Text

Requests Comments on Encl Draft Final Rept Reevaluation of Regulatory Guidance on Modal Combination Methods for Seismic Response Spectrum Analysis. Comments Requested to Be Provided by 980911
	ML20154D977
Person / Time
Issue date:	08/06/1998
From:	Murphy A; NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES)
To:	Bagchi G, Wessman R; NRC (Affiliation Not Assigned)
Shared Package
ML20154D982	List: ML20154D977; ML20154D984;
References
	REF-GTECI-A-40, REF-GTECI-SC, RTR-REGGD-01.092, RTR-REGGD-1.092, TASK-A-40, TASK-OR NUDOCS 9810080004
	Download: ML20154D977 (54)
	v • d • e

pg g

t UNITED STATES s

j NUCLEAR REGULATORY COMMISSION t

WASHINGTON, D.C. 20555 4001 August 6, 1998 12 2

MEMORANDUM TO: Richard H. Wessman, Chief 6 I

' /. '

Mechanical Engineering Branch 8$M Division of Engineering 9 C3 C

Office of Nuclear Reactor Regulation 5 = <iTl

=

Goutam Bagchi, Chief

$O Civil Engineering and Geosciences Branch 3.:::

Division of Engineering

~

~~

Office of Nuclear Reactor Regulation FROM:

Andrew J. Murphy, Chief

/

Structural & Geological Engineering Branch Division of Engineering Technology Office of Nuclear Regulatory Research

SUBJECT:

REVIEW OF DRAFT FINAL REPORT " REEVALUATION OF REGULATORY GUIDANCE ON MODAL COMBINATION METHODS FOR SEISMIC RESPONSE SPECTRUM ANALYSIS" 1

Reference:

Memorandum from William T. Russell to Eric S. Beckjord,

Subject:

" User Need" Request for Research Related to Areas of Seismic, Structural, and Geotechnical Engineering, dated December 16,1994.

I am requesting your comments on the attached draft final report " Reevaluation of Regulatory Guidance on Modal Combination Methods for Seismic Response Spectrum Analysis." This report was developed in response to the referenced " user need" memorandum that requested the Office of Nuclear Regulatory Research to initiate a project that would provide the technical basis to revise Regulatory Guide 1.92, " Combining Modal Responses and Spatial Components in Seismic Response Analysis," Revision 1, February 1976.

The " user need" memorandum noted that a number of methods for combining the responses of closely spaced modes have been devised which are not as conservative as those specified in the current regulatory guide but are nonetheless considered to be more accurate and appropriate to use. The " user need" memorandum also noted that during the revision of Standard Review Plan Section 3.7.2 (as part of the resolution of USI A-40), the staff decided to

. incorporate the use of these altemative methods by reference (to be reviewed on a case by case basis) with an implicit understanding to revise the regulatory guide later in response to public comments (see NUREG/CR-5347, " Recommendations for Resolution of Public Comments on USI A-40").

Several members of your staff, for example, Mark Hartzman, John Fair, Kamal Manoly, and David Jeng have participated in project review meetings with the contractor. Their input has been incorporated into the draft final report.

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9810080004 980806

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PDR RECGD

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o 01.092 C PDR f( M -/Pr

2 Your comments on the attached report are requested by September 11,1998. Questions should be directed to Roger M. Kenneally (telephone: 415-6303, email: rmk).

After the report is published, Roger Kenneally will develop and distribute for your review and comment, draft technical positions that will be used in the <evisiun of Regulatory Guide 1.92.

You input will be elicited on all aspects of the regulatory guide revision.

Attachment:

As Stated cc: PDR s

e n,

I AP.PENDIX A 4

Description of BM3 Piping Model s

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MODEL 3---UNIFORM 1 % TIME HISTORY ANALYSIS--- tha_x_ mall _dl.inp CONTROL INFORMATION 38 NUM ER OF NODAL POINrS

=

2 NUM ER OF ELEMENT TYPES NUMER OF STATIC IDAD CASES 0

=

NUMBER OF DYNAMIC CASES 1

=

0 NUMER OF ANCHOR CASES

=

31 NUMER OF FREQUENCIES

=

SOIITTION MODE 00DEK) 0

=

.0, EXECUTION

.1, DATA CHECK 0

SS CALCULATION FLAG

=

EQ.0 NO EQ.1 YES 0

ASME CODE EVAIDATION FLAG

=

EQ.1 CLASS 1 PIPING EQ.2 CLASS 2 OR CLASS 3 PIPING ACCELERATION DUE 'IO GRAVITY

=386.4 BANDWIITTH MINIMIZATION FLAG

=

1

.0 NO E

.1 YES 0

ARBI NODE NUMBERING FIAG

=

.0 NO E

.1 YES

=

1 NUMB R OF SUPPORT GROUPS FLAG FOR NODAL COORD. INPUT UNITS =

0 EQ.0 CONSISTENT UNIT EQ.1 FEET 'IO INCHES i

LIST OF ANALYSIS 'IO BE PERFORMED IDAD CASE DISK FILE ANALYSIS TYPE 1

0 UNIFORM TIME HIS'IORY ANALYSIS 1 NODAL POINr INPUT DATA ONODE BOUNDARY CONDITION CDDES NODAL POINT OOORDINATES NUMBER X Y

Z XX YY ZZ X

Y Z

T 1

0 0

.000

.000 2

0 0

15.000

.000

.000 3

0 0

19.500

-4.500

.000

.000 4

0 0

19.500

-180.000

.000

.000 5

0 0

19.500

-199.500

.000

.000 6

0 0

19.500

-204.000 4.500

.000 7

0 0

19.500

-204.000 139.500

.000 8

0 0

24.000

-204.000 144.000

.000 9

0 0

96.000

-204.000 144.000

.000 10 0

0 0

0 254.000

-204.000 144.000

.000 11 0

0 0

0 333.000

-204.000 144.000

.000 12 0

0 0

0 411.000

-204.000 144.000

.000 13 0

0 0

0 483.000

-204.000 144.000

.000 14 0

0 0

'O O

O 487.500

-204.000 148.500

.000 15 0

0 0

0 487.500

-204.000 192.000

.000 16 0

0 0

0 487.500

-204.000 235.500

.000

17 0

0 0

0 492.000

-204.000 240.000

.000 18 0

0 0

0 575.000

-204.000 240.000

.000 19 0

0 0

0 723.000

-204.000 240.000

.000 20 0

0 0

S 0

0 727.500

-208.500 240.000

.000 s

21 0

0 0

0 727.500

-264.000 240.000

.000 1

22 0

0 0

0 727.500

-264.000 205.000

.000 23 0

0 0

0 727.500

-264.000 190.000

.000 24 0

0 0

0 733.500

-264.000 184.000

.000 25 0

0 0

0 753.500

-264.000 184.000

.000 26 0

0 0

845.500

-2 G4. 000 184.000

.000 27 0

0 0

851.500

-264.000 178.000

.000 28 0

0 0-0 0

851.500

-264.000 160.000

.000 29 0

0 0

O O

851.500

-264.000 142.000

.000 30 0

0 0

O O

851.500

-270.000 136.000

.000 31 0

0 0

0 851.500

-300.000 136.000

.000 32 0

0 0

0 727.500

-2L1.000 255.000 000 33 0

0 0

0 727.500

-264.000 270.000

.000 34 0

0 0

0 727.500

-264.000 306.000

.000 35 0

0 0

0 727.500

-264.000

~414.000

.000 36 0

0 0

0 739.500

-264.000 426.000

.000 37 0

0 0

0 847.500

-264.000 426.000

.000 38 0

0 0

0 955.500

-264.000 426.000

.000 1EQUATICXI NINBERS N

X Y

Z XX YY ZZ 1

1 2

3 4

5 6

2 7

8 9

10 11 12 3

13 14 15 16 17 18 4

19 20 21 22 23 24 5

25 26 27 28 29 30 6

31 32 33 34 35 36 t

7 37 38 39 40 41 42 8

43 44 45 46 47 48 9

49 50 51 52 53 54 10 55 56 57 58 59 60 11 61 62 63 64 65 66 12 67 68 69 70 71 72 13 73 74 75 76 77 78 14 79 80 81 82 83 84 15 85 86 87 88 89 90 16 91 92 93 94 95 96 17 97 98 99 100 101 102 18 103 104 105 106 107 108 19 109 110 111 112 113 114 20 115 116 117 118 119 120 21 121 122 123 124 125, 126 22 127 128 129 130 131 132 23 133 134 135 136 137 138 24 139 140 141 142 143 144 25 145 146 147 148 149 150 26 151 152 153 154 155 156 27 157 158 159 160 161 162 28 163 164 165 156 167 168 29 169 170 171 172 173 174 30 175 176 177 178 179 180

[

31 181 182 183 184 185 186 32 187 188 189 190 191 192 33 193 194 195 196 197 198 34 199 200 201 202 203 204 l

35 205 206 207 208 209 210 36 211 212 213 214 215 216 l

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MAXIMJM NUMBER OF MATERIAL

~

TEMPERATURE INPUT POINIS

=

1 NUMBER OF SECTION PROPERTY SEIS =

3 NUMBER OF BRANCH POINF NODES 0

=

MAXINJM NUMBER OF TANGENTS COB 90N TO A BRANCH POINT 3

=

1 FLAG FOR NEGLECTING AXIAL DEFORMATIONS IN BEND ELEMENTS

=

0 (EQ.1, NEGLECT)

IMATERIAL PROPERTY TABLES

(

1) 0 MATERIAL NtNBER

=

NUMER OF TEMPERATURE POINTS =

1)

IDENTIFICATION

)

=

POINT YOUNG'S POISSON'S THERMAL NUMBER TEMPERATURE MODULUS RATIO EXPANSION i

1

.00 2.900E+07

.300 6.440E-06 ISECTION PROPERTY TABLE SECTION OUTSIDE WALL SHAPE FACIOR WEIGHT /

MASS /

NUMBER DIAMETER THICKNESS FOR SHEAR UNIT LENGTH UNIT LENGTH DESCRIPTION 1

3.500

.2160

.0000 8.9800E-01 2.3240E-03 2

4.500

.2370

.0000 1.3580E+00 3.5145E-03 3

8.625

.3220

.0000 4.1870E+00 1.08362-02 ELEMENT LOAD CASE MULTIPLIERS I

CASE A CASE B CASE C CASE D X-DIRECTION GRAVITY

.000

.000 Y-DIRECTION GRAVITY

.000

.000 Z-DIRECTION GRAVITY

.000

.000 THERMAL DIS 1DRPION

.000 000

.000

.000 PRESSURE DIS 70RTION

.000

.000 1 PIPE ELEMENT INPUT DATA ELEMENT ELEMENT NODE NODE MATL.

SECTION REFERENCE DESIGN PEAK TEST END CODES NODE INPUT NUMBER TYPE

-I

-J NUMBER NUMBER TEMPERATURE PRESSURE PRESSURE PRESSURE END-I END-J INCREMENT TAG (BEND (1HIRD (X3-(Y3-(Z3-(BEND RADIUS)

POINT)

ORDINATE)

ORDINATE) DEGREE) 1 TANGENT l'

2 1

1 50.00 200.00

.00

.00 0

0 1

II 2 BEND 2

3 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

19.500){

.000) (

.000) ( 90.0001)

?

3 TANGENT 3

4 1

1 50.00 200.00

.00

.00 0

0 1

II 4 TANGENT 4

5 1

1 50.00 200.00

.00

.00 0

0 1

II 5 BEND 5*

6 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

19.500)(

-204.000)(

.000) ( 90.0001) 6 TANGENT 6

7 1

1 50.00 200.00

.00

.00 0

0 1

II 7 BEND 7

8 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

19.500)(

-204.000) (

144.000) ( 90.0001) 81 9

1 1

50.00 200.00

.00

.00 0

0 1

II 8 TANGENT 9 TANGENT 9

10 1

1 50.00 200.00

.00

.00 0

0 1

II 10 TANGENT 10 11 1

1 50.00 200.00

.00

.00 0

0 1

II 11 TANGENT 11 12 1

1 50.00 200.00

.00

.00 0

0 1

II 12 TANGENT 12 13 1

1 50.00 200.00

.00

.00 0

0 1

II 13 BEND 13 14 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

487.500)(

-204.000)(

144.000) ( 90.0001) 14 TANGENT 14 15 1

1 50.00 200.00

.00

.00 0

0 1

II 15 TANGENT 15 16 1

1 50.00 200.00

.00

.00 0

0 1

II 16 BEND 16 17 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

487.500)(

-204.000)(

240.000) ( 90.0001) 17 TANGENr 17 18 1

1 50.00 200.00

.00

.00 0

0 1

II 18 TANGENT 18 19 1

1 50.00 200.00

.00

.00 0

0 1

II 19 BEND 19 20 1

1 50.00 200.00

.00

.00 0

0 IC

(

4.500)

(TI)

(

727.500) (

-204.000) (

240.000) ( 90.0001) 20 TANGENT 20 21 1

1 50.00 200.00

.00

.00 0

0 1

II 21 TANGENT 21 22 1

2 50.00 200.00

.00

.00 0

0 1

II 22 TANGENT 22 23 1

2 50.00 200.00

.00

.00 0

0 1

II 23 BEND 23 24 1

2 50.00 200.00

.00

.00 0

0 IC

(

6.000)

(TI)

(

727.500)(

-264.000)(

184.000) ( 90.00011 24 TANGENT 24 25 1

2 50.00 200.00

.00

.00 0

0 1

II 25 TANGENT 25 26 1

2 50.00 200.00

.00

.00 0

0 1

II 26 BEND 26 27 1

2 50.00 200.00

.00

.00 0

0 IC

(

6.000)

(TI)

(

851.500)(

-264.000)(

184.000) ( 90.0001) f 27 TANGENT 27 28 1

2 50.00 200.00

.00

.00 0

0 1

II 28 TANGENT 28 29 1

2 50.00 200.00

.00

.00 0

0 1

II 29 BEND 29 30 1

2 50.00 200.00

.00

.00 0

0 IC

(

6.000)

(TI)

(

851.500)(

-264.000)(

136.000) ( 90.0001) 30 TANGENT 30 31 1

2 50.00 200.f0

.00

.00 0

0 1

II 31 TANGENT 21 32 1

3 50.00 200.00

.00

.00 0

0

'l II 32 TANGENT 32 33 1

3 50.00 200.JO

.00

.00 0

0 1

II 1 PIPE ELEMENT INPUT DATA

(

ELEMENT ELEMENT NODE NODE MATL.

SECTION REFERENCE DESIGN PEAK TEST END CODES NODE INPUT f

NUMBER TYPE

-I

-J NUMBER NUMBER TEMPERA'IURE PRESSURE PRESSURE PRESSURE END-I END-J INCREMENT TAG 1

(BENC

('INIRD (X3-(Y3-(Z3-(BEND RADIUS)

POINT)

ORDINATE)

DEGREE) 33 TANGENT 33 34 1

3 50.00 20'J. 00

.00

.00 0

0 1

II 34 TANGENT 34 ~

35 1

3 50.00 200.00

.00

.00 0

0 1

II 35 BEND 35 36 1

3 50.00 200.00

.00

.00 0

0 IC

(

12.000)

(TI)

(

727.500)(

-264.000)(

426.000) ( 90.0001) 36 TANGENr 36 37 1

3 50.00 200.00

.00

.00 0

0 1

II I

a

t 37 L TANGENT

-37

.38 1

3 50.00 200.00

.00

.00 0

0 --

1

'Ik 1 BANDWIDTH MINIMIZATION MINBND (BANDWIDIE CX)tfrPOL PARADETER) =.

1-OE TION NtNBERS AFTER BANDWIIYIH MINIMIZATION

-0 E SEQUENCE X

Y Z

XX YY ZZ 1

38 223 224 225 226 227 228 2

37 217 218 219 220 221 222 3

36 211 212 273 214 215 216 4

35 205 2% 207 208 209 210 5

34 199 20Q 201 202 203 204 6

33 193 194 195 196 197 198 7

32 187 188 189 190 191 192 8

31 181 182 183 184 185

'%J 9

30 175 176 177 178 179 140 10 29 169 170 171 172 173 '174 11 28 163 164 165 166 167 168 12 27 157 158 159 160 163 162 13 26 151' 152 153 154 155 156 14 25 145 146 147 148 149 150 15 24 139 140 141 142 143 144 16 23 133 134 135 136 137 138 17 22 127 128 129 130 131 1?2 18 21 121 122 123 124 125 126 19 20 115 116 117 118 119-120 20 19 109 110 111 112 113 114 21 18 103 104 105 106 107 108 22 17 97 98 99 100 101 102 23 15 85 86 87' 88 89 90 24 13 73 74 75 76-77 78 25 11 61 62 63 64

'65 66 26 9

49 50 51 52 53 54 27 7

37 38

.39 40 41 42 28 5

25 26 27 28 29 30 29 3

13 14 15 16 17 18 30 2

7 8

9 10 11 12 31 1

1 2

3 4

5 6

32 16 91 92 93 94 95 96 33 14 79 80 81 82 83 84 34 12 67 68 69 70 71 72

.35 10 55 56 57 58 59 60 36 8

43 44 45 46 47 48 37 6

31 32 33 34 35 36 38 4

19 20 21 22 23 24 OBANDWII7IH PRIOR 'lV RES ING=

72 t

BANDWIDIM AFTER RESE ING

=

18 1 EQUATION PARAMETERS j

'IUTAL NUMBER OF EQUATIONS 228

=

BANDWIIFIM

=

18 NtRCER OF EQUATIONS IN A BIOCK = 228 l

NLMBER OF BIOCKS

=

1 1 NODAL LOADS (S T A T I C)

OR MASSES (D Y N A M I C) y NODE ICAD X-AXIS Y-AXIS Z-AXIS X-AXIS Y-AXIS Z AXIS NUMBER CASE FORCE FORCE FORCE DOMENT M3EENT BEMENT 1 DYNAMIC ANALYSIS

Nodal Masses of the Model BM3 Unit:Ibs/in/sec'*2 Node No.

Mass 1

0.017 2

0.026

'3

0. :2 4

0.227 5

0.031 6

0.165 7

0.165 8

0.092 9

0.267 10 0.275 11 0.182 12 0.174 13 0.092 14 0.059 15 0.101 16 0.059 17 0.105 18 0.268 19 0.18 20 0.073 21 0.207 22 0.088 23 0.043 24 0.052 25 0.197

]

26 0.178 27 0.048 28 0.063 29 0.048 30 0.175 31-0.158 32 0.163 33 0.276

~

34 0.78 35 0.687 36 0.687 37 1.17 38 0.585 nodal _ mass.xis

l MODE CIRCULAR NUMBER.

FREQUENCY.

FREQUENCY PERIOD (RAD /SEC)

(CYCLES /SEC)

(SEC)

O.

l' 1.8268E+01. 2.9074E+00 3.4395E-01 1

0:

2 2.7565E+01 4.3871E+00 2.2794E-01

'O 3

3.4657E+01 5.5158E+00 1.8130E-01 0

<4 3.5840E+01 5.7041E+00 1.7531E-01 0-5 4.3841E+01 6.9775E+00 1.4332E-01 0

6 4.6141E+01 7.3436E+00 1.3617E-01 1

0 7

4.9509Et01 7.8796E+00 1.2691E-01 0

8 6.4727E+01 2 0302E+01 9.7071E-02 0

9 6.9466E+01 1.1056E+01 9.0450E-02 0

10 7.0578E+01. 1.1233E+01 8.9025E-02 0

11 7.2241E+01-1.1498E+01 8.6975E-02 0

12.

7.8104E+01 1.2431E+01 8.0446E-02 0

13<

8.7205E+01 1.3879E+01 7.2050E-02 0

14 1.0130E+02 1.6122E+01 6.2026E-02 0

15 1.1333E+02 1.8038E+01 5.5439E'02 1

0 '16 1.1700E+02 1.8621E+01 5.3702E-02 10.17 1.2265E+02 1.9521E+01 5.1227E-02 g

0 18 1.2291E+02 1.9561E+01 5.1122E-02 0

19 1.3707E+02 2.1815E+01 4.5840E-02 0

20 1.3935E+02 2.2179E+01 4.5088E-02 OL 21-1.4.161E+02 2.2857E+01 4.3750E-02 0

22 1.6305E+02 2.5950E+01 3.8536E-02 l

JO 12 3 2.4517E+02 3.9020E+01 2.5628E-02 O

24 2.4831E+02 3.9521E+01 2.5303E-02 0

25 2.5890E+02 4.1205E+01 2.4269E-02 0

26 2.8845E+02 4.5908E+01 2.1783E-02

-0 27 2.9688E+02 4.7249E+01 2.1164E-02 0

28 3.3072E+02 5.2636E+01 1.8999E-02 0

29 3.7253E+02 5.9290E+01-1.6866E-02 0

30 4.4039E+02 7.0090E+01 1.4267E-02 0

31 4.4421E+02-7.0699E+01 1.4144E-02 t

4

. ew H

l

1 MODAL PARTICIPATION FACTORS MbDE FREQ(CPS) X-DIRECTION Y-DIRECTION Z-DIRECTION 1

2.907

-2.3473E-02 2.7695E-01 3.5437E-02 2

4.387

-5.5673E-01 9.0279E-02

-1.0926E-01 3

5.516

-4.7719E-02

-9.4682E-01

-1.9203E-01 4

5.704

-9.9555E-03

-2.1351E-01 6.8419E-01 5

6.977

-1.5342E-01

-4.5827E-01 7.4423E-01 l

6 7.344 5.5364E-01 3.9813E-01

-3.5022E-01 7

7.880

-3.7119E-01 3.9334E-01 4.5914E-01 8

10.302

-4.0272E-01 1.1940E+00

-1.1466E-02 9-11.056 5.5995E-01 1.8119E-01

-7.0772E-02 10 11.233 9.2828E-02

-1.4703E-01

-2.7472E-04 11 11.498

-1.4351E+00

-1.9653E-01

-3.4271E-01 12 12.431 4.2956E-01

-2.4824E-01

-4.5713E-01 13 13.879

-3.2298E-01

-1.5196E-01

-5.3436E-02 14 16.122 9.7048E-02 4.1157E-01 2.9944E-01 15 18.038

-1.0796E-01 4.0371E-02 4.3020E-01 16 18.621

-4.2217E-01

-4.0211E-02 2.9074E-01 17 19.521

-3.2449E-02

-2.8297E-01

-2.1127E-01 18 19.561

-1.4330E-01 8.6254E-02

-6.1027E-01 19 21.815

-3.8758E-01 1.1586E-01

-5.3784E-02 20 22.179

-3.6788E-03

-9.8705E-01 7.5035E-02 21 22.857

-6.9162E-02 8.0846E-02 1.1833E+00 22 25.950 1.6498E-01

-1.8727E-02 1.6100E+00 23 39.020 1.2766E-01 7.3949E-01

-2.9101E-02 24 39.521

-2.4041E-02

-3.3916E-01

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i 25 41.205

-2.0186E-02 3.3298E-01

-1.9897E-02 26 45.908 4.7943E-01

-1.6665E-02

-6.2062E-02 27 47.249

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-2.6544E-01 2.6515E-02 28 52.636

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29-

59.290

-2.5127E-03 2.4614E-04

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30 70.090 4.5397E 1.7356E-03

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t 31' 70.699

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1.340-109.900 1.350 -90.300 1.360 -68.300 1.370 -45.000 1.380 -21.051 1.390 3.690 1.400 28.670 1.410 52.250 1.420 72.600 1.430 88.450 1.440 99.200 1.450 104.800 1.460 105.300 1.470 100.650 1.480 91.100 1.490 77.830 1.500 62.140 1.510 45.060 1.520 27.540 1.530 10.580 1.540 -5.064 1.550 -18.940 1.560 -31.120 1.570 -41.830 1.580 -51.170 1.590 -58.910 1.600 -64.200 1.610'-66.070 1.620 -63.940 1.630 -58.020 1.640 -49.480 1.650 -39.460 1.660 -29.300 1.670 -20.800 1.680 -14.520 1.690 -9.130 1.700 3.370 i 1 i 1.710 2.640 1.720 7.910 1.730 11.692 1.740 13.840 1.750 14.260 1.760 13.490 1.770 12.720 1.780 12.340 1.790 11.590 1.800 9.390 1.010 5.210 1.820 -1.060 1.830 -9.275 1.840 -18.870 1.850 -28.840 1.860 37.870 1.870 -44.660 1.880 -48.440 1.890 -49.200 1.900 -47.540 1.910 -44.430 1.920 -40.640 1.930 -36.040 1.940 -30.680 1.950 -25.310 1.960 -20.220 1.970 14.660 1.980 -7.530 1.990 1.474 2.000 11.810 2.010 22.690 2.020 32.910 2.030 41.160 2.040 46.760 2.050 49.810 2.060 50.900 2.070 50.890 2.080 50.370 2.090 49.540 2.100 48.330 2.110 46.500 2.120 43.980 2.130 40.770 2.140 36.980 2.150 33.160 2.160 29.580 2.170 25.820 2.180 21.754 2.190 18.100 2.200 15.660 2.210 14.300 2.220 13.650 2.230 13.840 2.240 14.633 2.250 15.120 2.260 14.350 2.270 11.800 2.200 7.500 2.290 2.070 2.300 -3.854 2.310 10.127 2.320 -16.677 2.330 -23.257 2.340 -29.545 2.350 -35.200 2.360 40.270 2.370 -45.040 2.380 -49.500 2.390 -53.310 2.400 -55.620 2.410 -55.050 2.420 -51.130 2.430 -44.950 2.440 -37.650 2.450 -29.410 2.460 20.100 2.470 9.890 2.480 .560 2.490 10.110 2.500 18.160 2.510 25.070 2.520 31.250 2.530 36 340 2.540 39.640 2.550 40.830 2.560 40.210 2.570 38.300 2.580 35.700 2.590 32.830 2.600 29.700 2.610 26.130 2.620 22.040 2.630 17.430 2.640 12.430 2.650 7.293 2.660 2.293 2.670 -2.550 2.680 -7.200 2.690 -11.440 2.700 -15.455 2.710 -19.818 2.720 -25.084 2.730 -31.340 2.740 -38.050 2.750 -44.450 2.760 -49.700 2.770 52.860 2.780 -53.530 2.790 -52.120 2.800 -48.890 2.810 -43.630 2.820 -35.990 2.830 -25.760 2.840 -13.340 2.850 .310 2.860 14.222 2.870 27.998 2.880 41.354 2.890 53.810 2.900 64.880 2.910 74.050 2.920 81.260 2.930 86.860 2.940 90.700 2.950 92.200 2.960 90.090 2.970 83.490 2.980 72.430 2.990 58.100 3.000 42.290 3.010 26.590 3.020 12.040 3.030 .760 3.040 -11.720 3.050 -21.500 3.060 -30.694 3.070 -39.100 3.080 -46.480 3.090 -53.110 3.100 -59.020 3.110 63.370 3.120 -65.550 3.130 65.650 3.140 -63.900 3.150 -60.550 3.160 -55.090 3.170 -46.680 3.180 -34.860 3.190 -19.850 3.200 -2.720 3.210 14.938 3.220 31.850 3.230 47.100 3.240 60.000 3.250 70.250 3.260 77.000 3.270 79.050 3.200 75.700 3.290 67.100 3.300 54.100 3.310 38.090 3.320 20.270 3.330 .830 3.340 -19.900 3.350 -41.140 3.360 -61.870 3.370 -80.710 3.380 -97.000 3.390-111.450 3.400 123.830 3.410-132.440 3.420-135.540 3.430-132.490 3.440-124.440 3.450-113.420 3.460-101.280 3.470 -88.960 3.480 -76.420 3.490 -63.210 3.500 -49.130 3.510 -34.416 3.520 -19.460 3.530 -4.590 3.540 9.570 3.550 21.940 3.560 31.600 3.570 38.390 3.580 42.880 3.590 46.040 3.600 48.730 3.610 51.340 3.620 53.910 3.630 56.150 3.640 57.860 3.650 58.800 3.660 58.620 3.670 56.650 3.600 52.200 3.690 45.550 3.700 36.660 3.710 25.204 3.720 11.500 3.730 -3.020 3.740 -16.340 3.750 -26.690 3.760 -33.270 3.770 -36.140 3.780 35.800 3.790 -32.590 3.800 -26.660 3.810 -17.920 3.820 6.674 3.830 6.614 3.840 20.760 3.850 33.940 3.860 44.630 3.870 52.060 3.880 55.970 3.890 56.520 3.900 53.760 3.910 47.570 3.920 38.150 3.930 26.060 3.940 12.130 3.950 -2.690 3.960 -17.930 3.970 -33.551 3.980 -49.341 3.990 -64.670 4.000 -78.600 4.010 -90.240 4.020 -99.270 4.030-105.970 4.040-110.800 4.050-113.900 4.060-115.210 4.070 114.590 4.080-111.900 4.090-106.970 4.100 -99.680 4.110 -89.450 4.120 -76.210 4.130 -60.730 4.140 -43.930 4.150 -26.260 4.160 -7.650 4.170 12.124 4.180 33.040 4.190 54.610 4.200 75.930 4.216' 95.900 4.220 113.500 4.230 128.010 4.240 138.790 4.250 145.480 4.260 148.210 4.270 147.310 4.200 143.810 4.290 138.900 4.300 133.100 4.310 126.400 4.320 117.580 4.330 105.240 4.340 88.790 4.350 68.820 4.360 46.740-4.370 24.450 4.380 3.020 4.390 -17.985 4.400 -38.970 4.410 -59.390 '4.420 78.560 4.430 -96.210 4.440-112.440 4.450-127.390 4.460-140.460 4.470-150.200 4.480-155.300 4.490-155.600 4.500-151.300 4.510-143.250 4.520-131.860 4.530-116.610 4.540 -97.660 4.550 -76.090 4.560 -53.190 4.570 -29.800 4.580 -6.244 4.590 18.100 4.600 42.890 4.610 66.470 4.620 86.900 4.630 103.150 4.640 114.700 4.650 120.700 4.660 120.700 4.670 115.300 4.680 105.710 4.690 93.570 4.700 80.350 4.710 G5.900 4.720 53,190 4.730 38.710 4.740 22.900 4.750 5.790 4.760 12.130 4.770 -30.350 4.780 47.780 4.790 -62.290 4.800 -72.300 4.810 -77.800 4.820 79.400 4.830 -77.130 4.840 -70.940 4.850 -61.060 4.860 48.520 4.870 -34.956 4.880 -22.100 4.890 -10.920 4.900 -1.644 4.910 5.828 4.920 11.809 4.930 16.510 4.940 20.270 4.950 23.770 4.960 27.212 4.970 29.890 4.980 31.140 4.990 31.300 5.000 31.340 5.010 31.767 5.020 33.100 5.030 36.332 5.040 41.640 5.050 48.030 1 5.060 54.406 5.070 60.560 5.080 66.810 5.090 72.899 5.100 78.840 5.110 85.550 5.120 92.980 5.130 99.230 5.140 102.200 5.150 101.160 5.160 96.100 5.170 87.260 5.180 75.150 5.190 60.370 5.200 43.860 5.210 26.840 5.220 10.090 5.230 -6.380 5.240 -23.250 5.250 -41.154 5.260 -59.950 5.270 -78.380 5.280 -94.900 5.290 108.610 5.300-118.910 5.310 125.050 5.320-126.600 5.330-124.140 5.340-118.560 5.350-110.380 5.360 100.000 5.370 -88.120 5.380 -74.640 5.390 -58.930 5.400 -40.500 5.410 -19.920 5.420 1.700 5.430 23.380 5.440 43.670 5.450 60.900 5.460 73.590 5.470 80.700 5.480 82.600 5.490 80.500 5.500 75.990 5.510 69.310 5.520 60.290 5.530 48.,950 5.540, 35.770 5.550 21.310 5.560 6.090 5.570 -9.037 5.580 -23.540 5.590 -37.540 5.600 -50.530 5.610 60.550 5.620 -66.000 5.630 -67.350 5.640 -65.170 5.650 -58.730 4 5.660 -47.990 5.670 -34.340 5.680 -19.900 5.690 -6.430 5.700 5.590 5.710 16.663 5.720 27.730 5.730 39.400 5.740 51.120 5.750 61.180 5.760 68.200 5.770 72.050 5.700 72.950 5.790 70.580 5.800 64.910 5.810 56.'900 5.820 48.100 5.830 39.740 5.840 32.480 5.850 26.270 5.860 20.580 5.870 14.830 5.880 8.700 5.890 2.164 5.900 -4.749 5.910 12.073 5.920 19.580 5.930 -26.450 5.940 -32.030 5.950 36.390 5.960 -39.620 5.970 41.330 5.980 -40.840 5.990 -37.730 6.000 -31.830 6.010 -23.420 6.020 13.170 6.030 -1.850 6.040 9.490 6.050 19.690 6.060 27.790 6.070 33.310 6.080 36.530 6.090 38.280 6.100 39.370 6.110 40.050 6.120 40.120 6.130 39.100 6.140 36.540 6.150 32.410 6.160 26.630 6.170 19.130 6.180 10.267 6.190 .742 6.200 -8.230 6.210 15.060 6.220 -19.150 6.230 -21.340 6.240 -22.740 6.250 -23.650 6.260 -23.810 6.270 -23.000 6.280 -21.259 6.290 -18.869 6.300 -16.310 6.310 13.879 6.320 -11.870 6.330 -10.620 6.340 -10.140 6.350 -9.950 6.360 -8.950 6.370 -5.970 6.380 .301 6.390 8.060 6.400 18.110 ) 6.410 28.230 6.420 36.780 6.430 42.760 6.440 45.750 6.450 45.640 6.460 42.950 6.470 38.750 6.480 34.000 6.490 28.850 6.500 22.870 6.510 15.470 6.520 6.080 6.530 -5.390 6.540 -18.610 6.550 -32.987 6.560 47.530 6.570 -61.120 6.580 -72.690 6.590 -81.420 6.600 -87.220 6.610 -90.970 6.620 -93.460 6.630 -94.480 6.640 -93.640 6.650 91.160 6.660 87.140 6.670 -80.840 6.680 -71.600 6.690 -59.680 6.700 -45.690 6.710 29.680 6.720 -12.126 6.730 5.750 6.740 22.550 6.750 37.410 6.760 49.890 6.770 59.850 6.700 67.540 6.790 73.550 6.800 78.250 6.810 81.660 6.820 83.550 6.830 83.470 6.840 81.160 6.850 76.910 6.860 70.890 6.870 62.720 6.600 52.170 6.890 39.700 6.900 26.120 6.910 12.040 6.920 -1.990 6.930 -15.190 6.940 -27.070 6.950 -37.960 6.960 -48.116 6.970 -57.360 6.980 -65.737 6.990 -73.330 7.000 -80.510 7.010 -87.850 7.020 -94.880 7.030 -99.600 7.040-100.100 7.050 -95.350 7.060 -85.640 7.070 -71.000 7.080 -51.800 7.090 -29.710 7.100 -5.780 7.110 19.780 7.120 46.806 7.130 74.450 7.140 101.430 7.150 126.350 7.160 147.900 7.170 163.750 7.180 172.600 7.190 174.250 7.200 168.300 7.210.T54.150 7.220 131.300 7.230 101.150 7.240 65.960 7.250 28.530 7.260 -9.040 7.270 -45.377 7.280 -79.830 7.290-112.040 7.300-141.300 7.310-166.450 7.320-186.400 7.330-200.200 7.340-207.200 7.350-206.900 7.360 199.400 7.370-184.800 7.380-163.900 7.390-138.200 7.400-109.200 7.410 -78.470 7.420 -47.140 7.430 -15.687 7.440 15.267 7.450 44.560 7.460 71.130 7.470 94.400 7.480 114.000 7.490 129.900 7.500 141.500 7.510 148.400 7.520 150.100 7.530 146.250 7.540 137.200 7.550 124.300 7.560 108.300 7.570 89.500 7.580 67.410 7.590 42.110 7.600 14.440 7.610 -13.927 7.620 -41.410 7.630 -67.080 7.640 -90.270 7.650-110.000 7.660-125.100 7.670-134.850 7.680-139.100 7.690-137.250 7.700-129.600 ~- 1 7.710 117.100 7.720 100.820 7.730 -80.770 7.740 -57.460 7.750 -32.620 7.760 -8.010 7.770 15.471 7.780 37.660 7.790 58.690 7.800 78.450 7.810 96.280 7.820 111.410 7.830 123.450 7.840 132.300 7.850 39'.600 7.860 139.200 7.870 136.400 7.880 129.500 7.890 119.250 7.900 3 70 7.910 92.010 7.920 75.840 7.930 57.720 7.940 37.920 7.950 i 14b 7.960 -2.520 7.970 -21.820 7.980 -40.510 7.990 -59.170 8.000 ' 1 8.010 -92.750 8.020-103.800 8.030-110.140 8.040 112.400 8.050-11# iia 8.060 104.900 8.070 -96.650 0.080 -87.100 8.090 -75.870 8.100 -62 is 8.110 48.260 8.120 -33.402 0.130 -17.496 8.140 .406 8.150 16.910 8.160 32.890 8.170 46.180 8.180 56.360 8.190 64.200 8.200 71.036 8.210 77.420 8.220 83.650 8.230 90.380 8.240 97.560 8.250 103.810 8.260 107.540 8.270 108.040 8.280 105.310 8.290 99.760 8.300 92.090 8.310 82.870 8.320 72.570 8.330 61.660 8.340 50.380 8.350 38.570 8.360 25.950 8.370 12.300 8.380 -2.220 8.390 -16.790 8.400 -30.740 j 8.410 -44.040 8.420 -56.790 8.430 -68.800 8.440 -79.670 8.450 -89.000 8.460 -96.990 8.470 103.440 8.480-108.550 8.490-112.400 8.500-114.750 i 8.510-114.970 B.520 112.500 8.530-107.260 8.540 -99.150 8.550 -88.090 8.560 -74.040 8.570 -57.180 8.580 -37.870 8.590 -16.850 8.600 5.270 8.610 28.380 8.620 51.910 8.630 74.260 8.640 93.720 8.650 109.450 8.660 121.100 9.670 128.200 8.680 130.400 8.690 127.700 8.700 120.100 8.710 107.250 8.720 89.000 8.730 66.080 8.740 39.720 8.750 11.180 8.760 -17.840 8.770 -45.240 8.780 -69.470 8.790 90.020 8.800-107.190 8.810-121.150 8.820-131.800 8.830-139.700 8.840-144.100 8.850-143.450 8.860 136.300 8.870 123.250 8.880-105.500 8.890 -84.000 8.900 -59.430 8.910 -32.320 8.920 -3.390 8.930 26.080 8.940 54.630 8.950 81.450 8.960 105.970 8.970 127.360 8.980 144.900 8.990 158.450 9.000 167.700 9.010 173.050 9.020 174.100 9.030 170.450 9.040 161.400 9.050 147.100 9.060 128,100 9.070 105.600 9.080 80.610 9.090 53.780 9.100 25.690 9.110 -2.990 9.120 30.840 9.130 -55.700 9.140 -76.180 9.150 -92.250 j 9.160-104.600 9.170 113.500 9.180-118.700 9.190-119.850 9.200-116.600 9.210-109.750 9.220,-99.700 9.230 -86.700 9.240 -70.720 9.250 -52.310 j 9.260 -32.300 9.270 -11.850 9.280 7.790 9.290 26.030 9.300 42.200 9.310 54.980 9.320 63.940 9.330 70.120 9.340 74.920 9.350 79.170 1 9.360 82.530 9.370 83.690 9.380 81.760 9.390 76.760 9.400 69.500 9.410 60.990 9.420 51.460 9.430 40.670 9.440 28.463 9.450 15.039 9.460 1.330 9.470 -11.480 9.480 22.580 9.490 -31.490 9.500 -38.230 9.510 43.220 9.520 -46.510 9.530 -47.720 9.540 -46.320 9.550 -42.020 9.560 35.140 9.570 26.160 9.580 -15.980 9.590 -6.030 9.600 2.830 9.610 10.790 9.620 18.210 9.630 25.000 9.640 30.620 9.650 34.830 9.660 37.550 9.670 38.360 9.680 37.530 9.690 36.150 9.700 35.000 9.710 34.080 9.720 32.680 9.730 29.760 9.740 24.720 9.750 17.990 9.760 10.240 9.770 1.470 9.780 -0.120 9.790 -17.364 9.000 -25.200 9.810 -31.990 9.820 -38.050 9.830 -43.630 9.840 -48.640 9.850 -52.920 9.860 -56.130 9.870 58.110 9.880 -58.775 9.890 -57.850 9.900 -55.430 9.910 -52.211 9.920 -49.047 9.930 -46.472 9.940 -44.559 9.950 -43.041 9.960 -41.607 9.970 -40.300 9.980 -39.340 9.990 -38.671 10.000 -38.390 10.010 38.660 10.020 -39.390 10.030 -40.220 10.040 -39.990 10.050 -37.020 10.060 30.160 10.070 -19.540 10.080 -6.290 10.090 8.012 10.100 22.020 10.110 34.520 10.120 44.600 10.130 52.290 10.140 57.070 10.150 57.350 10.160 52.400 10.170 43.200 10.180 30.860 10.190 14.480 10.200 -6.460 10.210 30.710 10.220 -55.900 10.230 -79.890 10.240-101.080 10.250-118.280 10.260-131.000 10.270-139.800 10.200-144.530 10.290-144.400 10.300-138.900 10.310-128,900 10.320-115.600 10.330 -99.380 10.340 -80.500 10.350 -59.300 10.360 36.700 10.370 14.033 10.380 7.322 10.390 26.120 10.400 41.880 10.410 54.940 10.420 65.430 10.430 73.060 10.440 77.580 10.450 79.020 10.460 78.150 10.470 76.330 10.480 74.550 10.490 72.620 10.500 70.240 10.510 68.080 10.520 66.548 10.530 64.570 10.540 60,990 10.550 55.970 10.560 49.720 10.570 41.880 10.580 32.073 10.590 20.610 10.600 8.510 10.610 -3.090 10.620 -12.810 10.630 -19.320 10.640 -22.380 10.650 -22.990 10.660 -22.430 10.670 -21.220 10.680 -19.420 10.690 -17.070 10.700 -13.870 10.710 -9.230 10.720 -2.640 10.730 6.040 10.740 16.190 10.750 26.350 10.760 35.050 10.770 41.830 10.780 46.702 10.790 49.470 10.800 50.360 10.810 50.340 10.820 50.408 10.030 50.610 10.840 50.520 10.850 49.790 10.860 47.700 10.870 43.270 10.880 35.840 10.890 25.300 10.900 12.090 10.910 -2.000 10.920 -18.566 10.930 -34.600 10.940 -50.370 10.950 -65.840 10.960 -80.335 10.970 -91.960 10.980 99.770 10.990-104.600 11.000-107.640 11.010-109.170 11.020-108.390 11.030-104.140 11.040 -95.750 11.050 -83.370 i 11.060 -67.750 11.070 -49.800 11.080 -30.170 11.090 -9.260 11.100 12.120 11.110 32.260 11.120 49.860 11.130 64.89^ 11.140 77.5.3 11.150 87.160 11.160 93.000 11.170 94.820 11.180 92.830 11.190 87.940 11.200 81.100 11.210 72.700 11.220 62.990 11.230 52.770 11.240 42.840 11.250 33.290 11.260 24.209 11.270 16.200 11.280 9.850 11.290 5.360 11.300 2.550 11.310 1.110 11.320 .640 11.330 .620 11.340 .990 11.350 2.150 11.360 4.400 11.370 7.500 11.300 10.880 11.390 14.340 11.400 17.730 11.410 20.259 11.420 21.710 11.430 23.080 11.440 24.980 11.450 26.965 11.460 28.160 11.470 27.938 11.480 26.599 11.490 25.182 11.500 24.690 11.510 25.360 11.520 26.670 11.530 28.040 11.540 28.850 11.550 28.360 11.560 26.130 11.570 22.440 11.590 17.770 11.590 12.500 11.600 7.220 11.610 2.620 11.620 .720 11.630 -2.350 11.640 -2.390 11.650 -1.610 11.660 . 617 11.670 .920 11.600 3.420 11.690 6.430 11.700 9.400 11.710 12.390 11.720 15.210 11.730 16.890 11.740 16.320 11.750 13.120 11.760 7.362 11.770 .919 11.780 -10.882 11.790 -20.966 11.800 -29.822 11.810 36.880 11.820 -42.439 11.830 -47.303 11.840 -52.194 11.850 -57.220 11.860 62.050 11.870 66.320 11.080 -69.500 11.890 -70.980 11.900 -70.340 11.910 -67.710 11.920 -63.690 11.930 -58.970 11.940 -54.260 11.950 -50.090 11.960 -46.440 11.970 -42.270 11.980 -36.400 11.990 -28.420 12.000 -17.840 12.010 -3.320 12.020 15.710 12.030 37.772 12.040 60.600 12.050 82.640 12.060 102.520 12.070 118.400 12.080 129.100 12.090 134.500 12.100 134.700 12.110 129.250 12.120 117.700 12.130 100.550 12.140 78.100 12.150 52.740 12.160 23.730 12.170 -7.277 12.180 -38.660 12.190 -68.190 12.200 -93.800 12.210-114.550 12.220-130.000 12.230-139.700 12.240-143.600 12.250-141.700 12.260-134.700 12.270-123.630 12.280-109.630 12.290 -93.380 12.300 -75.610 12.310 -57.470 12.320 -39.980 12.330 -23.450 12.340 -7.610 12.350 7.842 12.360 23.170 12.370 38.750 12.380 54.421 12.390 69.210 12.400 82.510 12.410 94.600 12.420 105.830 12.430 115.870 12.440 124.100 12.450 129.750 12.460 132.200 12.470 131.100 12.480 125.000 12.490 114.650 12.500 96.600 12.510 72.050 12.520 42.340 12.530 9.520 12.540 -24.570 12.550 -57.940 12.560 -88.900 12.570-116.450 12.580-139.620 12.590-156.300 12.600-165.400 12.610 167.950 12.620-165.200 12.630-156.800 12.640-142.900 12.650-124.950 12.660 104.590 12.670 82.260 12.600 -58.530 12.690 -34.850 12.700 -12.586 12.710 7.563 12.720 25.470 12.730 41.240 12.740 55.000 12.750 66.710 12.760 75.900 12.770 82.240 12.780 85.500 12.790 86.060 12.800 84.360 12.810 81.370 12.820 77.950 12.830 74.920 12.840 72.070 12.850 68.120 12.860 61.730 12.870 52.410 12.880 40.500 12.890 27.110 12.900 12.990 12.910 .938 12.920 -14.149 12.930 -26.120 12.940 -36.660 12.950 -46.100 12.960 -54.490 12.970 -61.040 12.980 -64.720 12.990 -65.070 13.000 -62.030 13.010 -55.390 13.020 -45.610 13.030 -34.020 13.040 -21.900 13.050 -10.038 13.060 1.220 13.070 12.043 13.080 22.360 13.090 31.390 13.100 38.670 13.110 44.500 13.120 49.380 13.130 53.370 13.140 56.070 13.150 57.010 13.160. 56.100 13.170 53.440 13.180 49.390 13.190 44.480 13.200 38.620 13.210' 31.340 13.220 22.296 13.230 11.442 13.240 .454 13.250 ~11.920 13.260 -21.780 13.270 -29.890 13.280 -36.501 13.290 -41.440 13.300 -44.720 13.310 -47.160 13.320 -49.350 13.330 -50.840 13.340 -50.970 13.350 -49.600 13.360 -46.800 13.370 -42.500 13.300 36.830 13.390 -30.440 13.400 -23.784 13.410 -16.857 13.420 -9.740 13.430 -2.940 13.440 3.030 13.450 8.275 13.460 13.000 13.470 16.894 13.480 19.980 13.490 23.010 13.500 26.500 13.510 30.180 13.520 33.240 13.530 34.870 13.540 34.550 13.550 32.320 13.560 28.560 13.570 23.711 13.580 18.285 13.590 12.739 13.600 7.634 13.610 3.535 13.620 .530 13.630 -1.720 13.640 3.700 13.650 -5.810 13.660 -7.823 13.670 -8.870 13.680 -8.390 13.690 -6.510 13.700 -3.800 i 13.710 .966 13.720 1.590 13.730 3.890 13.740 5.838 13.750 7.000 13.760 6.950 13.770 5.750 13.780 3.714 13.790 1.142 13.800 -1.768 13.810 -4.710 13.820 -7.390 13.830 -9.770 13.840 -11.591 13.850 -12.200 l 13.860 -11.340 13.870 -9.560 13.880 -7.475 13.890 -5.222 13.900 -2.745 l 13.910 .145 13.920 2.271 13.930 4.110 13.940 5.130 13.950 5.458 13.960 5.302 13.970 4.490 13.980 2.820 13.990 .550 14.000 -2.179 14.010 -5.599 14.020 -9.860 14.030 -14.591 14.040 -19.160 14.050 -23.000 14.060 -25.570 14.070 -26.500 14.080 -25.680 14.090 -23.220 14.100 -19.470 14.110 -14.900 14.120 -9.890 14.130 -4.550 14.140 1.170 14.150 7.146 14.160 13.180 14.170 19.140 14.180 24.620 14.190 29.140 14.200 32.248 14.210 33.650 14.220 33.560 14.230 32.700 14.240 31.680 14.250 30.620 14.260 29.190 14.270 26.900 14.280 23.390 14.290 18.700 14.300 12.990 i 14.310 6.485 14.320 .483 14.330 -7.492 14.340 -13.966 14.350 -19.350 14.360 -23.480 14.370 -26.700 14.380 -29.506 14.390 -32.120 14.400 -34.510 14.410 -36.650 14.420 -38.370 14.430 -39.190 14.440 -38.730 14.450 -36.870 14.460 -33.660 14.470 -29.350 14.480 -24.120 14.490 -18.070 14.500 -11.270 14.510 -3.950 14.520 3.510 14.530 10 750 14.540- 17.290 14.550 22.640 14.560 26.370 14.570 28.290 14.580 28.400 14.590 26.850 14.600 23.910 14.610 19.740 14.620 14.520 14.630 8.550 14.640 1.920 14.650 -5.490 14.660 -13.744 14.670 -22.525 14.680 -31.200 14.690 -39.120 14.700 -45.690 14.710 50.430 14.720 -53.190 14.730 -54.160 14.740 -53.630 14.750 -51.870 14.760 -49.020 14.770 45.030 14.780 -39.870 14.790 -33.640 14.800 -26.470 14.810 -18.410 14.820 -9.659 14.830 .613 14.840 8.164 14.850 16.090 14.860 22.800 14.870 28.240 14.880 32.530 14.890 35.820 14.900 38.130 14.910 39.330 14.920 39.300 14.930 37.950 14.940 35.310 14.950 31.400 14.960 26.350 14.970 20.500 14.980 14.210 14.990 7.653 3 3

=

6 APPENDIX C Potential Differences Between Dynamic and Static Mass Distributions for the Same Computer Model l i \\ The analysis of nuclear power plant structures, systems and components often requires that static analysis solutions and dynamic analysis solutions for the same computer model be combined to 1 produce the total stress state, for comparison to a design allowable stress. One such combination is deadweight plus seismic load. To produce accurate load combination results, it is important that the mass distribution for the static analysis and the dynamic analysis be comparable and sufficiently refined. During the course of the present study (see Section 2.1.5), convergence of the Lindley-Yow plus missing mass in-phase response component to the static analysis solution for total mass times ZPA was tested for the BM3 piping model used in the numerical studies. The first attempt to correlate these two solutions was unsuccessful, because the piping computer code, based on SAP V, treats the distributed mass of the piping model differently in static analyses and dynamic analyses. l The dynamic analysis treats the distributed piping mass as lumped masses at the node points, based on adjacent pipe element lengths. The static analysis. treats the distributed piping mass times acceleration as a distributed load along the length of the pipe element. The static analysis procedure is the more accurate representation for calculation of stresses and reactions at supports. However, for the dynamic anQsis solution, a finite number of mass degrees of freedom are defined by assigning the mass to the locations of the node points. In dynamic analysis, the definition of node points is primarily driven by the physical characteristics of the piping system (support points, branch points, and in-line components) and by the dynamic behavior which is to be predicted (frequency range and mode shapes). A third criterion for definition of node points should also be included: sufficient refinement for accurate prediction of stresses and support reactions. Tables C-1 and C-2 present a comparison between two static analysis solutions, for BM3 support reactions and pipe end moments, respectively. The column labeled " static /dyn" are the results for the dynamic lumped mass representation multiplied by ZPA. The column labeled " static /sta" are the results for the static distributed mass representation multiplied by ZPA. Both analyses utilized the same computer model and computer code. Examination of Table C-1 indicates that reaction forces F,, F,, F, are in reasonable agreement; however, moment reactions at the fixed support points (nodes 1,31,38) have significant differences (>10%). Table C-2 indicates that the pipe end moments for elements in the vicinity of the fixed support points also have significant differences (>10%). Two effects contribute to poor correlation of the moment. In the dynamic analysis, mass apportioned to the fixed supports (nodes 1,31,38) do not produce any moment effect. Also, concentrated loads (i.e., lumped masses) do not generally produce the same moments as a distributed load. Both of these effects can be minimized in the dynamic analysis by defming of a sufficient number of node points in'the vicinity of fixed supports. During development of the mathematical model, a sensitivity study should be conducted to ensure that the ntme point distribution is sufficient to produce an accurate static solution. Successive refinement of the node point distribution should be performed until there is reasonable correlation between results generated with the dynamic lumped mass representation and the static distributed mass representation. Some existing computer codes for piping analysis may be less susceptible to this condition because corrections have been built into the code. It is important for the analyst to understand the treatment of mass and ensure that an adequate lumped mass distribution has been defined for the dynamic analysis. l ~ m i I [ Support Reactions of Static Analyses due to Different Mass [ Dis,tributions i U Forces in Ibs 1 % Damping l Moments in in-lbs L a i NODE REAC. ZPA ZPA No. TYPE statiddyn statidsta I Fx -46.21 -50.50 1 Fy 0.66 0.66 1 Fz 0.26 0.01 1 Mx 10.53 32.80 1 My -329.25 -478.00 1 Mz 158.43 -823.00 4 Fx -102.75 -98.00 4 Fz -5.49 -4.26 7 Fy -0.01 -0.85 11 Fy -1.17 -0.12 11 Fz 37.65 36.00 15 Fx -474.26 -475.00 17 Fy 3.78 3.84 17 Fz -35.17 -34.40 36 Fy 8.92 8.70 36 Fz 31.68 34.80 38 Fx -767.29 -770.00 38 Fy -11.24 -11.20 38 Fz -31.17 -35.60 38 Mx -189.06 -195.00 38 _My -2202.83 -2520.00 38 Mz 804.58 799.00 23 Fx -304.11 -297.00 s 23 Fy -0.68 -0.15 31 Fx -56.51 -60.60 31 Fy -0.25 -0.92 31-Fz 2.24 3.49 31 Mx 219.27 344.00 31 My 580.84 502.00 31 Mz 1702.48 2200.00 Table C-1 i Pipe End Moments of Static Analyses due to Different Mass Distribution Resultant Moments in in-lbs 1 % Damping i ELEM. ZPA ZPA NO. static /dyn static /sta 1 357.75 961.37 2 450.75 808.88 3 995.58 1241.14 4 327.00 526.13 5 241.67 303.25 6 868.02 627.91 7 725.62 486.36 8 354.59 179.07 9 502.09 498.37 10 912.98 836.15 11 1617.18 1642.18 12 3951.97 3927.76 13 2956.60 2925.35 14 8613.31 8664.29 15 475.77 503.56 16 467.76 445.32 17 310.22 302.07 18 706.10 755.46 19 471.56 519.70 20 2028.91 2009.32 21 580.58 524.32 22 1971.49 1987.20 23 1167.59 1191.47 24 1123.54 1119.43 25 922.09 788.22 _ 26 657.25 536.09 27 340.76 257.71 28 600.98 502.43 29 710.13 578.43 30 1812.15 2279.14 e 31 7523.23 7288.41 32 9107.41 8828.37 33 10884.84 10502.09 34 2199.42 2703.06 35 4815.86 5424.33 36 1247.94 1402.92 37 2352.78 2649.47 1 Table C-2 e APPENDIX D Tabulation of Mode Correlation Coefficients for the Double Sum Combination (DSC) and Complete Quadratic Combination (CQC) Methods 4 m. M l 4 1 1% DAMPING f MODE CORRELATION COEFFICIENTS b f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 1.0 1.0 1.0 2 Hz 1.0 .95 .54 .36 .15 .91 .22 .12 .04 .87 .I1 .06 .02 5 Hz 1.0 1.0 1.0 1.0 .95 .32 .24 .15 .91 .10 .07 .04 .87 .05 .03 .02 10 Hz 1.0 1.0 1.0 1.0 .95 .23 .19 .15 .91 .07 .06 .04 .87 .03 .03 .02 20 Hz 1.0 1.0 1.0 1.0 .95 .19 .17 .15 .91 .06 .05 .04 .87 .03 .02 .02 30 Hz 1.0 1.0 1.0 ~ 1.0 t .95 .18 .16 .15 .91 .05 .05 .04 .87 .02 .02 .02 -t. 2% DAMPING MODE CORRELATION COEFFICIENTS f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 2 Hz 1.0 1.0 1.0 1.0 .95 .69 .58 .41 .91 .35 '.25 .15 I .87 .19 .13 .07 .83 .12 .08 04. 5 Hz 1.0 1.0 1.0 1.0 .95 .55 49 .41 .91 .23 .19 .15 .87 .12 .09 . 07 .07 .06 .04 .83 10 Hz 1.0 1.0 1.0 1.0 .95 .49 .45 .41 .91 .19 .17 '.15 .87 .09 .08 .07 .83 .06 .05 .04 20 Hz 1.0 1.0 1.0 1.0 .95 45 .43 . 41 .91 .17 .16 .15 .87 .08 .08 .07 .83 .05 .05 .04 30 Hz 1.0 1.0 1.0 1.0 .95 .44 .43 .41 .91 .16 .16 .15 + .87 .08 .08 .07 .83 .05 .05 .04 4 1 j 5% DAMPING j MODE CORRELATION COEFFICIENTS 3. a l f, f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i2 2 Hz 1.0 1.0 1.0 1.0 .95 .88 .86 .82 i .91 - .65 .60 .53 i .87 .45 .39 .33 .83 .32 .27 .22 l .79 .23 .19 .15 1 .75 .17 .14 .11 5 Hz 1.0 1.0 1.0 1.0 .95 .85 .83 .82 .91 .58 .56 .53 j .87 .38 .36 .33 4 83 .26 .24 .22 .79 .18 .17 .15 .75 .13 .12 .11 l 10 Hz 1.0 1.0 1.0 1.0 .95 .83 .83 .82 l .91 .56 .54 .53 .87 .36 .34 .33 .83 .24 .23 .22 .79 .17 .16 .15 .75 .12 .12 .11 I. 9 5 ,, ~.. _, J a i 5% DAMPING MODE CORRELATION COEFFICIENTS (CONT'D) 4 1 f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 20It 1.0 1.0 1.0 1.0 i .95 .83 .82 .82 .91 .54 .53 .53 .87 .34 .34 .33 i .83 .23 .22 .22 .79 .16 .16 .15 .75 .12 .11 .11 3 i 30 Hz 1.0 1.0 1.0 1,0 .95 .82 .82 .82 .91 .54 .53 .53 .87 '34 .34 .33 .83 .23 .22 .22 .79 .16 .15 .15 .75 .12 .11 .11 i i I. I i r wa i I i 10% DAMPING MODE CORRELATION COEFFICIENTS f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 2 Hz I.0 1.0 1.0 1.0 l .95 .96 .95 .95 .91 .86 .84 .81 .87 .72 .70 .66 .83 .60 .56 .52 i .79 .48 .45 .41 -I .75 .39 .36 .32 .72 .32 .30 .26 i .68 .27 .24 .21 i .65 .22 .20 .17 .62 .19 .17 14 i .59 .16 .15 .12 .57 .14 .13 .10 l .54 .12 .11 .09 .51 .1 I .10 .08 l l I. b 10% DAMPING MODE CORRELATION COEFFICIENTS (CONT'D) f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 5 Hz I.0 1.0 1.0 1.0 .95 .95 .95 .95 .91 .83 .83 .81 .87 . 69 .68 .66 .83 .56 .54 .52 .79 .45 .43 .41 ( .75 .36 .35 .32 l '.72 i .29 .28 .26 i .68 .24 .23 .21 .65 .20 .19 .17 i .62 .17 .16 .14 l .59 .15 .14 .12 .57 .13 .12 .10 l .54 .11 .11 .09 I .51 .10 .09 .08 l i e 1 i i 9 + 1 i 10% DAMPING MODE CORRELATION COEFFICENTS (CONTD) I f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 i 10 Hz 1.0 1.0 l.0 1.0 A .95 .95 .95 .95 .91 .83 .82 .81 .87 .68 .67 .66 .83 .54 .54 .52 .79 .43 .43 .41 .75 .35 .34 .32 .28 .28 .26 i .72 i .68 .23 .23 .21 .65 .19 .19 .17 .62 .16 .16 .14 .59 .14 .14 .12 .57 .12 .12 .10 i .54 .11 .10 .09 .51 .09 .09 .08 i I e l i I l JO% DAMPING 4 MODE CORRELATION COEFFICIENTS (CONT *D) f, f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i2 20 Hz 1.0 1.0 1.0 1.0 l .95 .95 .95 .95 .91 .82 .82 .81 i .87 .67 .67 .66 .83 .54 .53 .52 .79 .43 12 .41 .75 .34 .34 .32 .72 .28 .27 .26 .68 .23 .22 .21 .65 .19' .19 .17 .62 .16 .16 .14 j .59 .14 .14 .12 l .57 '12 .12 .10 .54 .11 .10 .09 a .51 .09 .09 .08 i (. I 4 t i i 1 i 10% DAMPING i MODE CORRELATION COEFFICIENTS (CONT'D) a I f f/f DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC i i2 30 Hz 1.0 1.0 1.0 1.0 .95 .95 .95 .95 .91 .82 .82 .81 .87 .67 .67 .66 .83 .53 .53 .52 I .79 .42 .42. .41 I .75 .34 .34 .32 .72 .28 .27 .26 .68 .23 .22 .21 .65 .19 .19 .17 .62 .16 .16 - .14 .59 .14 .14 .12 .57 .12 .12 .10 .54 .10 .10 .09 .51 .09 .09 .08 I. d APPENDIX E Comparison of Time History Solution Methods - Mode Superpositon vs. Direct Integration I 1 e Comparison of Mode Superposition vs. Direct Integration Time History In mode superposition time history analysis, the residual or " missing" mass effect must be calculated and algebraically summed with the modal responses. To utilize modal time history analysis efficiently, only modal responses with frequencies up to the ZPA frequency need to be calculated; th,e " missing mass" not participating in these modes should be treated as an additional mode,vith its peak ~~ response calculated from a static analysis ofshe system for ZPA times " missing mass" (as defined in SRP 3.7.2 Appendix A). The time variation for the missing mass mode is obtained by scaling this peak response to the input time history. Ck arly, knowledge of the response spectrum is very useful when mode superposition time history is being performed. The value of the response spectmm is perhaps even more significant when a direct integration time history analyn is being conducted. The spectrum provides useful information for (1) the definition of the integration time step and (2) the development of a - p damping coefficients to simulate constant modal damping. The integration time step must be sufficiently small to capture responses which oscillate at all frequencies below the ZPA frequency. In addition, knowing the i frequency range of spectral amplification can help in the selection of target frequencies to obtain a best fit a - p combination for approximating constant modal damping. For the direct integration time history analyses of BM3, a and p were determined by specifying the target modal damping at the frequency of the fundamental mode (2.9 Hz) of the BM3 model and at an intermediate frequency between this frequency andfy,(16.5 Hz). A frequency of 11.5 Hz was selected, in order to achieve the best fit over the 2.9 Hz to 16.5 Hz range ofinterest. This resulted in a minimum damping equal to 80% of the target modal damping, which occurs at - 5.7 Hz, and a maximum damping equal to 130% of the target modal damping, which occurs at 16.5 Hz. Sensitivity to variations in the damping value diminishes asfy,is approached; consequently, somewhat higher effective modal damping in the high frequency end of the amplified range of the spectrum has relatively negligible effect on the total dynamic solution. In the case of the BM3 model and loading, the comparison between direct integration and mode superposition solutions, for both 1% and 5% damping, indicate close agreement. See Tables E-1 through E-4. The missing mass contribution to the mode superposition solution is significant. Without accounting for this contribution, one might conclude that direct integration is somewhat inaccurate when compared to mode superposition because ofits inherent inability t' represent constant modal i damping. Table E-5 shows the missing mass representing the residual after 31 modes (> 70 Hz). By including the missing mass contribution, excellent correlation was obtained between the complete mode superposition solution and the direct integration solution. E-1 c - - ~ Based on this comparative study, it is plausible that criticism of direct integration methods may be rooted in three potential sources for differences between direct integration and mode superposition solutions: 1) integration time step too large in the direct integration analysis; 2) poor definition of a - coefficients in the direct integration analysis; 3) failure to include ~the " missing mass" contribution in the mode superposition analysis. i E-2 Comparison of Peak Support Reaction Forces of BM3 due to Different Approaches Moments in in-lbs I % Damping NODE REAC. MODAL TIME MODAL (31 m) TIME DIRECT TIME No. TYPE (31 modes) at sec. Plus M.M. at sec. INTEGRATION at sec. I Fx 8.40 7.34 43.71 7.36 43.63 7.35 'll !Fy 10'I4 '7.34 14.36 14.56 13.'35 N.69 1 Fz 1.60 7.48 1.60 7.48 1.59 7.35 -1 ' Mz 1 49.98 6.68 J49.88 "6.68 45.'14 M38 1 My 776.73 7.36 776.40 7.36 778.86 7.35 1-Mz' 193.46 4.54 1278.42 7.36

279.55 17.35 4

Fx 68.75 7.34 116.79 5.36 117.78 7.35 4 'Fr

19.55 7.34 C20.01
-l7.34

?21.03 'if735 7 Fy 13.31 12.90 13.27 12.90 13.78 12.89 .11 -Fy --13.31 12.72 L13.31 E '2.72 613.74 il;i23i3 l 11 Fz 81.14 7.36 81.34 7.36 82.49 7.35 15. Fx 713.32 7.36 73137 ~ 7.36 L 74.29 3735 17 Fy 25.61 11.00 25.60 11.00 27.45 11.01 .17 - Fz 64.26 7.34 265.36 J 7.34 E66.32 E755 36 Fy 46.31 4.80 46.69 4.80 47.26 4.80 ' 36'. JFz '48.85 7.18 24212 '3.88 M I4130 n%36 ~ 38 Fx 128.79 4.48 732.18 7.36 736.41 7.35

38L i Fy :

43.52 7.36 f4334 ^ 77.36 74f51,id6 ~ 38 Fz 31.40 4.48 29.95 4.48 29.49 4.48 i38l IMx1 -720'04 7.54 iL719.65 $7.34 S71f65 1563 38 My 2142.00 4.48 2084.97 4.48 2055.80 4.48 P38L IMx1

3088.90

. 7.36 J3085.87 27.36 3018'90 - 5 36 23 Fx 259.22 4.48 259.59 4.48 265.71 4.47 "23 -

Fy-26.04

!7.38

426.08 l!7.38 52430 4D8 31 Fx 23.24 7.34 55.05 7.34 55.78 7.35 F3fl

/Fyi .'14.21

10.56 sif4.i7 3 0.56 d4.2D IEf6.M 31 Fz 16.07 11.84 16.08 11.84 16.90 11.83

!:31 )Mx i "1136 90 '11.84 L113d0 I11.84 ?1173.'80 Eli34 31 My 608.43 7.34 612.38 7.34 620.26 7.35 s31-iMz ^ 1767.90 - 7.34 '1773.20 7.34 !177630 Y735 i Table E-1 Comparison of Peak Pipe Resultant Moments of BM3 due to Different Approaches Moments in in-lbs 1 % Damping MODAL TIME MODAL (31 m) TIME DIRECT TIME Ele # (31 modes) at sec. Plus M.M. at sec. INTEGRATION at sec. I 823.40 7.36 790.95 7.36 791.02 73 2-818.58 .7.36 836.77 7.36 '838.40 57.35 3 1412.22 7.36 1429.28 7.36 1441.59 7.35 4. ~ 762.56

7.36

-763.87 7.36 L764.19 ^: !7.35 5 598.91 7.36 597.98 7.36 602.54 7.35 6 2134.36 .7.36 2133.18 ?7.36 ' 2141.09 3735 7 1876.38 7.36 1877.81 7.36 1880.06 7.35 8 852.96 738 852.74

7.38

'832.43 ?4.65 9 1415.46 7.38 1415.21 7.38 1438.67 4.66 10 1945.32 7.36 ~ 1946.24 7.36 1967.00 47.35 11 3826.15 7.36 3829.46 7.36 3804.39 7.35

12 8440.47 736 18428.48
7.36 28437.39 M7.35 13 6741.68 7.36 6741.03 7.36 6738.07 7.35 14-13241.93

-7.36 13219.91 a.7.36 13373.34 4;735 15 2290.84 7.36 2293.75 7.36 2364.57 4.67 1698.69 10.56

1696.61 i10.56

'1697.84 10.56 17 2455.47 10.66 2455.62 10.66 2492.35 10.66 '18 2360.13

7.36
2363.76 1736

. 234931 A2735 19 1623.90 7.36 1625.77 7.36 1623.68 7.35 '20. .7539.21 736 .7544.81 h 7.36 m 47484.45 R7.35 21 1595.61 7.36 1590.16 7.36 1564.96 7.35 ~22 2042.84 10.56 2046.41 W 10.56 f2036.58 110.56 23 1800.75 10.56 1807.75 10.56 1802.73 10.56 151831

!10.56 5!

31514.11 F10.56 =24' 1522.11 110.56 4 25 926.28 7.34 922.51 7.34 915.35 7.34 26c 665.57

7.34 J677.35

= 7.34 674.46 h734 27 399.00 7.34 390.71 7.34 387.38 7.34

28

.74239 .7.34 737.42 .7.34 1734.60 17.34 29 824.73 7.34 830.76 7.34 825.51 7.34 -302 .1988.96 17.34

1994.56
. 7.34
204235 "n7.35 31 7437.70 10.56 7428.79 10.56 7377.43 10.56
32:

8931.45

7.34

?878532 M 7.34 . 8703.23 4:735 33 11421.16 7.34 10970.06 7.34 10866.98 7.34 -34.

6096.62 v7.36
5993.82 17.36
5966.82 67.35 35 6633.91 7.36 6788.22 7.36 6757.74 7.35
36 :-

1942.86 736

1874.84 a7.36 21892.65 17.35 37 3538.74 7.36 3501.06 7.36 3487.86 7.35 Table E-2

Comparison of Peak Support Reaction Forces of BM3 due to Different Approaches Forces in lbs Moments in in-lbs 5 % Damping NODE REAC. MODAL TIME MODAL (31 m) TIME DIRECT TIME No. TYPE (31 modes) at sec. Plus M.M. at.sec. INTEGRATION at sec. 1 Fx 7.87, 7.36 44.22 7.36 44.19 7.35 1 FFy -11.33 7.34 13.86 L7.26 42.24 41.39 1 Fz 0.97 7.50 0.97 7.50 1.02 7.49 1 Mi-19.11 9.04 m 518.89 l9.04 L18.84 2431 My 695.11 7.36 694.80 7.36 703.87 7.36 1

Mr ~

156.'85 7.38 2240.22 E.7.36 {239.67 N.s7 4 Fx 65.32 7.36 113.70 7.36 113.82 7.35 4 ' Fr ' 13.38 7.36 4 J1'.07 17.56 13.85 L 7.36 4 7 Fy 8.01 7.60 7.95 7.60 8.26 7.61

11 ~

Fy 17.13 12.74 17.13 212,74 l:7.54 :512!73 11 Fz 71.69 7.36 71.89 7.36 72.60 7.36 15 - Fx "672.79 i 7.36 7690.97 ii7.36 T692.1'9 57.36 17 Fy 20.15 7.38 20.14 7.38 20.26 7.38

17. :

' Fz ' o 58.91 7.36 E60.05 47.36 ?$0.27 $736 36 Fy 37.58 7.38 38.70 7.38 39.84 7.37 36-i Fz: 50.25 12.60 E42.33 d2,60 Ni[88 NTOi 38 Fx 136.14 7.34 740.68 7.36 744.53 7.35 38 Fy; 38.25 7.38 !38.18 s7.38

d9.69 i!:7.37

~~ 38 Fz 31.43 7.18 29.85 7.18 29.61 7.19 i466.39

7136 F465A2 > 47.36 T481.'87

$539 L381 LMilb 38 My 2143.30 7.18 2080.83 7.18 2065.30 7.19 i 2713.'20 R7.38 = 2820.50 ML7!37 / ~ i a38: Mz': 2715.60 ' 7.38 23 Fx 270.09 12.60 270.50 12.60 268.35 12.60 23" c Fy; (10.54

7.38 il10.58

-i7.38 E10.97 !!d7 31 Fx 22.57 7.34 54.46 7.36 55.10 7.35

!31 Fy!;

29.62 ~ 7.38 J9'76 i758 ^ l10 16 "Edd 31 Fz 8.66 7.28 8.66 7.28 9.02 7.28 !31' -: Mii ?559.28

12.60 E560!28 E124d L598.88 *!ds 31 My 578.29 7.34 582.28 7.34 586.45 7.35 31 Mz1 L 1679.80 7.34

!1685.20 L7.34

1685i60 id5

~ Table E-3 = j Comparison of Peak Pipe Resultant Moments of BM3 due to Different Approaches Moments in in-lbs 5 % Damping MODAL TIME MODAL (31 m) TIME DIRECT TIME Ele # (3 Imodes) at sec. Plus M.M. at sec. INTEGRATION at sec. I 745.97 7.36 713.05 7.36 721.44 7.36 2

746.03 7.36 765.02 1736 0772.40 - A736 3

1306.36 7.36 1323.65 7.36 1326.79 7.36 '4 683.45 .7.36 684.42 E7.36 V693.18 s7.36 5 529.22 7.36 528.66 7.36 538.01 7.37 '6: 'I894.42 7.36 1893.29

7.36 11919.87 47.37 7

1663.22 7.36 1664.64 7.36 1695.52 7.37 8 765.19 736

764.93
7.36 A781.15 ' M7.37 9

1320.17 7.36 1319.86 7.36 1362.63 7.37 10 1713.20 7.36 1714.12 1736 a S1730.62. 1736 11 3504.53 7.36 3507.84 7.36 3557.59 7.37 12 7662.41 7.36 7650.43 7.36 : 7733.23 r:1736 13 6075.86 7.36 6075.21 7.36 6149.10 7.36 -14~ 12509.76 7.36 12487.74

7.36 F 12525.94 ih7.36 15 2027.62 7.38 2030.00 7.38 2061.56 7.37 16 1057.72 7.26 1057.35
7.26 31032.69 17.26 17 1496.87 7.38 1495.69 7.38 1523.29 7.37 18 2023.46 7.36 2027.05 37.36 M2107.67
s737 19 1368.94 7.38 1370.50 7.38 1433.54 7.37 20-6607.12 7.36 6612.74

?7.36 56886.72. G737 21 1340.38 7.38 1337.38 7.38 1397.33 7.37 22 1789.34 7.36 1803.08 %736 c1822.39 %7.35 23 1339.89 7.36 1352.91 7.36 1368.99 7.36 24 1236.93 7.36 '1229.48 L7.36 .1232.76 17.36 25 962.11 7.34 958.32 7.34 952.48 7.34 26 677.97 7.34 L 689.97 2734 s687.08 !k734 27 328.88 12.6 322.88 12.6 320.59 12.6 23 630.00 .7.36 624.60 '7.36 .635.75 87.35 29 731.43 7.36 737.73 7.36 749.77 7.35 1777.89 7.34 1784.26 07.34 '1798.77 47.35 30 31 7221.12 7.36 7194.98 7.36 7240.74 7.35 32 8935.01

7.34 18768.03 i;7.34 L 8747.80
.7.35 33 11395.76 7.34 10912.71 7.34 10913.38 7.35 34 5486.76 7.36 5338.54

.7.36 . 5519.24 1.7.37 35 6167.02 7.36 6348.71 7.36 6378.37 7.36 36 1835.19 736 1755.84 ~ 7.36 "1763.67 6.7.36 3278.70 7.36 37 3308.12 7.36 3263.83 7.36 Table E-4 Nodal Masses Participating in 31 mode combination participating _ mass.xis Units:Ibs/in/sec**2 Node No. Total Mass Participating Mass of 31 modes Missed Mass after 31 modes 1 0.017 0.000 0.017 2 0.026 0.000 0.026 3 0.212 0.001 0.211 4 0.227 0.000 0.227 5 0.031 0.024 0.007 6 0.165' O.166 -0.001 7 0.165 0.168 -0.003 8 0.092 0.093 -0.001 9 0.267 0.270 -0.003 10 0.275 0.277 , -0.002 11 0.I82 0.I83 -0.001 12 0.174 0.174 0.000 13 0.092 0.092 0.000 14 0.059 0.053 0.006 15 0.101 0.008 0.093 16 0.059 0.055 0.004 17 0.105 0.106 -0.001 18 0.268 0.272 -0.004 19 0.180 0.I83 -0.003 20 0.073 0.073 0.000 21 0.207 0,170 3.037 22 0.088 0.066 0.022 23 0.043 0.038 0.005 24 0.052 0.053 -0.001 25 0.197 0.202 -0.005 26 0.178 0.189 -0.011 27 0.048 0.053 -0.005 28 0.063 0.072 -0.009 i 29 0.048 0.051 -0.003 30 0.175 0.165 0.010 31 0.158 0.000 0.158 32 0.163 0.150 0.013 33 - 0.276 0.280 -0.004 34 0.780 0.881 -0.101 35 0.687 0.090 0.597 36 0.687 0.008 0.679 37 1.170 0.007 1.163 38 0.585 0.000 0.585 Note: Negative Values for Missed Mass indicate that 31 Mode Mass Participation exceeds the Total Mass. Table E-5 4 l APPENDIX F i j Use of Response Spectrum Generation SDOF Oscillator Responses to Establish the Threshold Frequency for In-Phase Modal Responses i l i t J 4 l 4 Methodology for Identification of the Lowest-Frequency SDOF Oscillator which Responds In-phase with the Input Time History during Response Spectrum Generation Introduction A key element in the utilization of the Response Spectrum Method to predict peak dynamic response due to a seismic time history input is establishing appropriate phase relationship between the individual peak modal responses, so that the result of modal combination provides a reasonable estimate of the peak dynamic response. During the generation of a response spectrum from a ground or in-structure time history record, the complete time history of each SDOF oscillator response is calculated and processed to identify the peak response. This peak response becomes a single point on the Response Spectrum plot. Associated with each SDOF oscillator peak response is the time of occurrence and direction of the peak response. This information is typically not retained since it is not needed to generate the Response Spectrum. However, valuable conclusions can be derived from this information by comparison to the time and direction of the peak acceleration in the time history record. The lowest SDOF oscillator frequency (f,) for which the time and direction of peak response coincides with the time and direction of the peak of the input time history represents the onset ofin-phase response with the input, provided all higher-frequency SDOF oscillator responses exhibit the same behavior; i.e., for f 2 f,, all SDOF oscillator peak responses occur at the same time and in the same direction as the peak of the input time history. To further verify that in-phase response exists, a comparison of the crossings of the acceleration equal to zero datum between the input time history and SDOF oscillator time history response should be performed for SDOF oscillator frequencies in the vicinity of f,. Once j;, is established by this procedure, which can be fully automated and made a part of the response spectrum generation algorithm, it is str'aightforward to specify in the modrJ combination methodology that all structural modes with frequency 2 f, are to be combined algebraically. Application of this approach to the BM3 time history record for 1% and 5% damping produced the following results: f, f; 1% damping 9.5 HZ 16.5 HZ 5% damping 7.5 HZ 16.5 HZ

Previously selected, based on the BM3 Response Spectrum plots for 1% and 5% damping Algebraic combination of structural modal responses for f 2 f, is accomplished by the " missing mass" procedGre using the ZPA. In the frequency range f, to f,, individual structural modal responses are algebraically combined with the " missing mass" contribution. Consequently, the response of all modes with / 2 f, are combined algebraically.

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^ 3.45 1.90 7.86 20.00 0.55 7.34 ~ 3.60-1.68 7.53 22.00 0.55 7.34 3.80 1.39 7.52 25.00 0.54 7.34 4.00 1.42 7.51 28.00 0.54 7.33 4.20 1.33 7.36 31.00 0.54 7.34 4.40 1.31 7.35 34.00 0.54 7.33 4.60 1.25 7.35

ML20154D977

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