowledgements References PART III ROCKING STIFFNESS OF MOUNTING ARRANGEMENTS IN EECTRICA CABINETS AND CONTRO PANES. 5 Abstract Instruction Cabinet Mounting Arrangements Finite Element Analsis Stiffness of Configurations and Stiffness of Configuration Parametric Stud Summar and Conclusions.. 7 v

7 Acknowledgements References PART IV RITZ VECTOR APPROACH FOR STATIC AND DYNAMIC ANAYSIS OF PATES WITH EDGE BEAMS. 9 Abstract Introduction. 93. Governing Equations for a Plate with Edge Beams Free Vibration Analsis Equivalent Mass and Stiffness Matrices Selection of Ritz Vectors Static Analsis Summar and Conclusions.. 3 Acknowledgements. 4 References... 5 PART V SUMMARY AND CONCUSIONS 6. Summar and Conclusions.. 7. Recommendations for Future Research.. 30 References... 3 ADDITIONA REFERENCES vi

8 IST OF TABES Page PART II MODIFIED RITZ VECTOR APPROACH FOR DYNAMIC PROPERTIES OF EECTRICA CABINETS AND CONTRO PANES. Structural properties and configurations of cabinets PART III ROCKING STIFFNESS OF MOUNTING ARRANGEMENTS IN EECTRICA CABINETS AND CONTRO PANES. Structural and geometrical details of cabinets 77. Rocking stiffness K θ for the three tpes of mounting configurations.. 77 PART IV RITZ VECTOR APPROACH FOR STATIC AND DYNAMIC ANAYSIS OF PATES WITH EDGE BEAMS. Ritz vectors for idealized boundar conditions.. 7 vii

9 IST OF FIGURES Page PART II MODIFIED RITZ VECTOR APPROACH FOR DYNAMIC PROPERTIES OF EECTRICA CABINETS AND CONTRO PANES. Structural constraints in a cabinet Bo tpe cabinet, CAB Frequencies for local back wall mode, CAB Incabinet response spectra, CAB Bench board tpe cabinet, SSF Internal frame Vibration shape for internal frame members, PROT IV ocal coordinate sstem Global and local coordinate sstem in a bench board Global and local displacements in a bench board Internal frame Frequencies for local back wall mode, CAB Incabinet response spectra for instrument location on the back wall, CAB Incabinet response spectra in bench board cabinet, SSF (local mode onl) Incabinet response spectra in bench board cabinet, SSF (local mode plus global mode) 48 6(a). Incabinet response spectra in the internal frame, PROT IV (member 6(b). I).. 49 Incabinet response spectra in the internal frame, PROT IV (member II) Cabinet MS Mode shape for back wall mode, cabinet MS4706 ( = 8 in) PART III ROCKING STIFFNESS OF MOUNTING ARRANGEMENTS IN EECTRICA CABINETS AND CONTRO PANES. Structural details of the mounting arrangement in cabinet DGSB Structural details of mounting arrangement in cabinet MS Structural details of mounting arrangement in cabinet PROT IV Global rocking mode of a cabinet Deformation of base plate in the cabinet rocking mode Deflected shape for base plate uplifting in cabinet DGSB... 8 viii

10 7. Deflected shape for base plate uplifting in cabinet MS Cup-like deformation of base plate around an anchor bolt Equivalent model with vertical springs at anchor bolt locations 84 0(a). Outer frame structure having rotational stiffness K θ in cabinet PROT IV 84 0(b). Intermediate tubular beams having rotational stiffness K θ in cabinet PROT IV. 84. Simplified model for evaluating K θ in cabinet PROT IV Simplified model for evaluating K θ in cabinet PROT IV Variation of rocking stiffness K θ with base plate thickness t in cabinet DGSB Variation of rocking stiffness K θ with cabinet depth D in cabinet DGSB Variation of rocking stiffness K θ with bolt distance b in cabinet DGSB Variation of rocking stiffness K θ with base plate thickness t in cabinet MS Variation of rocking stiffness K θ with cabinet depth D in cabinet MS Variation of rocking stiffness K θ with bolt distance b in cabinet MS Variation of rocking stiffness K θ with triangular plate thickness t in cabinet PROT IV Variation of rocking stiffness K θ with cabinet depth D in cabinet PROT IV. 89. Variation of rocking stiffness K θ with bolt distance b in cabinet PROT IV. 90 PART IV RITZ VECTOR APPROACH FOR STATIC AND DYNAMIC ANAYSIS OF PATES WITH EDGE BEAMS. Rectangular plate with edge beams 8. Elastic plate with edge beams, clamped at corners Fundamental mode shape at =3m, plate with clamped corners 9 4. Fundamental mode shape at =6m, plate with clamped corners 9 5. Fundamental mode shape at =3m, plate with simpl supported corners Fundamental mode shape at =6m, plate with simpl supported corners Variation of fundamental frequenc with edge beam mass, plate with simpl supported corners 8. Plate with edge beam at =0, simpl supported on four edges Fundamental mode shape at =3m, simpl supported plate with edge i

11 beam at =0. 0. Plate with edge beam at =0, simpl supported on three edges.. Fundamental mode shape at =0.89m, simpl supported plate with beam at =0. 3. Variation of the fundamental frequenc with the torsional stiffness of edge beams (I=0.0m 4 ) Variation of the fundamental frequenc with the bending stiffness of edge beams (J=0-6 m 4 ) Displacement of beams subjected to concentrated load at mid-span Displacement of the plate at =3m, 00KN concentrated load at midspan location... 5

12 PART I INTRODUCTION Jianfeng Yang

13 . Introduction In critical industrial facilities like nuclear power plants, the safet related electrical instruments such as relas must continue to operate during an earthquake. These instruments, mounted on cabinets and control panels, are seismicall qualified b a shake table test. The earthquake input needed in a shake table test is defined in terms of incabinet response spectrum. The characteristics of the incabinet response spectra depend upon the dnamic characteristics of the cabinet, the location of floor where the cabinet is mounted, and the instrument location in the cabinet. Currentl used simple methods to generate incabinet motion can give unrealistic spectra that can be ecessivel conservative. Vibration testing and finite element analsis are ver sensitive to the mounting arrangement of a cabinet. Both these techniques are time and cost intensive. The eisting methods are described in detail in Gupta et al. (999a) and discussed later in this dissertation. Recentl, Rustogi et al. (998a) and Gupta et al. (998, 999a) presented a new method to rationall calculate the dnamic properties of the cabinets and evaluate incabinet spectra. The used detailed finite element analses of 6 tpical cabinets and verification of finite element results against eperimental data to illustrate that onl few (often one) modes contribute significantl to the response at a particular instrument location on the control panel. The significant mode can be either a local mode (of plate or frame) or a global cabinet mode. It can also be a superposition of the local and global modes. Gupta et al. (999a) represented the significant local and global mode shapes using mathematical functions known as Ritz

14 vectors which were then used to develop an equivalent two degree of freedom sstem in the Raleigh-Ritz method. The cabinet dnamic properties were evaluated from the solution of this equivalent two degree-of-freedom sstem. A computer program, INCABS (Rustogi et al., 998a, 998b), was also developed to implement the Ritz vector approach. INCABS together with CREST-IRS (Gupta and Gupta, 997) is used to generate incabinet spectra. INCABS uses simple equations to account for the fleibilit of mounting arrangement at the cabinet base. Rustogi et al. (998a) developed two such equations corresponding to two tpes of mounting configurations.. Objective imitations were encountered during the application of the Ritz vector approach (Rustogi et al., 998a; Gupta et al., 998, 999a) to actual cabinets. Modifications are needed in the Ritz vector approach (and INCABS) to improve its accurac and reduce the number of outlier cabinets. The objective of this research is described below: Torsional stiffness of supporting members: The method presented b Rustogi et al. (998a) and Gupta et al. (999a) requires selection of Ritz vectors for local mode shapes of plate or frame members. Selection of Ritz vectors is governed b the idealization of boundar conditions for the constraints in structural members. Often, these constrains can not be represented b idealized boundar conditions. For eample, a partial rotational 3

15 constraint ma eist due to the presence of a unistrut at the supports of a local member. Ritz vector approach will be modified to account for the torsional stiffness of constraining structural members. Bench board cabinets: Several cabinets such as bench boards have instruments mounted on panels that are inclined to the global cabinet aes. Transformations will be developed, for inclusion in the Ritz vector approach, to calculate the dnamic properties for such cabinets in their significant local and global modes. Internal frame members: Rustogi et al. (998a) assumed that all the horizontal (or vertical) frame members within an internal frame vibrate in the same mode shape. Same Ritz vectors were used for all the horizontal (or vertical) frame members. Different member ma behave differentl due to different boundar conditions and different mass distributions. Ritz vector approach will be modified to account for this situation and enable the use of different Ritz vectors for different frame members. Higher order Ritz vectors and lumped masses: Although the formulation proposed b Rustogi et al. (998a) and Gupta et al. (999a) accounted for lumped masses, the application eamples did not. For simplicit, the considered the total mass on a panel to be uniforml distributed. Consequentl, the emphasis was on accuratel evaluating incabinet spectrum at the location of maimum amplification in a local member. Such an approach simplifies the hand calculation but limits the evaluation of incabinet spectrum 4

16 to a single location on a local member. A computer implementation such as INCABS should not have similar limitations. On the contrar, it can have an abilit to evaluate accurate spectra at several locations without a significant increase in effort. The proposed approach will be modified to include the effects of higher order Ritz vectors and lumped masses. Base fleibilit: Reconciliation of finite element analsis results with in-situ and shake table test data has shown that the cabinet dnamic behavior can be significantl influenced b the tpe of mounting arrangement and the structural details at the cabinet base (lambias et al., 989; Gupta et al., 999b). Rustogi et al. (998a) developed simple equations to evaluate the stiffness of cabinet mounting arrangement for two tpical anchor bolt and base configurations. New formulations will be developed for additional anchor bolt configurations commonl found in switchgears and frame mounted cabinets. Computer program INCABS: A new version of the computer program INCABS will be developed to implement the proposed modifications. The modified program will be validated using suitable eamples. Application to building slabs with edge beams: Transverse vibration of plates with edge beams has been widel investigated in a variet of applications that include aerospace, civil, naval, and power plant structural sstems. Several formulations have been presented over the ears to evaluate the static and the dnamic characteristics of slabs 5

17 with elastic edge restraints. In most of the earlier studies, the bending and the torsional restraints are represented as stiffness coefficients with specified distributions along the slab edge. However, no formulations are presented to evaluate the distributions imparted b the presence of structural members such as beams along the slab edges. An attempt has been made to appl the Ritz vector approach to rectangular slabs in buildings that are supported on edge beams. The objective of this stud is to illustrate that some of the assumptions made in earlier studies on such slabs (Takabatake and Nagareda, 999; Warburton and Edne, 984) can be avoided b using the Ritz vector approach. 3. Organization This dissertation consists of primaril three manuscripts that the authors plan to submit for publication in the peer-reviewed journals. The first manuscript, part II of the dissertation, reviews the eisting methods for evaluating the incabinet response spectra needed in the qualification of safet related electrical instruments. The limitations of the Ritz vector approach proposed b Rustogi et al. (998a) and Gupta et al. (998, 999a) also discussed. The modifications to the Ritz vector approach and the application to actual cabinets are also presented. In the second manuscript, part III of the dissertation, the effect of the base stiffness on the cabinet dnamic behavior is investigated. Formulations are developed to calculate the base stiffness for three different tpes of mounting arrangements. These formulations 6

18 enhance the applicabilit of INCABS to cabinets in which the significant mode is a rigid bod cabinet rocking. The manuscript given in part IV of this dissertation provides a discussion on the eisting literature for evaluating the static and dnamic characteristics of rectangular slabs supported on edge beams. Ritz vector approach presented earlier in part II is simplified further for application to such structural sstems. Application of the Ritz vector approach to a variet of eamples is used to illustrate its advantages in avoiding some of the assumptions that have been made in the eisting literature. Finall, the summar and conclusions of this stud, and the recommendations for future studies are presented in Part V. 7

19 References. Gupta, A. and Gupta, A. K. (997) CREST-IRS, a Computer Program for Generating Instructure Response Spectra, Technical Report C-NPP-SEP 8/97, Dept. of Civil Engineering, North Carolina State Universit, Raleigh, NC.. Gupta, A., Rustogi, S. K. and Gupta, A. K. (998) Incabinet Response Spectra, Proceedings of the 7 th International Smposium on Current Issues related to Nuclear Power Plant Structures, Equipment and Piping, pp. XI---XI--8, Raleigh, NC, December Gupta, A., Rustogi, S. K. and Gupta, A. K. (999a) Ritz Vector Approach for Evaluation Incabinet Response Spectra, Nuclear Engineering and Design, Vol. 90, pp lambias, J. M., Sevant, C. J. and Shepherd, D. J. (989) Non-inear Response of Electrical Cubicles for Fragilit Estimation, Proceedings of th International Conference on Structural Mechanics in Reactor Technolog, Vol. K, pp , Anaheim, CA. 5. Rustogi, S. K., Gupta, A. and Gupta, A. K. (998a) Incabinet Response Spectra, Technical Report C-NPP-SEP /98, Dept. of Civil Engineering, North Carolina State Universit, Raleigh, NC 6. Rustogi, S. K., Gupta, A. and Gupta, A. K. (998b) INCABS, A Computer Program for Incabinet Spectra, User s Manual, Technical Report C-NPP-SEP 0/98, Dept. of Civil Engineering, North Carolina State Universit, Raleigh, NC. 8

20 7. Takabatake, H. and Nagareda, Y. (999) A Simplified Analsis of Elastic Plates with Edge Beams, Computers and Structures, Vol. 70, pp Warburton, G. B. and Edne, S.. (984) Vibrations of rectangular plates with elasticall restrained edges, Journal of Sound and Vibration, Vol. 95, pp

21 PART II MODIFIED RITZ VECTOR APPROACH FOR DYNAMIC PROPERTIES OF EECTRICA CABINETS AND CONTRO PANES Abhinav Gupta and Jianfeng Yang Submit to: Journal of Nuclear Engineering and Design 0

22 Modified Ritz vector approach for dnamic properties of electrical cabinets and control panels Abhinav Gupta and Jianfeng Yang Center for Nuclear Power Plant Structures, Equipment and Piping, North Carolina State Universit, Campus Bo 7908, Raleigh, NC , U.S.A. Abstract Often, the simple methods used for evaluating incabinet spectra needed in the seismic qualification of safet related electrical instruments ignore the dnamic characteristics of the electrical control panels and cabinets. Gupta et al. (999a) developed a Ritz vector approach for evaluating the dnamic properties of the cabinets and the incabinet spectra. This approach was based on the conclusions drawn from detailed finite element analses of several cabinets. In this paper, we illustrate the limitations of the originall proposed Ritz vector approach that were encountered during subsequent applications to actual cabinets. Modifications to the originall proposed formulations are presented and their accurac evaluated b comparison of results for actual cabinets with the corresponding results obtained from detailed finite element analses. The modified Ritz vector approach can account for actual rotational constraints

23 imparted b supporting structural members such as stiffeners. It can also be applied to bench board tpe cabinets in which instruments are mounted on plates or frames that are inclined to the global aes as well as to frames in which parallel frame members can vibrate in different vibration shapes due to differences in boundar conditions and nonuniform mass distribution.. Introduction In critical industrial facilities like nuclear power plants, the safet related electrical instruments such as relas must continue to operate during an earthquake. These instruments are seismicall qualified b a shake table test and are tpicall mounted on control panels and cabinets. The earthquake input needed in a shake table test is defined in terms of an incabinet response spectrum. The characteristics of the incabinet response spectra depend upon the dnamic characteristics of the cabinet, the location of floor where the cabinet is mounted, and the instrument location in the cabinet. in and oceff (975) and Stafford (975) describe some of the earl developments in the evaluation of cabinet dnamic properties and incabinet spectra. Djordjevic and O Sullivan (990) and Djordjevic (99) used the modal data from in-situ testing of 45 cabinets to develop two simple methods for evaluating incabinet spectra in cabinets that satisf a specified set of caveats. A detailed discussion of these methods was presented b Gupta et al. (999a) according to which these simple methods can give unrealistic

24 spectra that are ecessivel conservative. Conservatism is introduced b including amplifications at cabinet locations where an instrument ma never be mounted, b using unrelated high values of maimum pseudo participation factors, and b enveloping of several individual spectra. These methods are not intended to give precise incabinet spectra. Often, electrical instruments are qualified b using a standard spectrum shape recommended b ANSI / IEEE (978). These qualification studies are conducted with no consideration of the instrument location inside the cabinet, the tpe of cabinet on which the instrument is mounted, and the floor on which the cabinet is located, leading to ecessive conservatism (Bandopadha and Hofmaer, 986; Budnitz et al., 987). Alternativel, one ma generate the incabinet spectrum b a shake table test or a finite element analsis of the cabinet itself using floor response spectrum as input at the base of cabinet (Katona et al., 995). lambias et al. (989) and Gupta et al. (999b) have shown that the shake table testing of cabinets can be ver sensitive to the mounting arrangements of the cabinet due to nonlinear behavior ehibited b it. Both, the shake table testing and the finite element analsis are time and cost intensive. Recentl, Gupta et al. (999a) used detailed finite element analsis of 6 tpical cabinets to develop a Ritz vector approach for evaluating the cabinet dnamic properties and the incabinet spectrum at the instrument location within the cabinet. The earthquake input at the base of cabinet was defined in terms of the floor response spectra corresponding to the floor on which the cabinet is located. This approach was based on the conclusions drawn from the following observations: 3

25 Onl a few (often one) significant modes are needed to accuratel evaluate the incabinet accelerations at a particular instrument location in the cabinet. The significant mode is either a local mode (of the plate or the frame on which the instrument is mounted) or a global cabinet mode. It can also be a superposition of the local and the global modes. Gupta et al. (999a) represented the significant local and global mode shapes using mathematical functions known as Ritz vectors which were then used to develop an equivalent two degree of freedom sstem in the Raleigh-Ritz method. The cabinet dnamic properties were evaluated from the solution of this equivalent two degree-offreedom sstem. Present paper describes the limitations that were encountered during the subsequent applications of the originall proposed Ritz vector approach. Modifications needed to overcome the identified limitations are also presented. The modified Ritz vector approach is verified b comparison with corresponding results obtained from detailed finite element analses of actual cabinets.. Ritz Vector Approach The displacement u at a given instrument location can be epressed as a summation of Ritz vectors 4

26 n u(, η, t) = r ( t) (, η) () r= r where r (t) represents the r th generalized coordinate as a function of time t and r (,η) the Ritz vectors as a function of horizontal coordinate and vertical coordinate η. For rectangular plates and frames, the local mode shape r (,η) can be further simplified as r (, η) = hr ( ) vr ( η) () For convenience, we drop, η, t and rewrite Eq. as u = n r r = (3) hr vr Thus, the motion at a given instrument location is linearl transformed into an n-dof sstem using the above equation. Eigenvalue problem for this generalized n-dof sstem can be written as KX = ω MX (4) where K and M are the equivalent stiffness and mass matrices, respectivel, and X is the vector of generalized coordinates. Solution of the eigenvalue problem gives the frequenc 5

27 of significant cabinet mode and the corresponding eigenvector. If the significant cabinet mode is a superposition of a global and a local mode, Eq. can be simplified as u = g + (5) g l l in which g and l are the generalized coordinates; g is the Ritz vector for the global cabinet mode; and l is the Ritz vector for the local mode. Gupta et al. (999a) developed epressions for the elements of equivalent mass and stiffness matrices of the -DOF sstem. Epressions were developed for local plate as well as local frame modes. The two sets of epressions were then combined to formulate epressions for plates with stiffeners. 3. imitations of the Eisting Formulations Accurac of the dnamic properties calculated in the Raleigh-Ritz method depends on the selection of Ritz vectors. The selection of Ritz vectors for g and l depends on the boundar conditions of a cabinet and its components, respectivel. Several mathematical functions have been proposed as Ritz vectors b various researchers (Kim et al., 990; Ding, 995; Geannakakes, 995; Rajalingham et al., 996). Gupta et al. (999a) used simple and eas-to-use functions given b Blevins (979). For structural members having comple boundar conditions, evaluation of global mode g or local mode l ma require 6

28 more than one simple Ritz vector. However, use of multiple Ritz vectors in the formulations proposed b Gupta et al. (999a) ma not alwas be sufficient to evaluate accurate dnamic properties of the significant cabinet mode. To illustrate this effect, let us consider a bo tpe cabinet shown in Fig. in which the instruments are mounted on the back wall. A bo tpe cabinet fied at base does not ehibit a global bending mode and the significant mode needed in the evaluation of incabinet spectra is a local back wall mode. Selection of Ritz vectors needed in the evaluation of the local back wall mode depends upon the boundar conditions at the back wall edges. As seen in Fig. and in section A-A, the two side walls, the top plate, and the base plate restrain the out-of-plane translation at all the four edges of the back wall. In addition, an angle section stiffener is located at the corner between one of the side plates and the back wall whereas a unistrut is located as a stiffener at the corner between the other side plate and the back wall. Consequentl, the back wall plate is a stiffened plate with stiffeners at = 0 and =, respectivel. In general, a stiffener can impart a translational as well as rotational constraint on the plate deformation. The degree of these constraints depends upon the bending and the torsional stiffness of the structural member such as angle or unistrut. For the particular situation shown in Fig., the out-of-plane translation at the plate edges is restrained due to high in-plane stiffness of side, top and base plates and therefore, each stiffener can impart onl a rotational constraint. In the formulations proposed b Gupta et al. (999a) for stiffened plate, onl the bending stiffness of the structural members such as angle or unistrut is considered. The effect of torsional stiffness is neglected. The originall proposed formulations account for this effect onl through the estimation of 7

29 boundar conditions at the plate edges. The boundar condition at an edge that has angle (open section) stiffener is considered to be simpl supported. At the other edge that has a closed-section stiffener, the boundar condition is considered as clamped due to high torsional stiffness of a closed-section member. Such an approach gives accurate results when the torsional stiffness of the closed-section member is high as was the case in almost all applications described in Gupta et al. (999a). Often, the torsional stiffness of the closed-section member ma not be large enough and it ma impart a partial rotational constraint. Incorporation of multiple Ritz vectors in the originall proposed formulations cannot be used to represent partial rotational constraint because the torsional characteristics of the stiffeners are not considered. The cabinet shown in Fig. is not an actual cabinet but a modification of an actual cabinet in which different stiffeners were used at different locations. This was done to facilitate the discussion on the selection of Ritz vectors in the originall proposed approach. To illustrate the degree of inaccurac that can be introduced in the eisting formulations, let us consider an actual cabinet named CAB, shown in Fig., that has instruments mounted on the back wall and has unistruts at both the back wall edges. In the formulations proposed b Gupta et al. (999a), the boundar conditions at each edge can be estimated as either simpl supported or clamped depending upon the torsional stiffness of unistruts. We conducted a parametric stud in which the torsional constant for the two stiffeners was varied from a ver low value (simpl supported edge) to a ver high value (clamped edge). The variation of frequencies for the local back wall mode calculated from finite element analses is shown in Fig. 3. The formulations proposed b 8

30 Gupta et al. (999a) give the frequencies that are equal to 3.70 Hz for simpl supported boundar condition and 6.75 Hz for the clamped condition. For a particular case in which the unistruts have J = 0.05 in 4, Fig. 4 illustrates the inaccurac that can be introduced in the incabinet spectra at an instrument location on the back wall obtained from the originall proposed Ritz vector approach with respect to that evaluated from a finite element analsis. Another limitation of the eisting formulations is that the can be used to evaluate the dnamic properties of significant local modes in onl those cabinet tpes in which the local components such as plates or frames are parallel to one of the three orthogonal global aes. During the application of the originall proposed approach, we encountered another tpe of cabinets called bench boards, shown in Fig. 5, in which the instruments are mounted on local plates or frames that are inclined to the global aes. Modifications are needed in the formulations proposed b Gupta et al. (999a) to enable application of the Ritz vector approach to such cabinet tpes. For instruments mounted on an internal frame, Gupta et al. (999a) considered a generalized frame of the tpe shown in Fig. 6 to develop formulations needed in the Ritz vector approach. Further, it was assumed that the complete frame vibrates as a rectangular panel and that the displacement shapes for the parallel frame members are same, i.e. same Ritz vector can be used to represent the displacement shape in all the parallel (vertical or horizontal) frame members. The originall proposed formulations were found to give accurate results for evaluating incabinet spectra in specific members of a frame shown in Fig. 7 (a). However, it was later found that in a similar et different 9

31 frame, the different vertical members could vibrate in different displacement shapes due to differences in boundar conditions and non-uniform mass distribution. Fig. 7 (b) shows the deformation shapes for various members in such a frame. Consequentl, modifications are needed in the Ritz vector approach to account for this effect. Finall, the effect of higher order Ritz vectors is also illustrated in this paper. 4. Torsional Stiffness of Supporting Members In this section, new formulations are proposed for Ritz vector approach in which the effect of partial rotational constraint imparted b the stiffeners is accounted for. The kinetic energ T of a stiffened plate can be epressed as T = T + T + T + T (6) p hb vb m where subscript p denotes the contribution of the plate; b that of the stiffeners and m that of the lumped masses. The subscripts h and v represent the horizontal and the vertical stiffeners, respectivel. If the plate s mass densit is denoted b ρ and the mass per unit length for stiffeners b µ, we can write 0

32 = η η η ρ p d d u h T = j j hj hb d u T ), ( η µ = i i vi vb d u T ), ( η η η η µ = q q q q m u m T ), ( η in which ( - ) and ( η - η ) are the plate and the stiffener dimensions in and η directions, respectivel, as shown in Fig. 8. It should be noted that the plate and the stiffeners ma have different dimensions in the above epressions. Similarl, the potential energ U can be epressed as vb hb p U U U U + + = (8) in which ( ) ( ) η η η ν η ν ν η η d d u u u u u Eh U p = 3 4 (9) η η η d GJ d EI U r j vr hr r j hj r j vr hr r j hj hb ) ( ) ( + = (0) η η η η η η η d GJ d EI U r vr i hr r i vi r vr i hr r i vi vb ) ( ) ( + = () (7)

33 where E, G and ν are the Young s modulus of elasticit, shear modulus and Poisson s ratio for the plate material, respectivel. The moment of inertia for stiffeners is represented b I and the torsional constant b J. Differentiating T with respect to r s ields the elements m rs of the equivalent mass matri whereas differentiating U with respect to r s ields the elements k rs of the equivalent stiffness matri. For n Ritz vectors considered in the analsis, we get n equations of the following form. T r = n s= m rs s n U ; = k r s= rs s r= n () m rs can be written as the summation of the contributions from the plate m prs, the horizontal stiffeners m hrs, the vertical stiffeners m vrs, and the lumped masses m mrs. m = m + m + m + m (3) rs prs hrs vrs mrs η m = ρh d dη (4) prs hr hs η vr vs = hjvr ( η j ) vs ( η j ) j m hrs µ d (5) hr hs

34 = vihr ( j ) hs ( j ) i η m vrs µ dη (6) η vr vs m mrs = m q hr ( q ) hs ( q ) vr ( ηq ) vs ( ηq ) (7) q r= n; s= n Similarl, k rs can be written as the summation of the contributions from the plate k prs, the bending stiffness of the horizontal and vertical stiffeners k hirs and k virs, and the torsional stiffness of the horizontal and vertical stiffeners k hjrs and k vjrs. k = k + k + k + k + k (8) rs prs hirs hjrs virs vjrs [ k + k + k k ] 3 Eh k prs = prs prs prs3 + prs4 (9) ( v ) hr hs η k prs = d vrvsdη (0) η k prs η hr vs hs η vr = ν hsd vr dη ν hr d vsdη + η η () η η 3

35 4 ( ) = 3 η η η η η ν vs vr hs hr prs d d k () = 4 η η η η η vs vr hs hr prs d d k (3) η η d EI k hs hr j vs j j vr hj hirs ) ( ) ( = (4) η η η η d GJ k j vs j vr hs hr j j hjrs ) ( ) ( = (5) η η η η η d EI k vs vr i hs i i hr vi virs ) ( ) ( = (6) η η η η η d GJ k vs vr i hs i hr i i vjrs = ) ( ) ( (7) r= n; s= n

36 5. Bench Board Cabinet The 6 tpical cabinets considered b Gupta et al. (999a) included bench boards in which instruments are mounted on plates (both stiffened and un-stiffened) that are inclined to the global aes. ike other cabinets, the significant mode in bench boards is also a local mode of the inclined plate or a superposition of the local plate and the global cabinet modes. However, a modification is needed in the originall proposed Ritz vector approach for application to bench board cabinets. The local and the global coordinate aes for such a cabinet are shown in Fig. 9. At a particular instrument location on the inclined plate, the relative directions of the local and the global displacements represented using Ritz vectors are shown in Fig. 0. According to Fig. 0, the displacement u n normal to the plane of the inclined plate at a particular instrument location can be written as u n = sin α + (8) g g l l Similarl, the in-plane displacement (tangent to the plate) u t can be written as u = cosα (9) t g g The total displacement, u T of the plate is given b 5

37 u = u + u (30) T n t The kinetic energ T of the plate can be epressed as T = T g + T l (3) where subscript g denotes the contribution of the global cabinet mode and l that of the local plate mode, respectivel. If ρ denotes the mass densit for the plate and µ g the mass per unit length of an equivalent column used to represent the global cabinet behavior, we can write T g µ g = 0 g d (3) g g ρh η Tl= u Tddη (33) η In which ( - ) is the plate dimension in direction and ( η - η ) that in η direction. g is the dimension of the equivalent cantilever column used to represent the global behavior of the cabinet. Similarl, the potential energ, U, can be epressed as 6

38 7 U g U l U + = (34) in which d EI U g g g g g = 0 (35) ( ) ( ) η η η ν η ν ν η η d d u u u u u Eh U n n n n n l = 3 4 (36) where E, G and ν are Young s modulus of elasticit, shear modulus and Poisson s ratio for the plate material, respectivel. The moment of inertia of the equivalent cantilever column representing global behavior is denoted b I g. Differentiating T with respect to g and l, we can write l gl g gg g m m T + = (37) l ll g lg l m m T + = (38)

39 in which m gg, m gl, m lg and m ll represent the elements of equivalent mass matri M. The epressions for these elements can be written as m gg g = µ g gd + ρh 0 η η ddη g (39) m gl ( α) ddη η = m = ρh sin (40) lg η g l m ll = ρh η η ddη l (4) Similarl, differentiating U with respect to g and l ields the elements k gg, k gl, k lg and k ll of the equivalent stiffness matri K. U g = k gg g + k gl l (4) U l = k gl g + k ll l (43) in which 8

40 9 ( ) ( ) η α η η η d d v Eh d EI k g g g gg g 3 0 sin + = (44) ( ) ( ) ( ) η α η η α η ν η η d d v Eh k k l g l g lg gl + = = sin sin 3 (45) ( ) ( ) η η η η ν η η d d v v Eh k l l l l l ll = 3 (46) Solution of the eigenvalue problem given b Eq. 4 ields the frequenc of the significant cabinet mode and the corresponding eigenvector X i. ] [ li gi T i = X (47) An earthquake input in z direction at the cabinet base results in significant displacements u z and u along the z and aes, respectivel, at a particular instrument location. These displacements, needed to evaluate the incabinet spectra, can be calculated as α sin l li g gi z u + = (48) cosα l li u = (49)

41 6. Internal Frame Net, we present a modification needed to account for different boundar conditions at the supports of different members in an internal frame of the tpe shown in Fig. and discussed earlier. The displacement in the local mode shape at an instrument location on the vertical frame member can be written as u l n = i= vi f i ( ) vi + hh (50) in which n is the number of vertical frame members, vi is the Ritz vector for the i th vertical frame member and h that for the horizontal frame members. The function f i () is given b f i (5) ( ) = n j j= i j j i 30

42 such that f i ()= at = i and f i ()=0 at i. It should be noted that f i () varies onl with and can be viewed as a Ritz vector in the horizontal direction. Consequentl, we can rewrite Eq. 50 as u n = + i= l i (5) hi vi in which vi i =... n i = (53) h i = n + ( ) fi i =... n hi = (54) h i = n + vi i =... n vi = (55) i = n + Eq. 5 has the same form as Eq. 3. Therefore, the formulations for the equivalent mass and stiffness matrices presented in section 4 can be used to calculate the dnamic properties of the internal frame. 3

43 7. Results We applied the modified Ritz vector approach to several actual cabinets discussed earlier in section 3. The structural properties and configurations of these cabinets are summarized in Table. To account for a partial restraint (translational or rotational) in the proposed formulation, two Ritz vectors are needed to represent an intermediate boundar condition. For eample, let us once again consider the cabinet CAB shown in Fig.. The partial rotational constraint that ma be imparted b the unistruts can now be accounted for b considering two Ritz vectors in the horizontal direction, one for the simpl supported boundar condition and the other for the clamped condition. Fig. compares the frequencies of the local back wall mode calculated from the proposed approach with the corresponding values obtained from the finite element analses. If the unistruts have J = 0.05 in 4, the incabinet spectrum for % damping calculated from the proposed approach is compared to that evaluated from the finite element analsis in Fig. 3. The cabinet damping ration is taken as 5%. The input spectrum for % damping is also shown in Fig. 3. As discussed earlier and shown in Fig. 4, the originall proposed Ritz vector approach gave significantl different spectra for this particular case. In the bench board SSF shown in Fig. 5, the instruments are mounted on the 36 in 65 in front panel that is inclined at 56.3 degree to the horizontal direction. The significant mode in this bo-tpe cabinet is a local mode of this front panel. The frequenc of the significant mode calculated from the proposed approach is 6.4 Hz compared to 6 Hz calculated from the finite element analsis. The incabinet response 3

44 spectrum for 3% damping calculated from the proposed approach is compared to that evaluated from the finite element analsis in Fig. 4. The cabinet damping ratio is taken as 5%. The input spectrum for 3% damping is also shown in Fig. 4. For a bench board cabinet SSF that is identical to the cabinet shown in Fig. 5 but does not have side plates, the significant mode is a superposition of the local mode for the 36 in 65 in front panel and a global cabinet mode. The frequenc of this significant mode calculated from the proposed approach is 0.7 Hz compared to Hz calculated from the finite element analsis. The incabinet response spectra for the two cases are compared in Fig. 5. The cabinet damping ratio is taken as 5%. The input spectrum is also shown in Fig. 5. For the internal frame PROT IV shown in Fig. 7 (b), the vertical frame members I and IV are considered to be fied at their ends whereas members II and III are considered to be simpl supported. The frequenc of the significant mode evaluated using the formulation proposed in section 6 is 9. Hz which is same as that evaluated from the finite element analsis. Figs. 6(a) and 6(b) compare the incabinet response spectra for 3% damping evaluated b the two methods for instrument locations on members I and II, respectivel. The cabinet damping ratio is taken as 5%. The input spectra are also shown in Figs. 6(a) and 6(b). et us consider a cabinet MS4706, shown in Fig. 7 that has instruments mounted on its back wall. The significant mode for this bo-tpe cabinet is a local mode of the back wall plate. Several lumped masses are located non-uniforml along the height of the back wall. For simplicit, one can consider the total mass of the back wall including 33

45 lumped masses which can then be used to evaluate an equivalent densit for the plate in the Ritz vector approach. Such a simplification with single Ritz vector in the vertical direction for pinned-pinned condition gives a frequenc of 40.5 Hz for the back wall mode, compared to the 34.3 Hz given b finite element analsis (Gupta et al., 999a). i.e. the single Ritz vector considered in Gupta et al. (999a) was π h = sin (56) πη v = sin (57) η Emphasis in Gupta et al. (999a) was on the accurac as well as simplicit in evaluating maimum amplification on the back wall mode. Though the maimum amplifications from the two sets of analses are close to each other, the mode shapes from the two analses are ver different. Consideration of non-uniforml distributed lumped masses in Ritz vector approach ma require inclusion of higher order vectors for the displacement shapes. The accurac of the dnamic properties increases with the number of higher order Ritz vectors considered in the analsis. However, the computational compleit increases with increasing number of Ritz vectors considered. The local mode shape of the back wall in vertical direction for cabinet MS4706 as calculated from the finite element analsis is shown in Fig. 8. Such a mode shape cannot be represented accuratel b a 34

46 single Ritz vector. Fig. 8 also shows the local mode shapes evaluated from the Ritz vector approach b considering different number of Ritz vectors in the following summation n n i v ppi i πη = = sin i (58) i= i= η It should be noted that the accurac improves significantl b considering five Ritz vectors in the particular case considered. However, the number of Ritz vectors needed for the evaluation of accurate incabinet spectra ma be much less depending upon a particular scenario. For eample, if onl one incabinet spectra is evaluated for qualification of all the instruments mounted on the back wall, a single Ritz vector ma be sufficient as the maimum amplifications are close to those evaluated from the finite element analsis even though these maimum amplifications occur at different values of η. On the other hand, if the incabinet spectrum is evaluated at a particular instrument location such as η = 40 in, more than one Ritz vector will be needed. As in finite element analsis, a convergence stud ma be performed in the Ritz vector approach to evaluate the accurac in cabinets with significant non-uniform distribution of the lumped masses and the stiffeners on the plate. 35

47 8. Summar and Conclusions Gupta et al. (999a) presented a Ritz vector approach based on the observations from detailed finite element analsis of 6 tpical cabinets to evaluate incabinet response spectra needed in seismic qualification of electrical instruments such as relas. These formulations use Raleigh-Ritz method to calculate the dnamic properties of the significant cabinet modes which are then used for evaluating incabinet response spectra. A few limitations were encountered in the subsequent application of this method to actual cabinets. In this paper, modifications are presented to overcome the identified limitations and improve the accurac of the Ritz vector approach. These modifications are Incorporation of the bending as well as the torsional stiffness of the constraining structural members such as stiffeners to account for the partial constraints imparted b them. Transformations for application to bench boards in which the instruments are mounted on plate or frame that are inclined to the global aes. Additional transformations for application to internal frames in which different but parallel members ma have different boundar conditions and mass distributions. Incorporation of higher-order Ritz vectors to improve the accurac for non-uniform distribution of lumped masses. Application of the modified approach to several actual cabinets is used to illustrate the improvement in the accurac of calculated dnamic properties and incabinet response spectra. 36

48 Acknowledgements This research was partiall supported b the Center for Nuclear Power Plant Structures, Equipment and Piping at North Carolina State Universit. Resources for the Center come from the dues paid b member organizations and from the Civil Engineering Department and College of Engineering in the Universit. 37

49 References ANSI / IEEE, 978. IEEE Standard Seismic Testing of Relas. C Bandopadha, K. K., Hofmaer, C. H., 986. Seismic Fragilit of Nuclear Power Plant Components (Phase I). NUREG/CR-4659, US Nuclear Regulator Commission. Blevins, R., 979. Formulas for Natural Frequenc and Mode Shape. Van Nostrand Reinhold Compan, New York. Budnitz, R. J., ambert, H. E., Hill, E. E., 987. Rela Chatter and Operator Response after a arge Earthquake. NUREG/CR-490, US Nuclear Regulator Commission. Ding, Z., 995. Natural frequencies of elasticall restrained rectangular plates using a set of static beam functions in the Raleigh-Ritz Method. Comput. Struct. 57, Djordjevic, W., 99. Amplified response spectra for devices in electrical cabinets. In: Current Issues Related to Nuclear Power Plant Structures, Equipment and Piping, Proceedings of the Fourth Smposium, December, Orlando, F. Djordjevic, W., O Sullivan, J. J., 990. Guidelines for Development of In-Cabinet Amplified Response Spectra for Electrical Benchboards and Panels. Report. Stevenson & Associates, Inc. Geannakakes, G. N., 995. Natural frequencies of arbitraril shaped plates using the Raleigh-Ritz Method together with natural co-ordinate regions and normalized characteristic orthogonal polnomials. J. Sound Vibrat. 8 (3), Gupta, A., Rustogi, S. K., Gupta, A. K., 999a. Ritz vector approach for evaluation incabinet response spectra. Nuclear Engineering and Design, 90,