MODAL SEISMIC ANALYSIS OF A NUCLEAR POWERPLANT CONTROL PANEL AND COMPARISON WITH SAP IV

by

M. R. PamidiSargent and Lundy Engineers

Chicago, lllinois

and

P. R. PamidiComputer Sciences Corporation

Hampton, Virginia

SUMMARY

This paper deals with the application of NASTRAN to seismic analysis byconsidering the example of a nuclear power plant control panel. A modal analy-sis of a three-dimensional model of the panel, consisting of beam and quadri-lateral membrane elements, is performed. Using the results of this analysisand a typical response spectrum of an earthquake, the seismic response of thestructure is obtained. ALTERs required to the program in order to compute themaximum modal responses as well as the resultant response are given. The re-sults are compared with those obtained by using the SAP IV computer program.

INTRODUCTION

Current government and industry regulations [References 1 and 2] requirethat the safety-related systems, structures and components of nuclear powerplants be designed to withstand specified seismic excitations without loss ofcapability to perform their safety functions. This requirement is necessaryin order to ensure

(a) continued operation of the reactor without undue risk to the healthand safety of the public during an Operating Base Earthquake (OBE),and,

(b) shutdown of the reactor and its maintenance in a safe shutdown con-dition following a Safe Shutdown Earthquake (SSE).

The design of such equipment to withstand seismic disturbances involvesdynamic analysis, testing or a combination of both. Seismic qualification byanalysis alone is deemed sufficient provided the safety function of a structureor component is assured by its structural integrity. Thus, for instance, mostmechanical equipment, such as heat exchangers, tanks, pressure vessels, etc.,are usually qualified by analysis. Qualification by testing is recommended

J-LJ

https://ntrs.nasa.gov/search.jsp?R=19770013565 2018-05-26T13:18:36+00:00Z

in those cases where functional operability is not necessarily assured bystructural integrity. Thus, most electrical equipment, such as switchgears,motor control centers, control panels, etc., are usually qualified by testing.However, in many instances, testing is impractical either due to the size ofthe equipment involved or due to the prohibitive cost entailed by such testing.In such cases, a detailed dynamic analysis of the equipment, such as a controlpanel, is performed and the various associated electrical instrumentation anddevices are then tested to acceleration levels determined by the analysis.

In this study, the application of NASTRAN to seismic analysis has beendiscussed by considering the example of a nuclear power plant control panel.A modal analysis of a three-dimensional model of the panel, consisting of beamand quadrilateral membrane elements, is performed. Using the results of thisanalysis and a typical response spectrum of an earthquake, the seismic responseof the structure is obtained. The results are compared with those obtained byusing the SAP IV computer program.

FORMULATION OF THE SEISMIC ANALYSIS PROBLEM

Obtaining the Modal Responses

The dynamic behavior of a system having multiple (n) degrees of freedomand subjected to seismic excitation is described by a set of differentialequations represented by

[M] {u} + [C] {u} + [K] {u} : - [M] [D] {uf} (1)

where [M], [C] and [K] are the (n x n) mass, damping and stiffness matricesrespectively of the system; {u} is the (n x I) displacement vector of interest;

{uf} is a (3 x I) vector that represents the time-dependent floor accelerationin the three component (X, Y and Z) directions; and [D] is an (n x 3) directioncosine matrix consisting of ones and zeroes that selects the masses that areinvolved in the motion in the three directions. The negative sign in thisequation merely indicates that the effective load due to the seismic distur-bance is opposite to that of the floor acceleration.

The displacement vector {u} in Eq. (I) may be expressed in terms of thenormal coordinates as

{u} : } "i : {w} (2)i =I

where {w} is an (n x I) vector that represents the normal (or generalized) coor-

dinates wi and [@] is an (n x n) matrix whose columns {@i } are the n eigenvectors

of the free, undamped system given by

[M] {u} + [KJ {u} : 0 (3)

516

By using the orthogonal properties of the eigenvectors and utilizing therelationships that exist among the generalized mass, damping and stiffness

quantities at any mode, Eqs. (l) and (2) can be combined to give a set of nuncoupled equations in the nomam coordinates as follows [Ref. 3]:

{_i + 2 _i miwi + m_ wi = - {Fi}T{uf }' i = I, 2, ..... , n (4)

where _i is the damping ratio and mi is the natural frequency of the i th mode.

{F i} is a (3 x 1) vector whose elements represent the so-called modal parti-

cipation factors in the three component directions for the i th mode and is givenby

{Fi}T= I_!xl T = {_i }T [M] [D]F!Ylz {@i} T [M] {@i} (5)

The denominator on the right hand side of the above equation is the generalizedth

mass for the i mode.

Equations (4) can be solved for significant modes by direct integration if

the time-history of the floor acceleration {_f} is known. The solutions for the

various significant modes can then be superimposed as per Eq. (2) to obtain thetotal solution. However, from a design point of view, it is simpler and oftenmore convenient and economical to obtain the maximum displacements in any givenmode by response spectrum analysis. This approach involves the use of designspectra derived from past earthquake data.

re reenhesoera s acemens,or,ehmo e_diy

(}_diz

for excitations in the X, Y and Z directions. The corresponding spectral velo-

cities {Svi} and spectral accelerations {Sai} are related to the spectral dis-

placements by

{Svi} {Sai}{Sdi} - = _ (6)

The maximum response for the i th mode due to the individual spectral dis-

placements Sdi x, Sdiy and Sdi z in the X, Y and Z directions respectively is

D.I./

given by [Ref. 3]:

{Ui}xma x {@i } Fix Sdi x(7a)

{u i = {@i }}Ymax Fiy sdiy(7b)

{u i} = {@i } Fiz Sdi zZmax

(7c)

The maximum response for the i th mode due to simultaneous seismic exci-

tations in the three component directions is obtained by combining Eqs. (7a),(7b) and (7c) and is represented by I

• + {u i + {u i{Ui}max = {Ul}xmax }Ymax }Zmax(8)

or {Ui}ma x = {@i } Gi (9)

where Gi is a scalar quantity given by the product

Gi = {Fi}T{Sdi} (10)

Eq. (9), which gives the maximum response for the i th mode, can be genera-lized to give the maximum response for any of m modes (I !m _n) by the singlematrix equation

[U']max = [@'] diag[[F']T[s_]] (11)

where

[U']ma x = [{Ul}ma x {U2}ma x ..... {Um}ma x] (12a)

Eq. (8) gives a conservative estimate for the maximum response for the i th mode.It is acceptable to the regulatory authorities to compute this response by tak-ing the square root of the sum of the squares of the maximum responses in thethree component directions [Ref. 4].

518

and

[@'] = [{@l } {@2 } ..... {@m }]

[F'] = [{Fl} {F2} ..... {Fm} ]

[S_] = [{Sdl} {Sd2} ..... {Sdm} ]

Eq. (ll) can be rewritten as

(12b)

(12c)

(12d)

[U']ma x = [@'] [G']

where [G'] = diag[F']T[s_]

l 0 ..... 0-

G2 ..... 0

• _ e• •

• Io. •

0 0 ..... G_-m

The elements of the (m x m) diagonal matrix [G'] are given by Eq. (10).

(13)

(14)

Combining the Modal Responses

The total response of the system can be obtained by combining the maximum

responses of the individual modes involved as given by Eq. (13). The exact

manner in which these modal responses are combined is, however, a matter of

judgment and there is no one way for obtaining the total response. When the

modes are not closely spaced, 2 current regulatory practice [Ref. 4] requires

that the resultant response (whether it be displacement, stress or other quan-tity) be obtained by taking the square root of the sum of the squares (SRSS)

of the corresponding maximum responses for the individual modes involved. When

the modes are closely spaced, it is required that the total response be computed

by any one of three methods acceptable to the regulatory authorities. Referredto as the Grouping Method, the Ten Percent Method and the Double Sum Method, the

details of these methods are given in Reference 4.

Two modes are said to be closely spaced if their frequencies differ from each

other by I0% or less of the lower frequency.

519

ANALYSIS

Analysis by NASTRAN Program

NASTRAN cannot directly perform seismic analysis, but can be adapted forthe purpose. The procedure employed here corresponds to the theoretical devel-opment presented in the previous section. The method essentially involves two

major steps as indicated below3:

Step I. Perform a normal mode analysis using Rigid Format 3 and obtain the

significant modes."

Step 2. Once the significant natural frequencies are known, input the appro-

priate spectral displacement matrix [S_] [see Eq. (12d)] and the

direction cosine matrix [D] [see Eq. (I)] via DMI bulk data cardsand repeat the normal mode analysis using the following ALTERs toRigid Format 3 (Level 16) and employing N_RM = MASS (normalizationto unit value of the generalized mass) on the EIGR bulk data card[Ref. 7]:

ALTER 106 $

MPYAD PHIA,MAA,/DUMMYA/C,N,I $

MPYAD DUMMYA,DIRCOS,/MODEPF/C,N,O $

MPYAD MODEPF,SEISMIC,/DUMMYB/C,N,O $

DIAGONAL DUMMYB/DUMMYC/C,N,SQUARE $MPYAD PHIA,DUMMYC,/PHIASS/C,N,O $

ALTER I08 $

SDRI USET,,PHIASS,,,GO,GM,,KFS,,/PHIG,,QG/C,N,I/C,N,REIG $ALTER 121 $

TRNSP PHIASS/PHIASST $

MPYAD PHIASS,PHIASST,/DUMMYD/C,N,O $

DIAGONAL DUMMYD/SRSSDISP/C,N,COLUMN/C,N,O.5 $

MATPRN SRSSDISP,,,,// $

ENDALTER $

The adaptation of NASTRAN for seismic analysis has also been discussed byother users [Refs. 5 and 6].

For seismic analysis purposes, a mode is considered significant if its fre-quency is less than or equal to 33 Hz.

520

The data blocks used in the above ALTER package have the following corres-pondences to the matrices mentioned in the previous section:

PHIA - [¢']

MAA - [M]DIRC(_S - [D]

M(_DEPF _= [F']T

SEISMIC _ [S'd]

DUMMYC z [G']

PHIASS _ [U']max

The so-called eigenvectors printed out by the above analysis actually repre-

sent the maximum modal responses (displacements) as given by Eq. (13). The

stresses obtained correspond to these displacements.

The output data block SRSSDISP in the above ALTER package is an (n x l)

vector that represents the resultant response obtained by taking the squareroot of the sum of the squares (SRSS) of the maximum modal displacements. The

corresponding SRSS stresses can be obtained by combining the individual modalstresses.

Analysis by SAP IV Program

SAP IV can perform seismic analysis directly without the need for a separateintermediate run just to obtain the significant modes. The proportions of the

seismic excitations in the three directions are specified and the spectral infor-

mation (displacements or accelerations) is input as a table of spectral values

versus period. The maximum modal displacements and the resultant (SRSS) dis-

placements and stresses are automatically output. The details of the method

are explained fully in Reference 8.

FINITE ELEMENT MODEL

The basic details of the finite element model of the control panel consideredare shown in Figures l, 2 and 3. Beam elements of three different cross sections

and quadrilateral membrane elements of two different thicknesses are used to model

the structure. The beam elements are represented by the CBAR elements in NASTRAN

and by the three-dimensional beam elements (element type 2) in SAP IV; the mem-

brane elements are represented by the CQDMEM elements in NASTRAN and by the plane

stress quadrilateral membrane elements (element type 3) in SAP IV. A total of

265 active degrees of freedom are involved in the analysis. The complete detailsof the model can be obtained from the authors.

521

RESULTS

Using essentially identical input data, the finite element model describedabove was analyzed by both NASTRAN and SAP IV programs following the procedureoutlined earlier. Seismic excitations of equal magnitude in the three componentdirections were assumed. The spectral data used was based on the El Centro(Calif.) earthquake of 1940. A damping of 2% (see Reference 9 for guidelinesin this regard) was assumed.

The results of the analysis are presented in Tables I, 2 and 3. Table 1lists the significant natural frequencies obtained by the two programs. Table2 gives the corresponding modal participation factors. Table 3 shows some re-presentative resultant (SRSS) displacements of significant magnitude.

It can be seen from Table 1 that the agreement between the significantnatural frequencies obtained by the two programs is excellent. The modal part-icipation factors given in Table 2 agree well too, except when their magnitudesare small; this is due to small differences that exist in the various eigenvectorcomponents (not shown) obtained by the two programs. The representative resultant(SRSS) displacements of significant magnitude shown in Table 3 also agree well,but the same is not true when they are of smaller magnitude. This is due notonly to the small differences in the modal participation factors involved, butalso to the different manner in which the spectral data is input to the twoprograms. The agreement in the results, on the whole, is quite good.

SUMMARY AND CONCLUSIONS

The application of NASTRAN to seismic analysis has been discussed by con-sidering the example of a nuclear power plant control panel. A modal analysisof a three-dimensional model of the panel, consisting of beam and quadrilateralmembrane elements, is performed. Using the results of this analysis and a typi-cal response spectrum of an earthquake, the seismic response of the structureis obtained. ALTERs required to the program in order to compute the maximummodal responses as well as the resultant (SRSS) response are given. The resultsare compared with those obtained by using the SAP IV program. The agreement, onthe whole, is quite satisfactory.

The paper demonstrates the adaptability and suitability o£ NASTRAN for seis-

mic analysis. The greater choice of elements offered by NASTRAN as well as the

availability of such desirable features as the CNGRNT capability (which can re-

sult in significant reductions in running times particularly for large problems)[Ref. 7] and Guyan reduction [Ref. lO] make this versatile program an attractivetool for seismic analysis of large structures.

522

l •

Q

•

•

•

•

•

8.

go

lO.

REFERENCES

Regulatory Guide 1.29, Seismic Design Classification, U. S. Atomic EnergyCommission, Directorate of Regulatory Standards, Washington, D. C., Revision

l, August 1973.

IEEE STD 344-1975, IEEE Recommended Practices for Seismic Qualification of

Class IE Equipment for Nuclear Power Generating Stations, IEEE, New York,1975.

Wiegel, R. L., (ed.), Earthquake Engineering, Prentice-Hall, EnglewoodCliffs, N. J., 1970.

Regulatory Guide 1.92, Combining Modal Responses and Spatial Components in

Seismic Response Analysis, U. S. Nuclear Regulatory Commission, Office of

Standards Development, Washington, D. C., Revision l, February 1976,

Go, J. C., "Seismic Analysis of Nuclear Power Plant Structures", NASTRAN:

Users' Experiences, NASA TM X-2893, September 1973, pp. 243-249•

Setlur, A. V. and Valathur, M., "Some Studies on the Use of NASTRAN for

Nuclear Power Plant Structural Analysis and Design", NASTRAN: Users'

Experiences, NASA TM X-2893, September 1973, pp. 585-594•

The NASTRAN User's Manual, (Level 16.0), NASA SP-222(03), March 1976.

Bathe, K. J., Wilson, E. L., and Peters•n, F. E., SAP IV: A Structural

Analysis Program for Static and Dynamic Response of Linear Systems, Uni-versity of California, Earthquake Engineering Research Center, Berkeley,California, June 1973.

Regulatory Guide 1.61, Damping Values for Seismic Design of Nuclear Power

Plants, U. S. Atomic Energy Commission, Directorate of Regulatory Standards,

Washington, D. C., October 1973.

The NASTRAN Theoretical Manual, (Level 16.0), NASA SP-221(03), March 1976.

J_J

Table I. Natural Frequencies of Significant Modes

Modeno.

1

2

3

4

5

6

7

8

9

I0

II

12

13

Natural frequencies (Hz)

NASTRAN results

0.6352E+01

0.6563E+01

0.6699E+01

0.I013E+02

0,I180E+02

0.1401E+02

0.1648E+02

0.1652E+02

0.1665E+02

0.1706E+02

0,2371E+02

0.2578E+02

0.3082E+02

SAP IV results

0.6352E+01

0.6563E+01

0.6700E+01

0.I013E+02

0.I180E+02

0.1401E+02

0.1649E+02

0.1652E+02

0.1665E+02

0,1706E+02

0.2370E+02

0.2579E+02

0.3082E+02

524

IU

.I-

N

O ¢_ O C:)O O C) O4- 4- 4- 4.I.ul Li./ LI.I I.l.I

00 C0 I_CM q" CM O'/I.¢) e-_ ,--. O

c; c; c; (_!

o 8m 4-

ur)c0 c0(M coq- cM

i

P- i-- r=.o o O o

I ! ! Ila.I td LhlCO ,--. C_I -

I I

CM

! ÷ IlaLJ ILl LIJ

CM r--

! !

0Z

U

NS..0

Ii-

0

U

e-0

L

0

_4

I---

0

U

"G°e.-

s,.

• 0 0 •

! I

:g

e-o

u

°_

iN

CM P- .N

. . . . ,,,, ,,', - + ,hi I.IJ

P1 O _ '_" tO _ _ el NP" I._ _O N O'/ ,-- N I-" N

c:; c; c; _ (:; _; c; c; c;I I, I I

0 0 0 0 0 0I I,,, ,,, - . + ,hl hl

• • * • o .0 0 {3 (D {3 O

I I

(:3 Oa Oa _0 0 _0 (:30 0 0 0 0 0 0÷ _ m ; ÷ I ÷

Lr; _ 00 o co 00 _;p

I I I I I

0 0 O 0 _ 0C_ 0 0 0 0 0 0+ 4- + + : 4- 3LLI I.¢.1 _ ILl ILl ILl _ t_l

03 p O I_ CO r_

l I

3 =o _

!

O O O (_ OI,,, - . ,+, -

0 CM 0 01 N

• • 0 , •0 0 0 0 0

I I I I

"" P N P- No o o o o o _ _ _ o F_. . . . . ,,, - . . +,., ,.,,

I I I I I

¢-.O

(J

S..

_3

0 0 0 0 0 0 0 0

. . . . + ÷,,, ,,,, ,.,,

! I !

(M O_ 0 CO 00 0 0 0 0

- ,,, + ,,,, +LIJ

c; c; c; c; c;I I I I

e_"_ ° P" N Cv) _1" L¢') IJ_ I"% CO (_

r_J_J

e-

E

°r-

(--QJ tD_

E(I) o')

O O O O O O O OO O O O O O O O

+ 4- + + + + + +

O _ CO _ _ _ _ I_

oO

v

4-_e-

4-_

4_O

(I)

°_

r_

v

Z

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0+ + + + + + + +

_ _S" CO _ O4 (_J OCO O O_ _ _ _O _O _O

°_

c-

m_

o4

O

c- c-O O

O

_E

.r--

c-o_O

t-

°r-

N N X N N X N N

O_ 0 0 '_" _ _ O_ O

526

• X

I

--I,_ _ I'_ _I I'_--

527

528

X

529