ML20195B853

Rev 0 to Version 1.1, Equipment Dynamic Analysis Software Package:Theory & Verification Manual

ML20195B853
Person / Time
Site:	Catawba
Issue date:	09/01/1986
From:	STEVENSON & ASSOCIATES
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References
PROC-860901, NUDOCS 8811020236
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STEVENSON

& ASSOCIATES o structural mechanical consulting engineering firm ,

9217 Midwest Avenue 10 State Street Cleveland, Ohio 44125 Woburn, MA 01801 (216) 587 3805 (617) 932 9580 TELEX: 5106015834 TELEX: 494 0995 HQCM FAX: (216) 587 2205

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EDASP. VERSION 1.1 EQUIPMEtiT DYNAMIC ANALYSIS SOFTWARE PACKAGE THEORY AND VERIFICATION MANUAL REVISION O September 1. 1986 Prepared by STEVDISON & ASSOCIATES 10 State Street Woburn. Massachusetts 01801 CONTR01. COPY hWiBER _

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t TABLE OF CONTENTS Page i 1.0 STRUCTURAL MODEL ..................................... 2 1.0.1 Background ........................ ............. 2 1.0.2 Case Description.................................. 2 1.0.3 Program Results .................................. 2 2.0 BASE EXCITAT;0N M00EL................................. 6 2.1 TIME HISTORY ......................................... 6 2.1.1 Background ....................................... 6 2.1.2 Case Description.................................. 6 2.1.3 Program Results .................................. 6 2.2 RESPONSE SPECTRA ..................................... 10 2.2.1 Sackground ....................................... 10 2.2.2 Case Description .. .............................. 10 2.2.3 Program Results .................................. 14 ,

2.3 POWER SPECTRAL DENSITY FUNCTION ...................... 15 2.3.1 Background ....................................... 15 2.3.2 Case Description ......... 4..................... 15 2.3.3 Program Results ................................. 16 2.4 RS/PSD CONVERSION .................................... 18 2.4.1 Background ...................................... 18 2.4.2 Case Description ................................ 19 2.4.3 Program Results ................................. 22 2.5 RS BROADENING ....................................... 26 2.5.1 Background ....................................... 26 2.5.2 Case Description ................................. 26 2.5.3 Program Results .................................. 26 3.0 RESPONt ANALYSIS .................................... 33 3.1 TIME HISTORY ANALYSIS ................................ 33

, 3.1.1 Background ...................................... 33 3.1.2 Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 <

3.1.3 Program Results ................................ 36 3 . 2 P S D ANAL Y S I S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.1 Background ...................................... 66 3.2.2 Case Description ................................ 67 3.2.3 Program Results .................................. 70 4.0 STRUCTURAL MODIFICATION .............................. 105 4.0.1 Background ..................................... 105 4.0.2 Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.0.3 Program Results .................................. 107 i

... .. - - _ _ __-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___ _ - __ -_A

Page 5.0 REVISION 1 VERIFICATION ......................... 114 5.1 CORRECTION #1 ................................... 114 5.1.1 Background .................................. 114 5.1.2 Case Description ............................ 114 5.1.3 Program Result ........................ ... 114 5.2 CORRECTION #2.....................-.............. 115 5.2.1 Backgr ound .................................. 115 5.2.2 Case Description ............................ 115 5.2.3 Program Results ............................. 115 5.3 CORRECTION #3 .................................. 115 5.3.1 Background .................................. 115 5.3.2 Case Description ............................ 115 5.3.3 Program Results ............................. 116 5.4 CORisECTION #4 ................................... 116 APPENDICES ........................... . - 123 ii

F-1.0 STRUCTURAL MODEL 1.0.1 Background The structural model used for the verification of EDASP is described in this section. The input for the structural model includes

- Number of nodes

- Number of modes

- Active degrees of freedom

- Node coordinates

- Mass at each node

- Node connectivities

- Frequencies

- Mode shapes 1.0.2 Case Description The structure used for the verification is an eccentric frame shown in figure 1.0.1. This particular frame has all six degrees of freeaom active at each node, and the directions are cross-coupled in that an excitation in any patticular direction forces the frame to respond in all six directions. Hence,the frame can be regarded as a general three-dimensional structure and its response should enconpass various issues that may arise when analyzing a complicated Structure. Nodal masses and beam t'ement properties for the frime are sumnarized in Tables

' 0.1 and 1.0.2.

To obtain the modal properties, i.e., frequencies and mode shapes for the input of EDASP, program STARDYNE , which is a general purpose dynamic finite element package at United information Service, was used.

All the 24 modes were solved using STARDYhE. However, only the first 10 moues were used for EDASP analysis. The frequenci* and mode shapes are shown in Table 1.0.3. These modal properties wert fed into EDASP .The generalized masses and participation factors were then calculated, although they were given in STARDYhE also 1.0.3 Program Results The results obtained from STARDYhE eigenvalue analysis are included as Appendix 1.0.1. The geometry data and modal data generated by EDASP are included in Appendix 1.0.2.

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Element Node i Node j Al A2=A3 11 12=13 1 1 5 2 1.78 1.5 0.75 2 2 6 2 1.78 1.7 0.85 3 3 7 2 1.78 0.6 0.3 4 4 8 2 1.78 0.3 0.15 5 6 6 2 1.78 0.6 0.3 6 6 7 2 1.78 0.6 0.3 7 7 8 2 1.78 0.6 0.3 8 8 5 2 1.78 0.6 0.3 Units: A(in**2) 1(in**4)

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L _ .c04228922 .000679479 Table 1.0.3 Frequencies and Mode Shapes for the Frare Podel 5

2.0 BASE EXCITATION MODEL Base excitations used to verify EDASP response analysis are described in Sections 2.1 to 2.3 for time history, response spectrum, and power spectral density function respectively. Section 2.4 discusses and verifies the RS/PSD conversion process. Section 2.5 verifies the broadening process available in the base excitation module.

2.1 TIME HISTORY 2.1.1 Background Base excitation acceleration historit, used for verification of E0 ASP response analysis are described in this section. Three random time histories were generated using a randum number generator with Gaussian distributions. Parameters in a time history file are

- Number of points in the history

- Time step

- Acceleration at each step 2.1.2 Case Description Three independent time histories were generated using the random number generator. The magnitude of each has a Gaussian distribution with zero mean and a unit variance. They are stored in files randx -

x-direction base excitation randy - y-direction base excitation randz - z-di,ection base excitation The number of points for each time histcry is 601. A time step of U.025 is used in the analysis. The three time histories have absolute maximums of 2.91, 3,48, and 3.59 for the x, y, and z excitations. They are shown in Figures 2.1.1 to 2.1.3.

2.1.3 Program Results The listing of the three time histories, randx, randy, and ra'id2, using EDASP are included in Appendix 2.1.1. The same time histories used in the STARDYNE verification analysis are included in Appendix 3.1.1 to 3.1.3 as part of the output files for time history response analysis.

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2.2 RESPONSE SPECTRA 2.2.1 Background EDASP permits the user to f pecify response spectra as the input base excitation and performs PSD response analysis. However the input response spectra must be converted to power spectral density functions (PSD) before performing the response analysis. The conversion can be made using the base excitation module in EDASP.

The response spectra used for the verification of PSD response

, analysis is described in this section. Parameters required to define a response spectrum are l

- Number of points (2 to 400)

- Damping ratio of the RS (0 to 0.99)

- Frequencies (Hz)

- Accelerations (g) l The converted P50s will be described in Section 2.3. Details and verifications of the conversion process will be described in section 2.4 2.2.2 Case description The response spectra used for the verifi

ation analysis are the ones in "USNRC Regulatory Guide I.C0 -- Design Response Spectra for Seismic Design of Nuclear Power Plants" Revision 1. December 1973 Each spectrum is defined by six po:nts as shown in Tables 2.2.1 and 2.2.2. Since these spectra were as>ured to be linear in the log (frequency)-log (scceleration) domsin, while the EDASP program is interpolating the spectra linearly in tha log (frequency)-(acceleration) domain, a much closely spaced frequency points is required, the seventy-five f requency points recommended in "USNRC Reguistory Guide 1.122 -- Development of Floor Design Response Spectra for Leismic Design of Floor-Supported Equiprent or Components", Revision I, February 1976 h ed to interpolate the spectra. A SA51C program was written to a fc m the interpolation, w seventy five points response spectra are stored in files nrcit0h rs - a- and y-direction t>ase t.acitation K5 nrcl60v rs -

z-direction base excitation R5 where nrcl60h.rs correspond to the USMC herizontal response spectrum scaled to 1g ZPA and nrcl60vors correspond to the USNRC vertical response spectrus scaled to 19 PA. The spactra are shcwn in Figures ?.t.1 and 2.2.2.

10

Freq.(Hz) 0.5% 2% 5% 7% 10%

0.2 0.471 0.368 0.302 0.277 0.250 0.25 0.736 0.575 0.471 0.432 0.391 2.5 5.95 4.25 3.13 2.72 2.28 9 4.96 3.54 2.61 2.27 1.9 33 1 1 1 1 1 34 1 1 1 1 1 1

Table 2.1.1 NUREG GUIDE 1.60 Horizontal Response Spectra (1g ZPA)

Frea.(Hz) 0.5% 2% 5% 7% iO%

0.2 0.313 0.246 0.202 0.184 0.165 0.25 0.49 0.384 0.315 0.287 0.26 3.5 5.67 4.05 2.98 2.59 2.17 9 4.96 3.54 2.51 2.27 1.9 33 1 1 1 1 1 34 1 1 1 1 1 Table 2.2.2 NUREG GUIDE 1.60 Vertical Response Spectra (19 2PA) 11

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n 2.2.3 Program Results These files can be viewed and plotted using EDASP program base excitation module. The results are included in Appendix 2.2.1.

14

2.3 POWER SPECTRAL DENSITY FUNCTIONS 2.3.1 Background The base acceleration power spectral density functions (PSD) for the verification of EDASP response analysis are described in this section.

Yhey are converted from the response spectra described in Section 2.2.

The parameters required to define a PSD file are

- Excitation duration (secs)

- Probability of exceedance

- Number of points

- Frequencies (Hz)

- Power spectral densities (g**2/Hz)

The probability of exceedance and duration for the PSD are associated with the response spectra. The results are not sensitive to these two parameters if the user specifies a response spectra as input, since the output RS will be consistent with the input RS. However, when a PSD is specified as input, the user must determine an appropriate probability of exceedance to be used. A value of 0.15 has been widely used since it's consistent with the development of NRC spectra.*

2.3.2 Case Description Twn PSDs were converted from the 5% response spectra in files nrcl60h and nrcl60v. The PSDs are stored in files nrci60x -

x- and y-direction excitation PSD nrcl602 -

z-oirection excitation PSD l

The parameters used in the conversion process are Probability of exceedance = 0.15 Excitation duration = 15 Divisions / octave = 24 Convergence criteria = 0.005 Maximum # of terations = 10 Use of 24 divisions / octave results in a PSD defined by 179 points and a RS defined by 35 points. Due to the relatively small given convergence criteria, both solutions ran through 10 iterations ano didn't converge.

The converted PSDs are shown in Figures 2.3.1 and 2.3.2.

2.3.3 Program Results The Resulting PSDs can be viewed and plotted using the EDASP base excitation module. The Rtsults are included in Appendix 2.3.1 ON.M. Newmark, J. A. Blume, and K.K. Kapur, "Seismic Design Spectra for Nuclear Power Plants," Journal of the Power Division, ASCE, Vol.99, PD2, November 1973, pp. 287-303 15

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• • "".?.?.*%

?. " .

....?- *: -

.*.**..~T ?* *

-a;g T-2 .".* T .

.~.

- ".O ~. : .i"4

., ?.* - .** : ..J~.:.*.'. *

..f~ ; . .n.-,J. .":

J - ..* &; ::::1 a  : :

~.

._.  :.*7". . . * :..  :

• .*?*

.  ?.*.  : ~"'". **. 7.

~~. *

:"  : *.* . ~ .

.. . . - - ' . ..... ~ . .: :.: . :?..

.. ?.: .:

J *J. .~. :. *. :.

• l T ?:. *;.*......:?.7..*..". *-

• ~.

.:: : 7.*.".?. . :27

.;.. *:.' :.? .-

a  :. J.

, . : . r. . . . . .- - . ..

.-- .* ; *: . : ::r::

?"  ?. P. *

..  :* .~.

1. . :.':.:::.:

... ~-~

. ~ . * .

a. -:.. ,

. "p ._ ,

W N

• .. ?'.*: -

_~-.

, =.

.*.*.=.*., L

.~.

• ~ .::: ' . . ~ . .*

?! T.r:

~.~ . **-**

1'  ?.  : * ?.  :* . ~ . ' " *. . ~ .

*:" *:** ~ . ~ . .~.  : . " * , ~ . .~.

r:.. :7:r:

-  : . r: :7:

0 *: ::  : :: :*

l

.--- .!--. - : . I:

.!.7:T: . 7:

T9

*.*.T. ff!

7. 7. ?. : .  :**

7

::r.:.:.

Figure 2.3.2 Power C.pectral Density Function for NRC 1.60 Vertical 5% Damping Response Spectrum x-- - -

E 2.4 RS/PSD CONVERSION 2.4.1 Background The relation between response spectrum and the associated PSD can be derived from random process theories.* The response spectrum at a certain frequency is the maximum response of the single 00F system. The distribution of the maxima can be computed knowing the energy content or PSD of the random process.

The equations used in EDASP are sumarized in the following.

PSD to Response Spectrum. To determine response spectrum from a PSD.S( d, the following formula is used:

R(w)={-2mintf(

g g )b in(1-r)]}b (2.4.1) where m

n * *n("o) "

• N (w)l S(w)da, n = 0,2,4 (2.4.2) o 4

w 0 + 4w0 2g 2w2 e

l Hg (w)l 2=

(2.4.3)

(wg 2.w2 )2 + 4w92c 2 ,2 T = effective earthquake durction r = probability of exceedance This procedure is straight forward tG app'.y but involves an approximatio.. -

It deviates from the exact solution wttn.

m \

N' (m]y)T (2.4.4) is small or when r =(1-m2/mmr.)*

g (2.4.5) is very close to 1.

1F.7. Kaul, "Stochastic Characteritotion of Earthq' lakes Through Their Response Spectrum," Earthquake Engineering and Structural Dynamics, Vol.6, pp. 497-509, 1978 18

/

0 I'

Response Spectrum to PSD . The PSD is determins d from the response spectrum '

by:

- 2( 2 S(w). --* R (w) f-2 in[ "- In(1-r)])~1 (2.4.6) ww wT i= .y

,. where L

l

(, = ( + 2/wT = damping ratio taking into account finite earthquake duration g = damping ratio of R(a) i To ensure proper representation of the PSD from the.res  ;

solution is iterated, i.e...a new response spectrum R(w)ponse spectrum, is determined from the i Eq.(2.4.1) by applying S(w) from Eq.(2.4.6) The PSD is updated by:

"}

S(u) pg~= S(w)9 I

R(w)4 (2.4.7)

' The process is repeated until S(w) converges.  !

4 NumericalintegrationofEq.(2.4.2) form o and m2 is carried  !

.ut by exact analytical solution. A constant PSD value, which is equal to the average of the two end points, is assumed within each frequency interval for integration. The interval size is determined by the 2

l l specified number of divisions / octave.  :

< In evaluating RS from PSD using Eq.(2.4.1), coarser frequency spacing is used for R5, since the response spectra are usually much less t i spiky than the associated PSD. The ratio of the nunber of i divisions / octave of an RS to a PSD is set to approximately 3/16 in EDASP.

2.4.2 Case Description .

Two cases for RS/PSD conversion were investigated to verify the l EDASP RS/PSD conversion module. The NRC response spectra described in i i Section 2.2 were used for both cases. The first case compares the EDA 5P '

results with the published results by Kaul cited in Section 2.4.1. The

, second case consists of converting the P50; generated from the 5% damped

RS back to RSs with other damping values. 7he results are then compared j- with the original NRC response spectra set, i t i

1 19 i

i i  !

Case 1 The 0.5%, 2%, and 5% damping curves of the NRC horizontal response spectra in file nrcl60h were converted to PSDs. The parameters used are e Excitation duration = 15 sec Probability of exceedance = 0.15 '

Maximum i of iterations =0 Divisions / octave = 24 The parameters are identical to those used by Kaul. The EDASP results at selected frequency points are summarized in Table 2.4.1.

Case 2 The PSDs in files nrc160x and nrc!60z generated from 5%

damping NRf response spectra are converted back to RSs with dampings of 0.5%, 2%, 5%, 7%, and 10%. The parameters used are Probability of exceedance = 0.15 Excitation duration = 15 Divisions / octave = 24 4

20

1 l

l l

l Freq. (Hz) 0.5% 2% 5% )

(Kaul 1) T1 (Kaul 2) T2 (Kaul 3) T3 0.20 17.3 x 10'4 12.0 x 10-4 10.0 x 10'4 0.25 22.8 16.1 13.7 I 0.30 24.0 17.3 15.2 0.50 20.2 15.4 14.4 l 1.01 17.0 14.7 15.0 1.51 16.1 15.2 16.2 2.46 15.8 16.9 18.8 4.02 6.06 7.61 9.02 6.02 2.82 3.96 4.90 10.12 0.87 1.47 1.99 15.16 0.18 0.41 0.68 Table 2.4.1 E0 ASP Results to Compare with Kaul's 21

2.4.3 Program Results The EDASP results to compare with Kaul's are included in Appendix 2.4.1. The EDASP results for Case 2 are included in Appendix 2.4.2.

Case 1 Comparison with Kaul's approximate solution (so called) are shown in Figure 2.4.1. The EDASP results are close but somewhat larger than that of Kaul. The difference occurs because that the EDASP interpolates the input RS linearly in the log (frequency)-acceleration domain while Kaul interpolates the input RS linearly in the 109-109 domain. The EDASP results are based on more conservative assumptions with the same input RS points without knowing what's happening between input points. If the user has an input RS that is known to vary linearly in the 109-109 domain, a fine frequency spacing must be used.

Case 2 Comparison with the original NRC response spectra for 0.5%,

l 5%, and 10% damping curves are shown in Figures 2.4.2 and 2.4.3 for i

horizontal and vertical spectra. The results show good matches for 5% and 10% damping curves up to near ZPA frequencies. For the light damping curve, EDASP is giving higher response at frecuencies greater than about 1.3 Hz and vice versa at lower frequencies. Since the NRC response spectra set is not necessarily consistent in itself, the difference can be considered to be insignificant.

l l

[

l i

22 i

SUBJECT._ . _ _ _ _ .__ ., JOB No _ _ _.. .._.SHFFT__ OF

_ f% -_.. -..

-. . - - - . . - . . . . - . 9

\

/

$ \

STEVENSON - - - - - - -

5 f --~ ~~~' -

& ASSOCIATES

<m w -

E N

f

!li: . II 25 ..t i i l r- -

u -

l

, x en . - ...

gg ~

s. 'N

[/  ;

10 ---1 1 - 4 ZAr7/Il-3f

.. - ll

• ---H' --- - - -

-o -

. IL ,r; i

. . . t f '

A2 'f /

y A.

---.j .

L.,

- A3 I x 3 \ '

g gs .- C2 :. .. -_ .. . .

\;y N

72 - ' - - -

I i-I i i

3 ,,5 . E 3- . ,  ; .tj

.i

$, j. 0 . - - . . .. .-

a. -

.7,_

i 6 -

t .: -

i .

-..._I.- s .' -

,.37 3 . . . . .. ._,;

4 __ .- .-;.. .. _

l.

,4 T2

,3 _.

77 .

IA ' ,##

• 25 -

.'I 6 1 2 5 5 4 5 <> " 'I >

255 2 ' * * "l o ' t ' h5 FREQUENCY (eps)

Figure 3. Derived acceleration power spectra (using equivalent oscillator dampiog)

Figure 2.4.1 Comparison with Kaul's Results 23

,. :e.

10 .

.~~-. - _ _-

' -K

' N a 5- f

\r

, . ... ,N .

/ / to 1.,

,/ ,.'

i' ' s %s,s .y

, , ' , .a' / Q 1

jd i ,

f

/

/

/ia,a - utc it a ih swa wenu Ab

• Resp n s< spo a k gw h4- Av~ L PSD C on &h 4 $se Sf MAC Ao ,je&J s/L<-.%

./

o_ g ,

/

l l

I,)

e

- e l 's>

e l

toe (f e .

Figure 2.4.2 Comparison with NRC 1.60 Horizontal Spectra l 24 l

10 n - . . .. - . j

/

,/

'y\)

[1) [ r/, -

i s.

J.;.:.

\

\

.l j.:

. I * */ . %, .

/

/ ,

NN -8

1. .

/

f

/

/

/

/. J

/

,' limes : W C J(,o verhJ spr e Ica.,

'/

, dah: Respoest spec k gee n d e s fr w tk PSD e'I -

Con wh ln~ 24 M!c s- w & c d spee %

f

/

i. o 1 0I i sc oc.

(!!, )

Figure 2.4.3 Comparison with NRC 1.60 Vertical Spectra

~

25

1 2.5 RS BROADENING 2.5.1 Background t EDASP allows the user to'sroaden response spectra using the base excitation module. In addition the same broadening algorithm has been built-in in the time history response analysis and PSD response analysis modules, so that the output RSs from response analysis are broadened if broadening factors other than zero have been specified in the instruction set. in this section, broadening in the base excitation module is verified.

To broaden a response spectrum, the user must specify the following

- Damping, if nore than one curve is present in the RS file

- Broadening factnr

- Filename to contain the broadened RS 2.5.2 Case Description Six RS files, generated from time history analysis, rs51x, rs52x, rs53x, rs81x, rs82x, and rs83x, each containing one RS curve, were broadened using the base excitation module. A broadening factor of 0.15 was used. The broadened RS were stored in files bsbix, bs52x, bs53x, bs81x, bs82x, and bs83x.

2.5.3 Program Results Listing of the unbroadened RSs using EDASP can be found in Appendix 3.1.4 as part of the results of time history analysis. Listing of the broadened R$s ar's included in Appendix 2.5.1.

Comparison of these files can be made readily by using EDASP ARS/TRS comparison capability af ter an appropriate instruction set was created.

The results are shown in Figures 2.5.1 to 2.5.6. The broadening prccess is verified by inspection.

l 26 i

~ -

.. .' .**?.,"J.*.:*..".. y

~:*

.. . . . . _ ........1**".*".**...

. .  % ; ::t.:*

t  !?.

... . . . . . . . . . .* * '..A..,... *

- .... ... . - . . . . . . . . . . . . .. ... . .J

- . ..... :....J: J

..2

f. * .

t...

..= . . ....

.= . ..

. . ::T.*

i

• : .r2:

. . *7.

.., . ==""

FN)

• N

~..".-

~

.e e,. ..

. ~..

a.-

. . _M-t* 3*,. :2 ' --

.7h"

. . .... *? ** *

1. 2

..~.; . .: * :

.. .. .._. ....... 1. *. :.::-: * . --. .

= ...

.. =. _.....

1 2.....

2: ::: if

... .S. t.

Figure 2.5.1 Broadening of RS rs51x in the Base Excitation Module

~.. . .

...-. , .*, . ,:, . : *..** r . ?.

• e. =,

.'L*..".*..I.?.**.P2.

. . .. .. .~ . . . . ... ...

.. .., .... ". . . t. : : t.:

...t.3

.-. . . ..t.

L-*-

..-..".4..*."".***..*f

.L .~ ?.

.... .y, -

L*2.. . . .

2....

4. ..

. ;:* 6

.L.*.~.*.. .

.M e

.= .. ..

. ;E ..*

N

p. .~....

....2..

g . .

-il

~

..........- .uui.u.uui

< - . T:T:--

3 .

- . ......I.

... - . . .1 - .

• J5

e. ::""..: .,.*

• I T*" " * *....: u.

_.1.."..'?:*.*."".*.**

.:... . :-.._.u.

Figure 2.5.2 Broadening of RS rsS2x in the Base Excitation. Module

-_ ..~ _ ~. .. . .- . ~ . . + n - a a a -

• a .' " . " . * . . .* * : . .". J. * ..

_ 1.".."..~.~*.*.a. . . .

., . - .:2."".!.!".'",*.*P..* . . " . . .* :::* .. 4.. : :. .: .: .

. za ::.*. ;* ; : *:. {

. .2....  :

.. ? ;: :

~......:* - ~7 7 * * * *

2 .: u*.._::2 a : -::.

-: : .-_. .:-_ :- +:::: . . - - - a:

- n :,

-.-a._- *

-~.g

.1:  :

." s P- .*2

.~. . ~ . .

. ~ . . ~ . . ~ .

.r  ::u : u .-

1. .~.. T:-" : *.:.: .-

.~~

.:. : = -

r N  ;

D ..

.~ :"

~. .

.i 55 --~. 5

• .*e.*..,

• . ~~

.P " --

~~~

.=.*.

• 55 3555- +

.~.  ?..~.

*::  :.~..~..~. .~.

. : 7

..- . - 2 7.* *1 **:::: f..,~..

** {.~..'"

. : *:2

. .. F.J ::: r .. T:  : --.

77'.7

3 *.

• • * * : : I". :* *

• 3  : : : *7 :

-n -: ._ u.:.:.: .:.,.

.._.u. . :: . :

l l

Figure 2.5.3 Broadening of PS rs53x in the Base Excitation Module

. . . . .: . . u : : .: .-

-.. . ::u:3 4. -:.

2 . ...:..  : . . --. . _ . * .- : 1 ..

. : . . ~ .. L* :: * : .-

- - ~ a.::::::  :. : ~..

,a:*.. -. .^ . :: . . - 'a * :. : J : ..

L:  :*i..

. . - - ~ ::: - -.. -..-  : - .: . . .* ::". . . . *: *. . *.:":,

• *.i :.*.

.: - .: .:. : . . a .: :: .:. :

.. . ~ . . . .~..~..~.

• :.*=.

.A* . -.:::.- ..

• ~~.,*,

• 4. .". '""""

W .

O

=: .- .:

.~. :"

.' 12. :2. .

.:. : ._- : J b **'~

.. . ~ .* . " .

"* * * ;; ~

.~. .K : .. ,R

• :: ::* *: : :. ~;' ..~.

• i:: .Rs"..~.

.  :'  : .~. .. **

. ~ . . ~ . :~.

: .g:: **: .~.

A"..~..~.

. := ::;:

. , .-=: :-:

Figure 2.5.4 Broadening of RS rs81x in the Base Excitation Module l

l l

l

.u.

~

~.

-a'+'..-.';...

• * .

• r. *~.~.p...7..~ .

. a. ..-

e

- - - . *::".. .-".C.

.". !. .::;. :.g*. ,; . .- ..,.

. g:

~

~

• i * *. .~. .

..** * . . - *". *. ;". 7; *... - * . - -* .:; : ~.:;:: .. ..

-- - . . " ' ' ~ . * . --..:": '. *. . .

? _

r . . . : r.:r.:

.-. * * ::*. ? .- *: .

.,..g..g,. **.."?.*/

. ,. .j g , ,

1 .*":. 'P. 7. 2 .*. :"

.*a

- ::4:

, __;; : --. . ' . Y: * -.

-

• 4::.:*. .

.. .g. 1

~. T.

• 7 -

'. ~ , . , '

, r .~

e y.

.'~

r.

u .I- -

e*

~

~.  %

.~.P.

l; *

= - .

.* * ~ .*~n

.h E ~

?.

p . r.: 7.

. . ~*

.~. 2 .~.7..". .~.

..: .. . 2:

. T_:~. 7.: .. 7.72

. 7 *.** :

. .::. ~ ::

. s*..~.

y::: ::

7.

f. :.~. .~. .~. ::

.~.

• 7.: T.* I 7:T :

f.* .-*7*

. 7 :T.* .. ?.!

N :. .:. . c:..r...

.: . ._ 4 : :. :. .:. .. ._.

Figure 2.5.5 Broadening of RS rs82x in the Base Excitation Module

., _ _ - . _ , _ _ - - - , _. ,7_.

."..N

• 4 "". 7. ? a  ;; * : .. ,g y, p. .

.:. .. = . _...; =.

: .i .

-. ..?.*.."."*.*'.".*:.

. . . - g *r., i. .: .7 . . . .

2 ; *. *. t : ; :g

,. 7 * : *. := ; *

• . - " .4 .* ..' l. . . . _.1...-  ;* **,,,,. 4:; ......... . .. .

a-*::..* ...g -

? 4:- -

.;; =

p

• ; A g a - .

' ; L .g **

.,.l*.***

.**J

-f..*'g".'y=,

4 ."* . . . " , ;

I :2"* .-.a,,

• • "*7

?.  :;;

• 4 * *. . :"-

?. :* l

^

4: *4 ~4 L .

47.' 7 *

's ' *.;

N N

W N

s

~

L.

_ a-- . 9

-7.b.

.~2*' ,

. ~ .*

55 ..325

--=7=-

• 2

!22:*

1* .-.P.

g! *

• 2 ** .. ."*-  ; /*. . . .~.

: ' *. *. 2......

T.* -.

-. .'T :.: -. .. T :::"

• & "*3*O  :* * "J:2 **.~.

V . ,. -

~. *.*. .**. 77 : : .*. P. .". : * .' **

4 ""7 Fioure 2.5.6 Sroadening of RS rsB3x in the Base Excitation Module

E 3.0 RESPONSE ANALYSIS

'3.1 TIME HISTORY ANALYSIS The response analysis capability for both time history analysis and PSD analysis is verified in the following two sections.

3.1.1 Background The time history response analysis in E0 ASP performs the time integration in the modal coordinates, i.e., solving the differential equation for each mode

+ + =

E jk 2 83 aj jk * 'Yjk -T jk "k (3.1.1) where j = Jth mode, j=1 to m k = active excitation direction, k = 1 to 3 y = relative modal displacement a = input base acceleration T = participation factor 8 = damping of the structure model The absolute acceleration response at the ith 00F is calculated by combining all the modes and the input base acceleration m

ij E' (3*I'2)

O ' d

• ik q D

j4

where i

• fth 00F where response is requested ua absolute ar.celeration o = the dirertion of 00F 1, q = 1 to 3

& = mode shapes The ce3ponse spectrum at DOF i is then computed by solving Ei ki

• 28"1 iki + " "ikl "*b ik (3.1.3)

where r = response of the single DOF system 6 = damping of the single DOF system ej= frequency of the single DOF system o

as p4g , the response spectrum value for the ith DOF at frequency)rDefining theby is calculated peak of o' gg i

b S) j

= I ph) (3.1.4) 1 k

l 33

s 9

The SRSS procedure used herein is based on the assumption that the input excitations are uncorrelated.

Eq.(3.1.1) and Eq.(3.1.3) are integrated by exactly solving these differential equations assuming that the forcing accelerations vary linearly between the time points at which they are defined. Although the solution is exact and independent of the time step used, the user should not use too large a time step since the responses are calculated at the same time points only. Thus if the the time step defining the input is too coarse the response peak may be missed.

3.1.2 Case Description Ten runs were made with EDASP to verify the time history response analysis. The first three cases have single direction input excitations in the x, y, and z directions. Cases 4 through 7 check the responses due to multi-direction excitations, i.e., check the SRSS process. Cases 8 and 9 check the enveloping process when more than one 00F is specified. Case 10 checks the broadening routine in the time history analysis module.

The analysis was performed using the frame model described in Section 1.0 with 5% structural damping. The response spectrum at each specified output 00F is calculated at 54 frequencies varying from 0.1 to 100 Hz. Dampings for the response spectra are set to 5%. No broadening was performed for the first nine cases. Excitations generated in Section -

2.1 were used as the base input. Sumary of the ten runs is shown in '

Table 3.1.1. ,

Run 1 Excitation randx was applied at the base in the x direction.

Response Spectra at node 5 and node 8 in all three directions were calculated. The results consist of six RSS which were stored in files rs51x, es52x, rs53x, rs81x, rs82x, and rs83x. Similar runs were made i using STARDYNE and the response spectra were stored in files rf51x, j rf52x, rf53x, rf81x, rf82x, and rf83x.  ;

Run 2 Excitation randy was applied at the base in the y direction.

4 Response spectra at node 5 and node 8 in all three directions were .

j l' calculated. The results cor.sist of six RSs which were stored in files t rs51y, es52y, rs53y, rs8iy rs82y, and rs82y. Similar runs were made

• I using STARDYNE and the response spectra wm a stored in files rf51y.  :

l rf52y, rf53y, rf81y, rf82y, and rf83y. j i

i

! Run 3_ Excitation randz is applied at the base in the 2 direction. l 1

Response spectra at node 5 and node 8 in all three directions were  !

t d calculated. The results consist of six R$s which were stored in files l rs51r, rs52r, rs53z, rs81z, rs82z, and rs83z. Similar runs were made  ;

i using STAR 0YNE and the response spectra were stored in files rf51z, l l ef52r, rf53r, rf81z, rf82z, and rf83z.

> l t

i l

! 34 i

I

,. l I

V ,

7 Run Excitation ARS name (node /DOF) 1 x rs51x(5/1) rs52x(5/2) rs53x(5/3) rs81x(8/1) rs82x(8/2) rs83x(8/3) 2 y rs51y(5/1) rs52y(5/2) r353y(5/3) rs81y(8/1) rs82y(8/2) rs83y(8/3) 3 z rs51z(5/1) rs52z(5/2) rs53z(5/3) rs812(b/1) rs82z(8/2) rs83z(8/3) 4 xy rs51xy((5/1) rs52xy(5/2)rs83xy(8/3) rs81xy8/1) rs82xy(8/2) rs53xy(5/3) 5 y,z rs51yz(5/1) rs52yz(5/2) rs53yz(5/3) rs81yz(8/1) rs82yz(8/2) rs83yz(8/3) 6 x,z rs51xz(5/1) rs52xz(5/2) rs53xz(5/3)

., rs81xz(8/1) rs82xz(8/2) rs83xz(8/3) 7 x,y,z rs51xyz(5/1)rs52xyz(5/2)rs53xyz(5/3) rs01xyz(8/1) rs82xyz(8/21 rs83xyz(8/3) 8 x ersx1(5/1,8/i) t.rsx2(5/2,8/2) rsx3(5/3,8/3) 9 x,y,z ersxyz(5/1,5/2,5/3,8/1,8/2,8/3)

10 x brs51x(5/1) brsbh(5/2) brs53x(5/3) bes81x(8/1) br s82x',8/2) brs63x(8/3) i l Table 3.1.1 Sumary of Time History Runs l-i

+

1 l

l 35 i

. - - - - _ _ _ - - - _ _ - - - _ - ~

Run 4 Excitations randr and randy were applied at the base in the x and the y directions. Respor.3e spectra at node 5 and node 8 in all three directions were calculated. The results consist of six RSs which were stored in files rs51xy, rs52xy, rs53xy, rs81xy, rs82xy, and rs83xy. The resulting RSs should equal the SRSS of the RSs in Run 1 and Run 2.

Run 5 Excitations randy and randz were applied at the base in the y :

and the z directions. Response spectra at node 5 and node 8 in all th ee directions were calculated. The results consist of six RSs whien were stored in files rs51yz, rs52yz, rs53yz, rs81yz, rs82yz, and rs83yz. The resulting RSs should equal the SRSS of the RSs in Run 2 and Run 3.

Run 6 Excitations randx and randz were applied at the base in the x and the z directions. Response spectra at node 5 and node 8 in all three directions were calculated. The results consist of six RSs which were stored in files rs51xz, rs52xz, rs53xz. rs81xz, rs82xz, and rs83xz. The resulting RSs should equal the SRSS o. the RSs in Run 1 and Run 3.

Run 7 Excitations randx, randy, and randz were applied at the base  !

in the x, y, and z directions. Response spectra at node 5 and node a in '

all three directions were calculated. The results consist of six RSs i which were stored in files rs51xyz, rs52xyz, rs53xyz, rs81xyz, rs82xyz, and rs83xyz. The resulting RSs should equal the SRSS of the RSs in Runs 1, 2, and 3.

Run 8 Excitation randx was applied at the base in the x direction.

Three response spectra, each having two 00Fs to be enveloped, were calculated. The RSs were stored in files ersx1, ersx2, and ersx3 They i were checked to see whether the enveloping process in EDASP is correct Run 9 Excitation randx, randy and randz were applied at tne base it all th ,ree directions. The responses at node 5 ed nede 8 in all directions are enycloped and the resulting RS stored in file ersxyz. It was checked to see whether the enveloping process in EDASP is correct.

Run 10 Excitation randx was applied at the base in the x direction.

i The same six RS', as in Run 1 were calculated except that a bruadening f actor of 0.15 was specit ted f or each of the RS. The broadened RS were l 5tcred in f11es ers51x, brs52x, bes53x, bes81x, brs82x, and bes83x. They j were chu.ktd anainst the broadening process in the base excitation j ' nodule, which has been verified separately.

I 3.1.3 Program Results Run 1 STAR 0YNE results are included in Appendix 3.1.1. The response spectra from STAR 0YNE are converted to a format consistent with E0 ASP.

They are listed by EDASP in Appendix 3.1.2. The results of the EDASP i

response analysis are included in Appendix 3.1.3. To compare the results directly, the instruction set for the EDASP run was set up such that re I TRS filenames corresponds to the STAR 0YNE response spectra files. Fit -

l 3.1.1 to 3.1.6 shows the ARS/TRS plots. The E0 ASP results agree quite

! rell with those of STAROYNE.

35

_ _ _ _ . _ . _ _ _ _ _ _ . - - . - . . . - ~ -- -- -- - - '

.e.

U.*.

• s_:2 n. 8

v : _-.v.v.._~ .:.%. : .=.: u. us. . s -

z- _:

. . :=... !:.5. ., :- " . * .

2, L.r - .. : :u -

.  : - g.

. . : = :: ._ .a.

"..-*:.*. a *._-T*.:. ..:

: , _ .1-- _ - _ __ _.

r-------* :2 w- 4

v - 2: -

.I .Fr.

.~: .  ::r..

. '_i .i. .:-. .Udi:iU.i.a.

.  !.!U . A -2 '_"?. ii_--

2-------

i_ .I:r.i.i:34

s

::

u .=_ e . . _

~

f.

{ _:. - =-

-- +

= .: : - : _

w N

e .

-  :: :: - e, a

3 --- : i. -

.t 1 .

. l :=.

u. .

s *:. -

'g i .

_: .' 1.~.

_L :2:

. i , ?_= - '

r ..;

. a.

_?.: I i

_.---  ?

.+ -

• T r_; . _:.

..:: . . .4

: . ~ . . .

-..._: :::: _ . ::::: 1 !:. 1:. . . . . ..

.-__ _: :: ::  : :::= ::

:::_ _ =  ?

.-1:----

r: .: -::;:

=

e. :

= . . :s . . - : . .-

- _ -:.- :1 - _ . . _.. :- ._.

Figure 3.1.1 Co.nparison with STARDYNE Results at Node 5 D0F 1 l

g,.,e-w -,, --w,-- --,.w m.- .m ,--y- , ,---. .-.-:-- - - - - , , - , - - , ,,-,-m- , _ . - - . . - - - ~.--

. . - - . - ... .. .~ . . - . , - - . - . _ . -.. _ .- ...-.

4 :. - -

- :::. ;:z

:J. i. a m. .. :. J s

-n.: .a-_ :: 2 4 3.

2  ::

2. .a :. :::

=

. =._ g: : .-

2:u r . auu s . = . a  :.--g .: ..r .1..:

s ]-.  ::

r.. ::.: : , ::

:: . I -- :1. . . . _ _ . _ ~ _

3. 2..::

.........a . s..

-2:. . 1 .3 :.::1 2 : . : ::- ..

2.. 2.....-

. . . . .:u u t- J

_2 :. :: -- :  : : : ._:

.- tiii

. ... = _ . . . .

s t,

= -..i n . = -

w:1 .tr:.. g -

4 =

=. fi

... y

(,nB - *

+t co .. : .. t

.- 1.- H i . .

. . . =-

".* : "= ~ - $

e. . 5 as

s. .

e. .

r.

+. ..~

. . . . . . . _ -- -6

p. : t2= : -

- =. : a....

a :. .. - .- -

j 1 !!  : ":23 . - -.

r:: .--::::

- . : : =-

• [2 Si r.': .- . M:

• 2. 2 :2

, .:. . . am.: : =

lt m:% .".*:::: n : .* * :-

• n .: -:- =, _: :_: ._  :.:% *- 23 :. 8

1. =.

Figure 3.1.2 Comparison with STARDYNE Results at Node 5 00F 2

v:s Cli;I  :::::

c:r i

.. .. l .....

1 Cli;e I::i. . ca.e c:r.i ..

*:e :lC C!i;8 C!i;r

.c:::: rs:1 ..I.. ..'.

C!i;e i::::: !!:. c 1.:1 lhei.sl ej,gog .,ung .

..e h

I Il L

I, I' ', .

g t

t I r".I's.

i.,,i ..

n

. .1 C!i;I La-

,;.. . . - " '... s t .. : g na"",, II3': * -

, . . , m

,,,,.,, g.,;g ,

"!s::' ', . .... . i... . . ..  :::1::: .o  !

,,!i ..::: o L E

l e u i

"' $ . '..  : se !

's ' . . '. :

"n en .

....  :.t # w (

-nu: +;"-- ["p;  ;

3 i:1.s o

..; cc -

lhei:5 y x

..,..a,. C!i;s p.i.r'o, o

CC C!i; <

8l . ~ H m

.c M

C!i;t C!i;e

• f.t":e 1.t":e 3 C!i;s Cli;l C o,

C!i;I Cf;;i +

s.

. . . 9 i I::ri I::re I c:: ::::

...... U '

i :::i.,..

.II

. . .ll.

re:i cc m. ,

i a::;;: i::'::4 -

I;'!!;i .

C!i;s m -

I

.. . . ..i.....--.. ..t... . - . s. '*'"' I o C!';8 C!!;) C!i;I 'I C!i:t .....C!i;I

" S-C!i;I a

l.r*:s Cli;I 4.i":' C!i;' '-- .. I;' S

>:: ::a:; c'.::1 . ......; r!!i;i C!;;s W c . :. e; . :, i 1.1" e t.C':s f.e*g a .e..g t l

t:: , ::i

..-. t I::P "

t! *;n C ':e **.**

i L:: ::;I::t::;

.c;t:: 6. .. .c:t:: s::s i:. e a:8.e e ~~e i t

39 '

6 t

f i

s

~. .. .. .

1.T 8 .- r-.::U :i .. i'g -s u s . r. n

• 14 ia.!i

u. :.s:u. : :: s .[:.:::

+ ...:....

.g :.: .- : ::t

.. :-.. t;. . as. . .

i -.~ i': V i.!

15-.

.::::::; = :=

.3  : .,- : s s ::. :-

.: . :-3::::

.:- s t-  :- . . _ . . .

4, usu:=  ; :: .: = =

1  : : :::: -.... -:

:: -3 =r =

=:,--1 ::,:::--J 4_7 -. . n_.

s

..i I

. .4  :

2 -

d . .... -

'l i

a 1

4 1

=

1 u -

\ o

.) .. 1. . : - .

.. . . . . ..1-  ::

• eu

. :* 2. *:. .? -

-in. - . n. - .

=.

Z

.-*.*v is is: . ..: .

-: r ::

:::: 2  ::::: .- -

I :Lu==

= : =:::  ::- 2: , :::: ::

-=

usAuu .:. v

. =J = ,u==us= =

s:: .

.=::

. - ir..r...:

.g:::r_:::..:

= r

...3 .: r.

Figure 3.1.4 Comparison with STARDYNE Results at Node 8 DOF 1

-~_- ,, - -

.,,r--w - > - - - - - - - . , , - w e- , - - - - - x- -- -n - - , - - -- , n.- , - , -

. : =:.:- :

u *

..y::j..

. .. r g a:.:

A.

.: .=. ::T .,.

, f:::: : = re: ::

= = :::: M. ; , . ;*. :

s : 2

. .: v. : :-

.. f* : : . .  ;*....

.. ~::: =:J = ==

x :::.:. :. : ::

- - . * - ~ ~: -

!"5

. =.  :: . . - 93 r ..r:: :2-ti::

= q g:.: = ..

2 22.: : .- a v : .~. =: - 4 .* = :. .~

.*. *. * . .*. 2.~..**.*.

• .~..~,

$ [" i3

. *. :. *: .:* I N

~". .

J. : %* V b :::=

=* g ~: ~. ; ,

.g. . s . .

. .I' 3

t t

• ~ .

• 4

.*  :""Id ;;.

-e.

2 :--: . ! ?: : 6 .

L:..

f,'

[ '-.

f!

s -

. 5 e

.~ . *: *.:

e. .., _. .

=

$ 4

~  : -

r .~ * :

• g . . . - i, '  : **..

% C=:

,  :, : .:u : * . T* *

~

.  ; k.

f - A l

h. ' '

• j n

=f t  : .;

E *

• ==:

.I : ._i E .h - .

.z. -

5 k

? .

=

::::- ~- - -

p:: .s ea:

. :::::: .- ,, .--::: ::=

.:= .

22, -

. :.  ; === =  :=

._===

=

== :

=1=

'  ::: s i

_i _:T9_0l.!.P. .!i.E_.*_i . _ .iii.. 7. !

Figure 3.1.5 Comparison with STARDYNE Results at Node 8 D0F 2

is

.. i.'i.' * .m .- U *. - r

-fi. : ..a. t. ss . : rr

• l.i. , fi.

_ :.. .8.i.

. , _2. - .

.. -. .. . . . . ..a . ..

f 2 .: ,.s

_.....r..f

. . ... _. !i:'..i" U . 34

_ s 2_: ...: . . -22.-..: : .z

2.r.i ;. 1: .

2

._ ::.: g......

. g .._..

~. .___ . :. . .

. a

+

.- :::r :

.. ::rt 2

4 b * ...

eu .- . ;: ..

..g.... .

1 -

.. . i ?

.2

3 6

.4.

?-

4.t

.?

2:.. *2 .I 2.4.  :

a. t

. c

' =.

.f -

• ."*.W.*Z 1

• :r.1 ?

I

s;::s

.=.. . _ i  :::::: - -

s ..-._ .._

..- z

_ _.. i. t ..

4 r_. .

Figure 3.1.6 Comparison with STARDYNE Results at Node 8 D0F 3 1

1

- . _. -..y, . , - - - , - , - - . -y- .- , - - - - - - - - - - - .r----y -.-y _- ----n- - -- - y ., . -- , - , - - - ,,

Run 2 STARDYNE results are included in Appendix 3.1.4. The response spectra from STARDYNE are included in Appendix 3.1.5. The results of the EDASP response analy:,Is are included in Appendix 3.1.6. Figures 3.1.7 to 3.1.12 show the comparison plots. Again, the results agree quite well.

Run 3 3TARDYNE results are included in Appendix 3.1.7. The response spectra from STARDYNE are included in Appendix 3.1.8. The results of the EDASP response analysis are included in Appendix 3.1.9. Figure 3.1.13 to 3.1.18 show the comparison plots. The results agree quite well between STARDYNE and EDASP.

Run 4 The results of the EDASP run are included in Appendix 3.1.10.

A simple BASIC program was written to cFeck the SRSS process. The results are included ir, Appendix 3.1.11.

Run 5 The results of the EDASP run are included in Appendix 3.1.12.

Same EA5TC program as in Run 4 was used to check the SRSS process. The results are Mcluded in Appendix 3.1.13.

Run 6 The results of the EDASP run are included in Appendix 3.1.14 Same EA5TC program was used to ckeck the SRSS process. The results are included in Appendix 3.1.15.

Run 7 The results of the EDASP run are included in Appendix 3.1.16.

Same EK5TC program was used to check the SRSS process. The results are included in Appendix 3.1.17.

Run 8 The results of the EDASP run are includeo in Appendiy 3.1.18.

The enveloping results are shown in Tables 3.1.2 to 3.1.4 at selected frequencies.

Run 9 The results of the EDASP run are included in Appendix 3.1.19.

The er.veloping results are shown in Tables 3.1.5 at selected frequencies.

Run 10 The results of the EDASP run are included ir Appendix 3.1.20. The broadening process are compared with that in the base excitation module in figures 3.1.19 to 3.1.24.

43

a T*. ."*.  ?* ; .

: r. ... . b -

.c*

L. .. . . -

. . ...-.. .s. . 1. T. .

b 3.: ..2..:a.

. 3 I 4.

• =..*.,,g g . ..

......g I." 1".;

• 3.~'.** *;

.  ; *J*.t J ...

....... - . g

• ,;*,*...  ?. . - -.

" 2 * *.; *. .. 1 I.'*

.J*..i .....

. . . . T.... I

3. .J 32: ..

3.J = . = . . ..... . 2..L.

• -.T .

. . . .~ =. ..

I .*

• ..t..t: -

J; * * ~ 2 : *" .g 2-*.e.

..J 3 Lt*. *J 2.**".".J..M.""

8 Z .*.

4 3 ..'*P... 7. r. .*.

2....

6

p* *g  : ?

.=* *:= .I ...

.Z..'.

u i

• -. .2 .

4 s.

. 1 3 L~. ...~.

L 3

I .

g .

r Cm .*. .

f i

• ==.

, r, .. , .

2 t

.g. ..

g 2 -

11

. =,.

.. s

.i..:.- . :

t.

• M-- umb Z

7.., . .. . .... .

. 5 ?." *.:

2 ....

J. ... .

i

• g-*

. . . .. *. g 2...

3*J J**

. .. 1. E. ..

. = =.

==

• Y.: ... :"3  :

..""....L*;

-* . ~ . ~ . .

g.

• U:

s,.,.-= J

_ ~ : -a _ : 2. . _-:. 3 2. .~ ". . g. 2 7 Figure 3.1.7 Comparison with STARDYNE Results at ' ra.: .: * - .1 t

- - , - , - , . . . ,--m--.,-n.- ---r-na-,-- ,a y,,- - -

.-,-,,-w - =' -~ ~ ' = -

-*.m .* c .t w . ~. r sA =-<e 1 -. --

__ _ _ _ _ . _ _ __ .. . ~. m . . . _ _ _ . - _. .-

- =....

"s. * * *

....gg .. . . . . .

.t

. .. .- . ..*:b. . :. *... 2 :.:" .: g..!

t

• +=..-. . 2.. 73 .. . {;:;

...r_.._...

g; .

h J I

.2.

t y

. m m

......e, S. :.

.. . .e..

. .e.o.. .

....3 o ~. ,, ...3 . 1...I. .

s,-. . .....:

en. =2....,

4~.

b ..

t . . . ..$

• * +g

...o. .

.,.. .3& .I .

. * +

. 4 -J $ y .." j ,

. i.

e *

. a

. ..m- ..- *, ,

? =*

g.. .

4.*e li e

I Wm M

l M* M g

!Z  :..g'. t

• 4 . . .

. : g . . .

i .3.::.::..:

i ....

T..*...

_i ._..::_ . ::  ::::_ ::

. .. 13.i. 1.

.. . . .. . ... ._. . _. ..._.z Figure 3.1.8 Comparison with STARDYNE Pesults at Node 5 00F 2 l

. - - --. - . - - .- - . . _ . _ . . . - - , . , - - - , . , _ - , . - -. - ,,- - . . - . - - . . _ , _ . . , ~,. --

i 4, : : .*. = = . .

g *=.1. * * * * * * = ,*

y**=..  :** 2

..* a ?. T " .J.*.t.,..=.=22 " " * * **

. . ~ . . .. .. . . . . . . _]**t..**

,.;*-*..? . .

.l.. * = . ..*= . . . *

• • "**.."*.t.-

. 1* I.

~ ~ * * * - =*..R.

=

J *1 . .... . ..... . ..= .

A ~. .' 1 *4...

. ....*i.*.....

2".._..

. _., . 1 :: :::. : :

1."..?.. *.* ~: .""

- .1 -.. .

A 1

f :::

.. J .... . . .

3. .

t..

L *

• • " ~.

. . e

f. .

i N. . , g ?_" .'

4 m

s

. .s.

. +

4 J.

O . .

.. *. L:..;

.~

e. .~

. "y ! *.* *.g * =

t.

3.;:' .*

r. "

......L.

.f .t.". 7-. *_.a.

} '" *y l ., " .*

-d

".j  : * *2 '

. 7:::

~

9 1

2. ::::::  :. := ::

1._... _. ...

w. . . . .: ..

.n . :.- .- : 2.-

..r:

v. .,.,: .. :. .i Figure 3.1.9 Comparison with STARDYNE Res>lts at Node 5 DOF 3

_ m - _ _ _ _ .- - m =m.-._ _ m - __ - _ ._. - __. - _ .a

22.

• 3 .. ,..

...'t.*

• 1.*. . . . . *

• f.* .*J 2 .t. 5.. 1.. " 12M. . t *t. .*f.

...3 _

...2.~.!.=. . . . . ....

• 3

. g.

-. . 31.

.. .SJ 4.2..

3 -.: *4* 2

.,...: * : I * .S. . .3 *.. . . . . 24L

. .. ...3...S

. ....!.. *J.

. . . , .3*.

..*:*.*.3...

**1-l

.3.-. ... .

3 .. .. ...........

. .. -...f 2 .

.I

. 2 ..

=,

. . 1.

i O 3- .*

N

• 4

• . ' ?.*: . .

• 2.

t l

l . .

4 i

, .L.

.g.

.T . .

.1 l

.J. + ..

l

..: d b. ' '

i J* 4

. 3 J

? .

e

• .._ I

~~~~ .

1,

~

l

=?.l.

1

• 2

.4 J. ..

.

• 23. .f .,.22.

2

.. . . tg

• 2.,. 2*

4

.... .? ....:*

• 2 2...... 3.***.2 T* f t: *

*,*..*. .*;! ! *. ?". .".12 .

.. * : * : 7 ,2 2.:-.::  :. f.:: : .. .: .:.: .

Figure 3.1.10 Comparison with STARDYNE Results at Node 8 00F 1

m

4

g. _..

. . . .8.'.*..

%.*. f.*' ' " * : *

..."M

-..=._.1.:.*I.J.".**. ....".:*

.2 f*..

g........

.... .. ..J. _. . .....  :

4 ::*:. . . t. *.  :. *3 .2.

..I

.. .. : i .... _..5.....

A J **J

* *."._' **. .g 1g: ... , 24.. .. .

. _. "*3***4*_* .. . . . . _ ..

. 4 *::: : J '*

2 ._. .. . .......

.J.

._..'_...TZ..

, a.

1.: . ::.. _

s. -

1_:. _

.:. _ : ..s _

1 .

r .

- u a l

r

. = ._-:._=.

.. . = _ .

. :===  : :: :: .

_ :.___. t

_..=_...=_.

_=.=..=._..=._. .

_ - . := _E.,r.... :..: . m:  :.: . s.: .. .s .: _:.. :

Figure 3.1.11 Comparison with STARDYNE Results at Node 8 00F 2 I

It. :**,12...

• ... . . .* . ".*. f* *.? ..

1* .%* / : **.*1*.*.*P.*g

• f =: ; . :. 7.:.  : : ".

.. 7: ... ;* : 7.: ...g .

! % 3.*

& :n*1

• i 7:

I

'*t.".* . T .:

2 2.

-! .. 7:

3

• . . . .. .. .. 1

........&..e 3 :i

. 3. 4:. 3 ^

! .. 3. ,...

1 ....

J .. ..-.....

. .J

.2......

-. . a. .

.-.L*".*

1.

.. 1... .

t*

O .

.~

@ . { *..: k 3, * .

..j2 .

. . . 3 *

. ....2. .

3 1 Z

.-** I*

3.* .

9

.t

• e.Te.

C . *?

a

....!. T. .

. 2. .

..2.**..

J

. . ..... .r I.

4

.?

e .. e

.? ... e N

.e

.Y

.. ..., . 4

. 9 7.: Z 2.: r.*

3.. .....

.. ... .. 4 3- .

. ..... .. .._.... ...J

s. _.... . .

. . . _ .s..... ... .

. 2:=..: : . , =..::. . ..... n....,..,

s. . . , .

Figure 3.1.12 Comparison with STARDYNE Results at Node 8 00F 3

~._. . . . ___

s UU 8 ...- hj ~. .

r. fj :u !s:.r.e_=. .t*.8!J p: Ii:- f _

s ::z :.!:i iiii:rV U . li

._: . ....- .: _r -

.:s. . . : :: ::: _.

- - . .: ::n ::: . - : : :_:

1j [a r;- s El dil.

~ -

~ ?: 5ld.5.5*l.I!.*4. : UmEd5

.I _I* E. .

.:.~..~.  : u : u:......

u .-

• h Ui LJUa U U b.

j 3

. . e E

t i

?4 2  :*

::n.

r_; . .: 3_ .

!;E i: : -

g r.  :

?

!7 j 's.'.

-  ; 2

• m

.: I.

o _ ., .. -

i .s .

., 6  ;; n :: r.

..a

.-

4 .

ge

.. e .

' r". .

F.: . W--

/.. _i

. i. .:

2. ~

.~

-.a t t

1. *5 a_.

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Figure 3.1.14 Comparison with STARDYNE Results at Node 5 00F 2

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g e*:s t!'!CI e::c:s a i::rw -

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m-.-% . m_..--,-mee- r- -, yy-

Enveloping of Response Spectra l

E0 ASP FILE: ERSX1 - Envelope of 5/1, 8/1  :

EDASP FILES: RSS1X, RS81X FREQ. RSS1X RS81X Envelope ERSX1 0.1 3.333E-1 3.336E-1 3.336E-1 3.336E-1 0.8 1.891 1.993 1.993 1.993 2.3 '9.434 1.498E+1 1.498E+1 1.498E+1 3.8 7.616 7.751 7.751 7.751 4.8 5.922 6.642 6.642 6.642 7.4 4.534 5.018 5.018 5.018 10.0 3.630 4.660 4.660 4.660 t 21.0 3.2 04 4.337 4.387 4.387 i 45.0 3.295 4.474 4.474 4.474 l

100.0 3.286 4.407 4.407 4.407 L

t I

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Table 3.1.2 Envelope of Nodes 5 and 8 DOF 1 Responses t

62

Enveloping of Response Spectra E0 ASP FILE: ERSX2 - Envelope of 5/2, 8/2 l EDASP FILES: RSS2X, RS82X FREQ. RSS2X RS82X Envelope ERSX2 l

0.1 1.109E-3 1.254E-3 1.254E-3 1.254E-3 l

0.8 8.355E-2 9.397E-2 9.397E-2 9.397E-2  !

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8.991E-1 1.053 1.053 1.053 10.0 8.127E-1 9.514E-1 9.514E-1 9.514E-1 21.0 7.575E-1 8.800E-1 8.800E-1 8.800E-1  :

45.0 7.778E-1 9.044E-1 9.044E-1 9.044E-1  !

100.0 7.587E-1 8.818E-1 8.818E-1 8.818E-1 l I

I Table 3.1.3 Envelope of Nodes 5 and 8 DOF 2 kesponses i

)

f I k i

I

{

63 ,

t

r-r Enveloping of Response Spectra EDASP FILE: ERSX3 - Envelope of 5/3, 8/3 EDASP FILES: RSS3X, RS83X  :

i r f FREQ. RSS3X RS83X Envelope ERSX3 0.1 3.924E-4 7.681E-4 7.681E-4 7.681E-4 0.8 3.629E-2 5.539E-2 5.539E-2 5.539E-2 2.8 8.861E-1 1.745 1.745 1.745 3.8 8.982E-1 1.707 1.707 1.707 4.8 8.746E-1 2.119 2.119 2.119 7.4 6.634E-1 9.364E-1 9.364E-1 9.364E-1 10.0 4.621E-1 6.742E-1 6.742E-1 6.742E-1 f 21.0 3.351E-1 5.974E-1 5.974E-1 5.974E-1 45.0 3.353E-1 6.066E-1 6.066E-1 6.066E-1 f 100.0 3.382E-1 5.951E-1 5.951E-1 5.951E-1 Table 3.1.4 Envelope of Node 5 and 8 DOF 3 Responses f

a L

i r

Enveloping Response Spectra EDASP FILE: ERSXYZ - Envelope of 5/1. 5/2, 5/3, 8/1. 8/2. 8/3 EDASP FILES: R551XYZ, R552XYZ, RSS3XYZ. RS81XYZ, RS82XYZ, RS83XYZ p-FREQ.

i R551XiZ R552XYZ R553XYZ R581XYZ RS82XYZ RS83XYZ Envelope ERSXYZ

.1 3.333E-1 2.926E.1 1.6t9E-1 3.3336E-1 2.928E-1 1.649E-1 3.336E-1

.8 3.336E-1 1.892 1.842 1.433 1.994 1.866 1.474 1.994 1.994 2.8 9.748 9.770 4.932 1.503E+1 1.109E+1 7.513 1.503E+1 3.8 1.503E+1 g 8.903 5.677 1.232E+1 8.011 6.813 1.963E+1 1.963E+1 1.963E+1 4.8 6.773 4.195 1.332E+1 6.936 4.713 2.830E+1 2.830E+1 7.4 1.830E+1 4.709 5.097 1.187E+1 5.141 4.006 9.793 1.137E+1 1.187E+1 10.0 3.800 3.012 7.432 4.728 3.459 7.775 7.775 21.0 7.775 3.339 2.855 4.679 4.435 3.162 5.977 5.977 5.977 45.0 3.434 2.888 4.650 4.524 3.206 6.079 6.079 6.079 100.0 3.420 2.845 4.718 4.456 3.168 6.059 6.059 6.057 l

Table 3.1.5 Envelope of Modes 5 and 8 Responses in All Directions -

f 3.2 PSD ANALYSIS r

3.2.1 Background The modal equation for the P50 analysis is the same as Eq.(3.1.1).

Take the Fourier Transform of both sides of Eq.(3.1.1) and sum the excitation directions, we have L

i I

r Yjk(u) a H3 (o) I T Ak I") 424 I where Y p(w) = Fourier transform of y (t) f Hj I") *

((wj/w) 1+218(ej/w))*I 3

r = participation factors Ak (e) =

fourier transform of a (t)

Define the PSD of the input motion a (t) as i, l

i e

SAk(o)

=

2E{AI")AI")}

k k (3.2.2)  !

= ensemble average whereE((.)

Ak w) = complex conjugate of A (e)

The PSD of the relative acceleration response can be found by l

S yj (u) (? jjj]g(g)p((a) t j p (rj ik S AkI "))

+ I(?jjtjj Re{R(u)R{(u))

j jg SAk(w))

I r r)t (3.2.3)

' )

.. t j where Hj (c) =

-w hj (w) l t = made shapes l

, j,1 = mode indices  !

i Finally, the PSD for the absolute acceleration response are found j assuming the: input motions are independent. The P50 of the absolute j acceleration at DOF i arc obtained by the following equation: l l

! l 66 l

i c

7 7 - --

lw a

Sgj(e) =

$ 4

$34)Re(H(e)Rf(w)}

j r g jk r SAk(")

+

(2 r)q 4p3 Re(A(e))

3 + 1) 54q(e) (3.2.4) 3.2.2 Case Description Ten runs similar to that in Section 3.1.2 were made to verify the PSD response analysis. They are summarized in Table 3.2.1. The first three cases were verified in two stages. Firstly, the elevated PSDs were compared with independent STAROYNE runs. Secondly, the conversion from PS0s to RSs were checked (for Case 1 only) against the RS/PSD conversion process in the base excitation module. Other runs were checked similar to that in Section 3.1.2. All response spectra were calculated at 21 frequencies varying from 0.2 to 34 Hz.

Run 1 The PSD excitation nrcl60x was applied to the structure in the x direction. Responses at node 5 and node 8 in all three directions were requested. The elevated PSDs were stored in files sp51x, sp52x, sp53x, sp81x, sp82x, and sp83x (renamed from ps*.-l files). The RSs from EDASP run were stored in file.s ps51x, ps52x, ps53x, ps81x, ps82x, and ps83x, The PSD files were converted te RSs using base excitation module and stored in files ss51x, ss52x, ss53x, ss81x, ss82x, and ss83x.

STAR 0YNE PSD output were convertad to EDASP compatible format and stored in files pf51x, pf52x, pf53x, pf81x, pf82x. and pf83x.

Run 2 The PSD excitation nrcl60x was applied to the structure in the y direction. Responses at node 5 and node 8 in all thre directions were requested. The elevated PSDs were stored in files sp51y, sp52y, sp53y, sp81y, sp82y, and sp83y (renamed from ps*.-l files). The RSs calculated from EDASP were stored in files ps51y, ps52y, pe,53y, ps81y, ps82y, and ps83y. STARDYNE output P50s were converted to EDASP compatible format and stored in files pf51y, pf52y, pf53y, pf81y, pf82y, and pf83y.

Run 3 The PSD excitation nrcl60z was applied to the structure in the z direc?.fon. Responses at node 5 and node 8 in all three directions cere requer,ted. The elevated PSDs were stored in files sp512, sp52z, 67

f i

Run Excitatinn ARS name (node /00F) 1 x ps51x(5/1) ps52x(5/2) ps53x(5/3) ps81x(8/1) ps82x(8/2) pm3x(8/3) 2 y ps51y(5/1) ps52y(5/2) ps53y(5/3) ps81y(8/1) ps82y(8/2) ps83y(8/3) 3 z ps51z(5/1) ps52z(5/2) ps53z(5/3) ps81z(8/1) ps82z(8/2) ps83z(8/3) 4 x,y ps51xy(5/1) ps52xy(5/2) ps53xy(5/3) ps81xy(8/1) ps82xy(8/2) ps83xy(8/3) 5 y,z ps51yz(5/1) ps52yz(5/2) ps53yz(5/3) ps81yz(8/1) ps82yz(8/2) ps83yz(8/3) 5 x,z ps51xz(5/1) ps52xz(5/2) ps53xz(5/3) ps81xz(8/1) ps82xz(8/2) ps83xz(8/3) 7 x,y,z ps51xyz(5/1) ps52xyz(5/2) ps53xyz(5/3) ps81xyz(8/1) ps82xyz(8/2) ps83xyz(8/3) 8 x epsxl(5/1,8/1) epsx2(5/2,8/2) epsx3(5/3,8/3) 9 x,y,z epsxyz(5/1,5/2,5/3,8/1,8/2,8/3) 10 x bps 51x(5/1) bps 52x(5/2) bps 53x(5/3) bps 81x(8/1) bps 82x(8/2) bps 83x(8/3)

Table 3.2.1 Sumary of PSD Analysis Runs 68 J

~

l sp53z, sp81z, sp82z, and sp83z (renamed from pse.-l files). The RSs calculated from EDASP were stored in files ps51z, ps52z, ps53z, ps81z, ps82z, and ps832. The RSs calculated using STAR 0YNE were converted to EDASP compatible format and were stored in files pfSiz, pf52z, pf53z, pf81z, pf82;:, and pf83z.

Run 4 The PSD excitation nrcl60x was applied to the structure in the x and the y directions. Responses at node 5 and node 8 in all three directions were requested. The elevated PSDs were stored in files sp51xy, sp52xy, sp53xy, sp81xy, sp82xy, and sp83xy (renamed from ps*.-l files).

They are checked against the sum 0) the P50s in Run 1 and Run 2 by a simple BASIC program.

Run 5 The PSD excitations nrcl60x and nrcl60z were applied to the structure in the y and the z directions. Responses at node 5 and node 8 in all three directions were requested. The elevated PSDs were stored in files sp51yz, sp52yz, sp53yz, cp81yz, sp82yz, and ap83yz (renamed from ps*.-l files). They are checked against the sem of the PSDs in Run 2 and Run 3 by the same BASIC program.

Run 6 The PSD excitations nrc160x ano nrcl60z were applied to the structure in the x and the z directions. Responses at node 5 and node 8 in all three directions were requested. The elevated PSDs were renamed from ps*.-l files and stored in files sp51xz, sp52xz, sp53xz, sp81xz, sp82xz, and sp83xz. They are checked by the same BASIC program against the sum of the PSDs in Run 1 and Run 3.

Run 7 The PSD excitations nrcl60x were applied to the structure in the x and the y directioins and nrcl60z was applied in the 1 direction.

Same responses were requested. The elevated PSDs were renamed from ps*.-l files and stored in files sp51xyz, sp52xyz, sp53xyz, sp81xyz, sp82xyz, and sp83xyz. They are checked by the same BASIC program to equal the sum of the PSDs in Run 1, Run 2, and Run 3.

Run 8 Excitation nrc160x was applied to the structure in the x direction. Three response spectra, each having two 00Fs to bt: enveloped, were calculated. The RSs were stored in files epsx1, epsx2, and epsx3.

They were checked to verify the SRSS process in EDASP.

Run 9 Excitations nrcl60x, nrcl60x, and nrcl60z were applied to the structure in the x, y, and z directions respectively. The responses in all three directions at node 5 and node 8 were enveloped. The resulting RS was stored in file epsxyz. It was checked to verify the SRSS process in EDASP.

Hun 10 Excitation nrcl60x was applied in the x direction. The same six R5s a Iin Run 1 were calculated except that a broadening factor of 0.15 was specified for each RS. The broadened RSs were stored in files bps 51x, bps 52x, bps 53x, bps 81x, bps 82x, and bps 83x. They were checked by plotting against the RSs broadened using the base excitation module.

69

3.2.3 Program Results Run 1 STARDYNE results are included in Appendix 3.2.1. The RSs from STARDWE~Tre listed by EDASP in Appendix 3.2.2. The results of EDASP analysis 'are included in Appendix 3.2.3. The elevated PSDs computed from STARDYtiE and from EDASP are compared in Figures 3.2.1 to 3.2.6. The plots were made by editing the PSD files to look like RS files and using ARS/TRS comparison in the response analysis module. Only slight difference was noticed near the spikes. This is due to different fcequency poin'.s used to define the PSh. For the 5% structural damping used, STARDYNE gave PSDs defined by 192 points while EDASP gave PSDs defined by 107 frequency points. The EDASP PSDs were converted to RSs using the base excitation module. The results are included in Appendix 3.1.4. Comparison of RSs generated from separated RS/PSD conversion modules shows excellent agreement (Figures 3.2.7 to 3.2.12).

Run 2 STARDYNE results are included in Appendix 3.2.5. The PSDs from 3TARUYNE are listed by EDASP in Appendix 3.2.6. The results of the EDASP analysis are included in Appendix 3.2.7. Figures 3.2.13 to 3.2.18 show the plots of PSDs computed from STARDYNE and from EDASP.

Run 3 STARDYNE results are included in Appendix 3.2.8. The PSDs f rom ITARUYNE are listed in Appendix 3.2.9. The results of the EDASP analysis are included in Appendix 3.2.10. Figures 3.2.19 to 3.2.24 show the comparison with STARDYNE.

Run 4 The results of the EDASP run are included in Appendix 3.2.11.

The results of x and y excitation combination checking are included in Appendix 3.2.12.

Run 5 The results of the EDASP run are included in Appendix 3.2.13.

The results of y and z excitation combination checking are included in Appendix 3.2.14 Run 6 The results of the EDASP run are included in Appendix 3.2.16.

The results of x and z excitation combination checking are included in Appendix 3.2.16.

3 Run 7 The results of the EDASP run are included in Appendix 3.2.17.

The results of checking the excitation combination in the x, y, and z directions are included in Appendix 3.2.18.

Run 8 The results of the EDASP run are included in Appendix 3.2.19.

, The enveloping results are shown in Tables 3.2.2 to 3.2.4 at selected frequencies.

Run 9 The results of the EDASP run are included in Appendix 3.2.20.

The enveloping results are shown in Table 3.2.5 at selected frequencies.

Run 10 The results of the EDASP run are included in Appendix 3.2.21. The broadening process are compared with that in the base excitation module in Figures 3.2.25 to 3.2.30.

70

• • . . ~ . * *

• -.J.....2.**.

. ... .1..**

. ..2

--,'.A.*

; 1:. , ?: J ..- .

"L** .

.=a

.m, N -

e*

--.. ; .7 :.

.~. ; -~

..-.. ~

2 :::** -

-:::: 4

..Z... --- -

T...... . . . . .  ;*....%.

-=

Figure 3.2.1 Comparing PSDs at Node 5 00F 1

{s t****:,

. C!i.I  ::s:'.*

e::i'..

. . . . .. . t f

i

. . t '.'lfil1 t::t... t,.....i ::t .. .

s ... , t . .,ll .. .. . l.,,s.

. . . .3

.'. ' ' f ..i.I t;:t : 6".l.' l lll!il1

"::: f'I' ' a;'.. 1.

ai *t b.t..I l e. . . "l p..g. .s.. l t.... .,3s. . .

......l

..l..;l

..l...

l ...I. i l.!..'

N f:l.i 6

"" . i. . . . .

. . , . . . . . . ...... . J [* ,

b e

If, W

,"n t

.i g

I.isi:s y o

l.eg"l

..si 2ll C!i.*l +J C!i;f so to

.....] g M

Q.

C..f U.l 1.'.!il1 t:n i ,. ,. f I... . .I

. g f .. .,i..8 ..#. 1 .,. i. .. ,l .c=

L r.".We r..i;. m o.

g

.. o v

.::t

i. . . t .::l'.s

.. r..

u.

., t::4...

11 . . ll. N.

rl:; . i: m e : ::, i:: ::: C!Gl 48 g...'.se "s L 3

,..t,,g g g g . .enal m

. 1.,,

. . . ..I ,.,.I ..

g u..s. i., p. ..i.

g . .g .w C":1:..

1..,... g i.,3:: i::1 i i:'..:I C!j.'l "i' i ' :'.' ' ' ."!;;t r.t:t s:";;r t*:'.3

'. .I a ..:t C!ib Cf;;i c':.i I:"i;t C!i;l f.t":i i c' e

ll% llc

.::: , . ri e:, i. e ..';' *

s. 4 ll 1; t.:,

.:;t:: 1 .. e::l:: . .:e  :.: i 1 e . "I 72

7

• "-i: ~.- **

U.~..~.* . i**.*.**..*:* if- . , . -

~ . ,_.. .

1, , - . . . ...

; .; a . 55

. . ._ .-.:..::_: :...g... .

• . I.1%*. * :". .": *

. .. . l_if f.

a..

.=

g

-- ......  :.'t g : ::.. ..

. .. ..==.=.=

. _. _..g -

• =.

-. - _.24.7

_

~ -

b)

- . }:

. 1, .;

r .

=*

9

.:. : = . ._.

= -.

-. .- . -= =. . .

. =..

z :.-... . . .- _....... _.. . . . .

- : : = . . _: .: =. :..

Figure 3.2.3 Comparing PSDs at Node 5 00F 3 4

, -n- n-v.- - - ,, w -

--,.w,, --r -- e - ---, -

gv

w A

• • 1.- ... *: * . *

. .. *".....1.....

..g 3..* :2*2*

.*. ;. * !.: I

~. ^ . .. . . . ...

..~..

= .< ..

.. . .. . .......I.

• ... . + . .

.".4 .". O.* ;

. '2"*

/,; A.:"" ~

4

(

l O. L . :.: .

I N

2h

• **a.

=

4

.  :=

2. :

-.2.

J * .

1 j .

w I

i .

+

n!

~

a T:. .

,I 1 n

• 7..

. Z**... "

""" 9 y L.Z..

22'I:..

L*:* .. . .* ::

T

.T..~.."

Z..... ... . a.Z. . .*.*:  :

2 1. .:.*:.. . . . " .

Figure 3.2.4 Comparing PSDs at Node 8 00F 1 4

__, ._ . . -~ . <

h i,

4*.

. . . .....'i.

. . -. . . . . -1..*.**..*r*.....

T.O.

....... .=. *..

3; ::: :. J

_ .=.

2

==.1.- ****. .. **

.. e

.."..". "..-*!t.*.

.... .... 2.*

• • T.*. :. ..J .::***. .7

2. 2:--

...2 r*r.:r* 4 T *: : 2:: *. ::::*:

.e..

.. *:..s

.-D...e . *" * .*: : *.".-. . -.

• i L L *.,;

3

.... .. -~. .... . ..

= ,.

. .I i

N

• 2..*T....".

l l

l l

.=..-.. .- -

a -

l

+

I l

1' .. . ..

. . .. ~ .-  :

,~. *:" ::*.

L* L*  : '

='. . .= ._~.. .

."

• 7.~

.. . 2:..*

I .. -! -7:7 .- *

-.. :::O*

• 2.a

..2 -7'-

T: T. :..::. .:

• 7:7:

. 7*

.I = s.. . . :: : . . :.: ..= : :;: *.-

Figure 3.2.5 Comparing PSDs at Node 8 DOF 2

,l

4 N

i: .*:s l'!!il1

. i::i::

t . -

......) . p.,f CIU' 1::1..i .';e:.  !

.i's .

. ::::: .g llll ) 5: ..

. tll!Ut

.r "t" "r

'.. I:::; 8

. ::::: 1:::::::'.:: ' ."..I:.';

1. l W ;.g ... ..g e

f ..tel I' !il1

......g t ;\

.,y., . . .

...  :::t.'s

e. i m

9...:e! o

~'

= . .~..

. . . c

i.e

" ~ -

e:

W t.1.e o.s v l ;fill D o

I.q Z

... e.I o, u,a I::..;;.:l

. se

. .. m

......i o m

G.

1:!!ill C!i;I .

m fl:!il l':!ill c ll !il1 ClilI S-ll!il1 C!Cs E.

.. . . y .. g 8

I::

I...,.'s

... ..i .::t.'s i...... g

. ..... . t

. ::e..

1p+ .ll" N.

m i

n::. : :::. i.

:r::: . . ". l o  ;

i . n. L.

1.' !i!l ;3

.f. .i.. ..t., .n.". l @

g::4,;.,. -

11":' i::!i;' t.r:: i::!.;i u.

I "+"I i::!;;i s;;;;;i i nli.i

.. ... a

':.'I':' C!i:n l':!!:: tlll;;s i::!;;i e

a:: e:'.*:e C!i # co;;,

" c"..;.,

' , "+

.. .. ;, ,~..,; ;

e ,;;i;:s

> 11.'l :ll:'

1. : r ca

.,.. [

':s i:..*:e . .

je....

..ag{a.u..g

" "* ! *$ *l

. 1,[. $,l 4 l,) e . .g 76

h5C;  :~. .- E -

% .:. .~. ; - . . : - ~~i Q *~ Q -

.r..- -

T%.a. ~- T:

• a- . . . ._ - ~ . . .- . :._: -

. : a . : . :.:. : --

._a : ..a4

u. ..:. . _.. .:-. .: . _.a

::: G-

_~

.. ^.._ * * -. . . *..**::

.*4=l: . . . . * .:.: * .:;_ -

.: *' ^ . ':.'"'4.'-

.7*.

. .*. t*

.'* L-; ,". ; * *= ; L7 : :--

a ? :a a T**" 4 :2"*

4* :*: = L

, ,_ * * *.""L*.." ,.".*.

• d.* * .* *.* *a

_?*:.

e.*.*

._i.-L..

"_I "J

& T

~ .

N i

ik l ~

~-  :-

Ei :1?i ;

^ .

.. _._' . : .a-..

.. 4. n : .  : :.. .-

2 :::*-*- -

_..a ..._

_.' . .. _L*: . .. . .

T*

-n _: :-_"..:;:..

.; t*. .

• • i: .:

t.

Figure 3.2.7 Comparison of RS/PSD Conversion at Node 5 DOF 1 ii l

-r -,, ,-

i:.. :s i:li:t i:::::s

t .

......l ......t l I:!!i:1

.. l.. ..I :::i .. "

s:..':s : ll::: i:l!i;t "

..,. r-  :-' #:'i

.s:: :: .e:: :: '. i ;

r::: ::::::i:.I.:t """!

l.a ..i 1.. ;. . ... ..

N A

O t....,

Q C!i;I e

. .. . i o l

...... 2:

s'l INI .*

. .s .::::: m g

.o

...g., .

t:..f m

g l ...... QJ

.i:1.

.; s c a

.::i'i y o

e*

.:t .e cs l ili:t.. .e sm Q.

..**. ly . LA l...i a:

.n. 8t

,eg . .. r.ui;e u w

.. n gl.. .....l e O

.M e*

1::!i;t C!i:1 '

1.0":t l.t" t to O-C!i 41::!i'i y

C!us C!;:t

.::t :::i's co.

i,...

.. l... N.

........... m as

.11... . . 31..

. . L.

s't:i a'I.; 3 t:. :: :: :: -

en p:ei;!.,

i.i;i , .-

w

,i s.a f

1. !?.,;a s'!.s;;, i!:'j:s 4:!'Ut 4)'!I l , c.

(. .,,. ,:: , . 7, ' , i ',. i .i.

p:: . . ":;l

. .:' ..:t .

I ' III.

e ~..:t ; ..:1 l I.t";8 i I";*

Ifg 1.e 's

• 5 e 1.s's I

I;;8 'l t..t::: 1. :: . '.

. ,7 p ; ,. . . ., , 1. .. . . . .Cl.'.

78 l

l f

a~..*. ** 7. ". .. .~.* *. .

4

• * ! *. 7.J. *.~. T . 7? * !  : :* . .. .

. .. .I .". . "4 *.: :., *4 *. ., r! . *  :  ;

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• :1.".:..,..*L 7* * : *. . = . , .

• • 7 7.

~?** T* .~. .~.* *~. .

..*.L*.*.**7..*J J*. . " ..  ::  ?:".:;

a  !*.*

..*L**.*. . ...*:

* %*

• 7 2 : :

• 4 ? *4*.* J

~ ...~.

• '::7.L*:

..~ ....

^ . . . . ~ . . ,* * * * ' . * . * .

2 ."* : *. .*.; .".*

.,7..*.

• J L.* .*

• ,,:: ; .- *:. .J . .t::.: ..

-. ** ? .

.2

_-- 4 7

7 N **

@ ** . *[

.~. *

.. *~2.*

.* ; .* *J L*" ~. ;

7.**. -

~.

~-***-

1.**."k* '

.* 2 ;; i : -_

?".*.*.*

L*. L*.*.".

%* * "J *d 1.*.

• : 2: 4

• 22:..**I ..

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C L* C

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• **l *2: 2 **

. . J * *.#

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L; .:*.

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• .; ..' .: .: .1 .*

Figure 3.2.9 Comparison of RS/PSD Conversion at Node 5 DOF 3

, '1

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• • !b..*  ; a :": : : g.:*::.. a: :: . ::

L-  :.- :;; ".""?. 4 1 **. *

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g: ggg L; ::::% :J

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J

b* 4 4 b* .6 ." -

J.*%.

02: L: :n ""'*"..412"""

7.

a ::* : *g : *g g ;  :: : ::::ll L

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t. :

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+ .

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=*

h I

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r.:  : := :: * *::: * * *::

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.~.

J *:

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.n .-:.- 2. :.:.r_ :. ::. _

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1 Figure 3.2.10 Comparison of RS/PSD Conversion at Node 8 D0F 1 ,

s

. **=

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J 23.

• J

'4~.*.

2 :* ~ !

• *  :

• 4 C:: * :::: **. :.*: : L7 ""*O*."."**. J .".

. . "J 12 .* ,"J *.*.! *P**.:_

"": ".,**J *g =..":.T.:

: t " *

!!*.. ,, -2' .- 7.:

• : .; = : - +:::

• *:. 2 *****1*T" 1  :*4 4 * "41J ? JJ *.;; : :"* 4 * : ** : *J  ; :.J:: = =

• . **J *J **

. .t."":...:. :.:.' -

J .~.* *

. .*. "*J.

. : J: .~.

• J !*.

• J ."*

.. ;2:__ .!. 7 7. .4 .:- .**:.!

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.d '.d. b ; -

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Co ..

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21

1. .: **

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Figure 3.2.11 Comparison of RS/PSD Conversion at Node 8 00F 2

$-. ..l  !.$. 55 .55

..".T. -..- .'$ !$

• "_ I2 n. .*.* I

. ~.2. *. .'. .i.. .a.a. .. . . .**.1.

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. . k.*

. . . I.E

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+

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7: . . .73. 7:

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• 7:.: 7: -

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.n:..._... ..:::

. . .. =.

Figure 3.2.12 Comparison of RS/PSD Conversion at Node 8 DOF 3 I

l 1

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.. -, ... .. .... .. ...~..

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t

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i.

Figure 3.2.13 Comparison of PSDs at Node 5 00F 1

!"".*. .~., ~.

M.

• • : . 7 .

-- ',"*_ .:..~:

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• • . 1 *.* * ?. ?:

f.*..*"." .

. 7_:

...~.

2_:T_:;;

.:-- _"L*. : : :

• 2 T "u". .". - .,

7.

.~..~...

. *. ::J.

. . .*. P. ; . ::*.

_.:.~...~.:,-.

.:-.;- -. .. ..*, . . ..  :: :::::=

. u. . : -

-..T * ""*

• - ; . M. . .M. g*

m.. .:. = _ .: .:..-.: -

2. .._- -

6 co u . -

..i.i -

il 4ij ~_ - ---"*,- t

__ r 4 m :....

:- . . .:::::=

~..~.  : .~.

_ r_

= ....rr::..:.. -

.2 : :

i i

_: _: :-- u. :.:..~ : .: :_. u. . :: :. :

Figure 3.2.14 Comparison of PSDs at Node 5 D0F 2

.::=.. . _=.....:.;

__. _.u:: . = . . : : : =: :-=..=..=.  :  :... .: . :: - .-::

.u ..

. . = .. :. : . =..

u ..=:- : :: - - =.:=_.=_..=:  :. ::

. = =.: :- :.

2 : :- : :-

. .-=.-:

. L: ".J* .* '..*%*

t

[

i u

w u,

a L

..g=

6

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. . . . .. ::=

.:. : ==

.=.

=.= : =

u

. .- -...u-::.

O ::~*::::::::**..

Figure 3.2.15 Comparison of PSDs at Node 5 DOF 3

P e::s C!ilt .' i::::::

.  ::;r..

.... ..m.

1 C!i;i 1::8.. (..e..I ::r.. .

11.i's *:1C' C!i;l '

C!i '

:: . ::::: -1 i' -l- "r- C!i;t  !

1:::ht 4::t :; :'e.:t ......l l.m . l ( .. ... ... ...l P

C!ilI I':!!;f 1:

. . .. g

..!':i!.

.l g

C3

, ...I .1 8: CO l .

m i:1.r v r

....... . . .  :. :i O

.......... 2:

..... ..u. cm ..

ps

• ~~- :t.i u

!::,, en 1:m:i ,

o C!i;I t.o C'-

r.'.'i'l C!i;l  %

i

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to C!i;# Cli:1 '"s, r.a..ii ". l l..'.s. .-l to g C!i"8 C!i'l c.

E 7

r. i.;;, r.i.e".,

o

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' !i;' Ci;a C!i.i C!i;n C'..: i

.t. .. e.e .

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t

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m_.. __m _ _ _ _ _ . _ _ . , ,_- u _ _ _ .

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Frequency (Hz)

Figure 3.2.17 Comparison of PS9s at Node 8 DOF 2

• s

1 a

s:'e :

C!i;l . a::Ca Cs**e

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I r"'s;;

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Ct;;

s i,::.i f. 8. ::l:: ."..r . i ii

.. . . . > :. 1 . .. . g 1.m.1 I;m.. ...'.: ...

C!i;I C!i:6

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, ..~..

l..; n

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% *:', . , co s ..

I 4 ... @

...'g V

- ,, .. .s:t.:

g...I o

g a n.** o .........

, ,,,,,,,,, cg.a

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....: 'm' g

ll.fi:q e c2 W

3.s

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a I.""'i;l '+-

O

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e

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C!!:t C!i:1 L c

C ; , . .;.,.!i;l C...!i t c.

.. e g

o C!i;s C!1:1 U t

e 1::t- i::t-

. (.... ..il . ...i

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i m fll , 3.::. .ll., .

re:g . : ::1 e

L

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I E'

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"'I't ::l't o i:' ' ;s 8.*:6 1'.'i;t c];;. ,, . 8;;

, 6;, , t;*';';

i:::::e. #:: ::s

.jg.

. . . .g s 4 u

l e$ l 9

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.ct::i~ -i e::::. :.:. .: . w. .i l

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n. _ ._.

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r ' r.,:

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Figure 3.2.19 Comf ariso, o# PSDs at Node 5 00F 1

_.__-- r. _.

. : .. u. ., . -.

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e .

O h

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r:  ::::::

_ r_. _--

- :_:...=.

Figure 3.2.20 Comparison of PSDs at Node 5 D0F 2

v.

. ::.~.-  :.  :- --..- .~. ..:...

= .--. -

a u.

. -. . .,,:e- .:. :_2. .

.. . _ ~. . _ . _ _ ::e  : . .;.,,- . . .

... . . :: _-:: .:.~...:

. . - .; - : a

.:. i '  :'a 1 i :.: .-

2 _-

0 c

w

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U 5/s . . - -

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.ib_i ..

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r ::u t : . : _. :-:.

Figure 3.2.21 Comparison of PSDs at Node 8 DOF 3

i

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r.:',.* s :s.?n i::r-t . - l t '..*  !

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r :..':e '*l!. r.!i;: , .,<

e .!:, .C.t: I

4. "4 e;:r', i.'.. .i .

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.-.. b.k..b.i.b# .: .A

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1 Figure 3.2.23 Comparison of PSDs at Node 8 00F 2 i

l

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...s .....q I v..!i:'

t':s .e (... .I e::r . .

c..':i ::1:::: #:N;s 4

s::::: : :: I e...,.. . .i i,.s.., , n; ..:

t..a '

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r..e. .e n.

g.= e, a..,g v.

1. 2!i;# o c

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to v=

I . ihl !..!s I (

g 1  ::!;;s #;N;#

! C:S:t i;'.',;t i g . .g .. 1.1 I. .4. . I Q

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Figure 3.2.25 Broadening in the PSD Analysis (Node 5 DOF 1) mws--.-wt yy - ' - - - . _ _ .= +mm- ' - P7- ---'M --- P'- T w- f- '-7-9vg ~ * - - - T y+ + - *1-' *-- -

m-

4'".*., P ?. *. . . . .

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g ' ./ . . . 2

:

2 . ** T. * : :*: I

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* .?.La*

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L.1* : a 'J 1 A: J

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27. .~.P.  : ?. T. 7.

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t*. P. 77* 3 ."1 T*. ."

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Figure 3.2.26 Broadening in the PSD analysis (Node 5 00F 2)

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Ia

J*~!J

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--- . : *, : r

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Figure 3.2.27 Broadening RS in the PSD Analysis (Node 5 D0F 3)

t C!;;#  ::::::

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i:...:e ;lll::: i;!!i;' ,.....

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. Figure 3.2.29 Broadening RS in the PSD Analysis (Node 8 DOF 2)

__ _ __-. - .__ - _ _ _ . . - _ _ _ _ . . . _. ..___ _- . _ . _ -_ _ _ . - - - - - ._, _ - _. - , _ . -_ _ . . ~ . .-. .

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Figure 3.2.30 Broadening RS in the PSD Analysis (Node 8 DOF 3) t

Enveloping of Response Spectra EDASP FILE: EPSX1 - Envelope of 5/1, 8/1 EDASP FILES: PS51X, PS81X FREQ. PS51X PS81X Envelope EPSX1

.2 3.050E-1 3.103E-1 3.103E-1 3.103E-1

.33 6.156E-1 6.285E-1 6.285E-1 6.285E-1

.56 9.845E-1 1.025 1.025 1.025

.93 1.681 1.849 1.849 1.849 1.56 3. 9 54 5.041 5.041 5.041 3.37 5.214 7.676 7.676 7.656 7.28 2.817 4. 004 4.004 4.0 04 12.77 2.510 3.703 3.703 3.703 20.34 2.432 3.615 3.615 3.615 34.0 2.4CB 3.584 3.584 3. 58 5 Table 3.2.2 Envelope of Nodes 5 and 8 D0F 1 Responses l

101

Enveloping of Response Spectra EDASP FILE: EPSX2 - Envelope of 5/2, 8/2 EDASP FILES: PS52X, PS82X FREQ. PSS2X PS82X Envelope EPSX2

.2 4.288E-3 4.772E-3 4.772E-3 4.77E-3

.33 1.184E-2 1.318E-2 1.318E-2 1.318E-2'

.56 3.554E-2 3.956E-2 3.956E-2 3.956E-2

.93 1.108E-1 1.233E-1 1.233E-1 1.233E-1 1.56 4.961E-1 5.524E-1 5.524E-1 5.524E-1 3.37 1.413 1.677 1.677 1.677 7.28 6.822E-1 7.808E-1 7.808E-1 7.808E-1 12.77 6.19E-1 7.023E-1 7.023E-1 7.023E-1 20.34 6.007E-1 6.802E-1 6.802E-1 6.802E-1 34.0 _.5.944E-1 6.727E-1 6.727E-1 6.729E-1 Table 3.2.3 Envelope of Nodes 5 and 8 00r 2 Responses i

j 102 1

i l

[

\

L

Enveloping Response Spectra EDASP FILE: EPSX3 - Envelope of 5/3, 8/3 EDASP FILES: PS53X, PS83X FREQ PS53X P583X Envelope EPSX3

.2 1.522E-3 2.496E-3 2.496E-3 2.497E-3

.33 4.24E-3 6.936E-3 6.936E-3 6.935E-3

.56 1.294E-2 2.104E-2 2.104E-2 2.105E-2

.93 4.167E-2 6.692E-2 6.692E 2 6.692E-2 1.56 1.965E-1 3. 046E-1 3. 046E-1 3. 046E-1 3.37 5.525E-1 1.114 1.114 1.114 7.28 4.049E-1 6.11E-1 6.011E-1 6.011E-1 12.77 2.741E-1 4. 564 E-1 4.564E-1 4.564E-1 20.34 2.512E-1 4.305E-1 4.305E-1 4.305E-1 34.0 2.455E-1 4.228E-1 4.228E-1 4.228E-1 i

Table 3.2.4 Envelope of Nodes 5 and 8 D0F 3 Responses l

1 103

]

Enveloping Response Spectra EDASP FILE: EPSXYZ - Envelope of 5/1,5/2,5/3,8/1',8/2,8/3 EDASP FILES PSS1XY?, PSS2XYZ, PS53XYZ, PS81XYZ, PS82XYZ, PS83XYZ F P _'O . PS51XYZ PS52XYZ PS81XYZ PS82XYZ PS83XYZ Envelope  :

FPS 53XYZ EPSXYZ i

g .2 3.052E-1 3.088E-1 2.015E-1 3.105E-1 3.103E-1 2.039E-1 3.105E-1 3.105E-1

.33 6.161E-1 6.246E-1 4.070d-1 6.288E-1 6.283E-1 4.121E-1 6.289E-1 6.289E-1

.56 9.855E-1 1.012 6.410E-1 1.027 1.024 6.535E-1 1.027 1.027

.93 1.684 1.792 1.014 1.859 1.840 1.052 1.854 1.854 1.56 3.974 4.655 1.729 5.068 4.939 1.891 5.068 5.068 3.37 5.962 7.133 5.937 7.808 8.087 8.661 8.661 8.661 7.28 2.976 3.888 6.312 4.096 4.157 6.461 6.461- 6.461 12.17 2.633 3.465 2.976 3.770 3.806 4.499 4.499 4.499 20.34 2.547 3.373 2.593 3.678 3.711 4.082 4.082 4.082 30.0 2.520 3.343 2.490 3. 64 6 3.678 3.955 3.955 3.955 Table 3.2.5 Envelope of Nodes 5 and 8 Responses in all Directions l

4.0 STRUCTURAL MODIFICATION 4.0.1 Background 1he equations of motion of the undamped structural system may be written as:

[H]k + X X=f (4.0.1) where [H] and [K) are the mass and stiffness matrices, X is the vector of displacement in physical coordinates, and f is the vector of applied forces.

In general, [M] and [K] contain non-zero oTf diagonal terms, thus Eq.(4.0.1) represents a set of coupled second order differential equations.

Using the coordinate transformation X = [6 ]Z, Eq.(4.0.1)

'- ~

may be written in modal coordinates as:

[I3k+[w3Z=C4Y_f,,

2 (4.0.2) where [ I ] is the identity matrix, and M ) and [6 ) contain the frequencies (eigenvalues) and mode shapes (eigenvectors) which constitute the solutions to:

([K] - w 2[M])[+] = 0 (4.0.3)

The mode shapes are assumed to be normalized so that:

[4]T[M) C+] = CI] and [4]Ttg3 t43 . tw 23 4

(4.0.4) d Note that when the equations of motion are in modal coordinates, a set of uncoupled second order differential equations results, thus easing the solution process.

In situ testinf, directly provides a subset of the system's frequencies and mode shapes, avoiding what can be a difficult process of analytically developing the mass and stiffness matrices required to solve Eq.(4.0.3). As the full set of eigendata of a structural system theoretically contains an l infinite number of frequencies and modes shapes, only a subset of that data is l obtainable either analytically or through testing.

I When the goal is to conduct a response analysis, the required size of the subset is determined by the spatial distribution and the frequency content of the applied forces. In the case when the set of modal data of a system is to i be used to predict the modal data of that system after modification, the

requirements on the original set of modal data are more stringent.

i 105 t

The nodal data of the modified system will have to meet the aforementioned requirments developed by an examination of the applied forces. The modal data of the original system must span a sufficiently large space so that linear combinations of the original modo shapes will approximate what are expected to be the significant modo shapes of the modified system.

Given that an adequate set of modal data for the original system exists, prediction of the modal data of the modified system proceeds as follows:

([H] + [aM])k + ([K] +[aK])X = f (4.0.5) where [H] and [K) are the mass and stiffness matrices of the original system ,

and [aM) and [$K) are the desired mass and stiffness modifications.

Again, using the coordinate transformation:

= [4]Z (4.0.6)

Eq.(4.0.5) can be, rewritten as:

[43'(CM] + AM)[,]Z_ +

[$[(EK) + [AK])[4]Z_ = [t[f (4,0.7)

Note that the mode shapes of the original system are used in Eq.(4.0.7). In effect, the assumption is made that tne original mode shapes are adequate to describe the behavior of the modified system.

Eq.(4.0.7) can be further reduced to:

([!)+[4[ cam)[e])[+([w2] + [4YCaKl[o])Z_ = [4[f_ , (4.0.8) or more simpl/:  !

[M*]f + [ K*ll_ = i* (4.0.9) where,

[M*]=I!1+[e[fAM][4] (4.0.10)

[K*1 = [w21+[s((AK1[4] (4.0.11) 106 i

l l

l

[_

Eq (4.0.9 written in) the modal coordinates of the original system.Note represents that Eq. the equations of

.9 can be written once modal data for the original system are obtained and the) mass and stiffness modifications are analytically defined.

Finally, the modal data of the modified system is obtained through the solution of the eigenvalue problem.

(CK*] - (w*)2[M*])1* = 0 (4.0.12)

Note that the above development does not include any damping effects.

Theoretically, damping can be included through the use of a velocity dependent term in Eq. 4.0.1 Decoupling of the equations of motion with non-complex mode shapes w(ould,)tiowever, occur only if the damping matrix was proportional to a linear combination of the mass and stif fness matrices, in practice, it is not usually possible to define the damping matrix. As a result, the eigenanalysis is performed ignoring damping.

J 4.0.2 Case Description Five structural modification cases were considered in this section to verify the EDASP structural modification routine. All modifications refer to the original structural model described in Section 1.0 with the first ten modes. The results were compared with STARDYNE runs which modifies the properties of the structure and perform the eigenvalue j analysis once more for each case.

Case 1 A spring was added to connect node 5 and node 7 with a stiffness of 14000 2.

Case 2 The masses at node 5 and node 6 were tripleo.

Case 3 The entire mass at node 5 and node 8 were taken out.

Case 4 Node 5 00F 1 were tied to the ground by a spring with stiffness of 12000.

Case 5 The modifications are 1) same change in stif fness as in case 1, 2) added stiffness of 120000 to node 6 00F 1, and 3) tr? pled masses at i

node 5 and node 6.

4.0.2 Program Results Case 1 The STARDYNE and the EDASP results are included in Appendices 4.0.1 and 4.0.2. Comparison of the nodifiea frequencies are shown in Table 4.0.1.

I i

107 [

l l

[

Mode EDASP STAROYNE 1 2.29 2.29 2 2.36 2.36 3 3.63 3.63 4 4.49 4.49 5 5.81 5.80 6 5.85 5.84 7 5.91 5.91 8 7.07 7.07 9 8.60 8.60 10 10.1 9.24 Table 4.0.1 Comparison of Modified Frequencies Case 1 108 l

l 1

Case 2 The STAROYNE and the EDASP results are included in Appendices 4.0.3 and 4.0.4. Comparison of the modified frequencies are shown in Table 4.0.2.

Case 3 The STAR 0YNE and the EDASP results are included in Appendices 4.0.5 and 4.0.6. Comparison of the modified frequencies are shown in Table 4.0.3.

Case 4 The STAR 0YNE and the EDASP results are included in Appendices 4.0.7 and 4.0.8. Comparison of the modified frequencies are shown in Table 4.0.4 Case 5 The STAR 0VNE and the EDASP results are included in Appendices 4.0.9 and 4.0.10. Comparison of the modified frequencies are shown in Table 4.0.5.

109

Mode EDASP STARDYNE I 1.76 1.76 2 1.80 1.80 3 2.78 2.75 4 3.25 3.25 5 3.63 3.55 6

4.67 4.67 7 4.77 4.77 8 5.47 5.47 9 6.12 5.56 10 6.90 6.11 Table 4.0, 2 Comparison of Modified Frequencies Case 2 110 1 - - - - _- - - - - - - -

{

Hode EDASP STARDYNE 1 3.30 3.30 2 3.58 3.58 3 5.67 5.67 4 7.65 6.46 5 8.51 8.38 6 41.1 9.66 ,

7 83.3 34.5 8 114 34.9 l 9 173 35.7 10 933 60.3

)

Table 4.0.3 Comparison of Modified Frequencies Case 3 111 l

Mode EDASP STARDYNE 1 2.32 2.32 2 2.52 2.52 3 3.98 3.98 4 4.49 4.49 5 5.35 5.31 6 5.83 5.83 7 5.89 5.89 8 7.07 7.07

, 9 7.09 7.09 10 8.61 8.61 Table 4.0. 4 Comparison of Modified frequencies Case 4 l

i 112 1

Hode EDASP STARDYNE 1 1.78 1.78 2 2.64 2.52 3 3.25 3.09 4 4.24 3.25 5 4.67 4.66 6 4.77 4.77 7 5.45 4.89 8 6.11 5.71 9 6.59 6.11 10 8.54 7.14 Table 4.0. 5 Comparison of Modified Frequencies Case 5 113

{

l

. i

i

5.0 REVISION 1 VERIFICATION  !

l 5.1 Correction #1 l 5.1.1 Background  !

i For cert 6in problem dimensions, the PSD/RS conversion process at the t

end of the PSD response analysis requires a scratch disk to be placed in i drive A. The Revision 0 version of EDASP 1.0 generated a "file not  !

found" system error whenever this situation arose. This problem was [

corrected in Revision 1. L 5.1.2 Case Description l

The PSD excitation nrcl60z was applied to the model frame in the l x-direction. The instruction set specified a model damping of 55, 75 ARS  :

frequency points, and the ARS at point 5 in the x-direction. The ARS was [

j named ps51X-1. This reqsired a scratch disk to be used for the PSD/RS  ;

4 conversion calculation.

5.1.3 Program Result The Revision 1 program executed withest generating the system error j encountered with the Revision 0 program. Results are included in i Appendix 5.1.1. Using the ARS/TRS comparison function, ps51x-1 was

compared with ps51x (ps51x was developed in Appendix 3.2.3. It is identical to ps51x-1 except that only 21, rather than 75, frequency 1

pointswerespecified). This comparison is shown in Figure 5.1.1. The frequency points common to both spectra nave the same acceleration l values. Note that the frequency spacing used for ps51x was not fine enough to resolve the spectral peak revealed by ps51x-1.

1 7

i I

2 t

l 114

_ c- w an

5.2 Correction #2 5.2.1 Background for certain problem dimensions, the transfer function calculation portion of the PSD response analysis process requires a scratch disk to be placed in drive A. The Revision 0 version of E0 ASP 1.0 generated a bad file number" system error whenever this situation arose. This problem was corrected in Revision 1.

5.2.2 Case Description The same analysis as described in Section 5.1.2 was performed, except the model damping was specified as 0.5% rather than 5.0%, and the 21 point ARS frequency spacing of Appendix 3.2.3 rather than the 75 point spacing of Appendix 5.1.1 was used. The resulting ARS was named ps51x-2. The small damping value used in this analysis resulted in a scratch disk being required.

5.2.3 Program Results The Revision 1 program executed without generating the system error encountered with the Revision 0 program. Results were included in Appendix 5.1.1. Using the ARS/TRS comparison function ps51x-2 was compared with ps51x, This comparison is shown in Figure 5.2.1. The two spectra differ only in the model damping used for their development -- 5%

for ps51X, 0.5% for ps51x-2. The expected amplitude difference is apparent in figure 5.2.1.

5.3 Correction #3 5.3.1 Background Given a response spectrum file which contains a set of spectra of different dampings corresponding to the same base motion, the spectra broadening functon of the base excitation module allows one spectrum of that set to be broadened and the result stored in a new file. The Revision 0 version of E0 ASP 1.0 incorrectly assigned that broadened spectrum the damping value of the first spectrum in the original set, rather than the damping value of the spectrum that was broadened. This problem was corrected in Revision 1, 5.3.2 Case Description The PSD corresponding to response spectrum rs51x (see Appendix 3.1.3) was generated using the RS/PSD conversion function of the base excitation module. The spectra corresponding to 1%, 3%, 5%, and 8%

damping were then developed using the P50/R5 conversion function. The broadening function was then used to broaden the 5% damped spectrum of this set, 115

5.3.3 Program Results Complete results are included in Appendix 5.3.1. The set of fou" spectra corresponding to 1%, 3%, 5%, and 8% damping are shown in Fig 9re 5.3.1. The broadened 5% damped spectrum is shown in Figure 5.3.2.

Comparison of these two figures shows that the correct spectrum of the set was selected for broadening, and the correct damping value was assigned to the broadened spectrum.

5.4 Correction #4 The Revision 0 versior, of EDASP 1.0 produced a garbled display of tabulated data when the number of data elements in a table corresponded to certain whole number multiples of fifteen. This anomaly was traced to a minor programming error and corrected in Revision 1. The garbled display produced by Revision 0 is shown in Figure 5.4.1. The correct display produced by Revision 1 is shown in Figure 5.4.2.

e b

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.t, i 4 1 FIGURE 5.4.1 121

F TJ ASP t.

abr Nodr- Cunne i eitio TC , ' : i 'u Node #1 Node #2 Node #1 Nodr' #2 teotic #1 Nodo #2 1 16. 1 1 31. 1 1 1 I 32, 1 1

2. 1. 1 17. 1 1 1

10. 1 1 'J 3. 1

3. 1 1 34, 1 .

4. t 1 19. 1 1

20. 1 1 35. 1 5, 1 1 1 21.- 1 1 36. 1

6. I 1

37. I 1

', i 1 22. 1 1

23. 1 1 3G. 1 1 C. 1 1

39. I 1 0 4 1 24. 1 1 1 ,

40. 1 to; 1 1 25. 1 1 1

26. 1 1 41. 1

11. 1 1 '

27. I 1 42. 1 t 1 1 43, 1 1 N. I 1 20. 1 1 44, 1

11. I 1 29. 1 1

30. I 1 45. t 1 1 it.

1 / 1 FIGURE 5.4.2 122

}

APPENDICES STAR 0YNE eigenvalue analysis for the frame model 1.0.1 1.0.2 EDASP structural model Listing of input time histories 2.1.1 Listing of input response spectra 2.2.1 Listing of input P50s 2.3.1 2.4.1 2,.4.2 EDASP results to compare.ation with **

Kaul'sConversion Broadening using base excitation module 2.5.1 STARDYNE time history analysis with x e 3.1.1 Response spectra from STAR 0YNE analysi. **

3.1.2 3.1.3 EDASP time history analysis with x excitatio 3.1.4 Response spectra from STAROYNE analysis **

3.1.5 3.1.6 3.1.7 E0 ASP time history analysis with y excitati 3.1.8 Response spectra from STARDYNE itations analysishistory a 3.1.9 EDASP tin 3.1.10 E0 ASP time history analysis with x and y exc 3.1.11 SRSS process check for x anditations y excitations 3.1.12 EDASP time history analysis itations with y and z exc 3.1.13 SRSS process check for y and z excitations 3.1.14 EDASP time history analysis d z with x and z exc excitations 3.1.15 SRSS process check for x and i s z excitations 3.1.16 EDASP time history analysis with x, y, an 3.1.17 SRSS process check for x, hhy.kkand (1)

(2) z excitat on 3.1.18 EDASP time historyheckanalysis for enve 3.1.20EDASPtimehistoryanalysisforbroadeningc STtROYNE PSD analysis with x excitation

• 3.2.1 3.2.2 PS0s from STAROYNE analysisEDASP PSD analys 3.2.3 3.2.4 RSs converted from EDASP PSDsSTAROYNE PS 3.2.5 PS0s from STAR 0YNE analysis 3.2.6 3.2.7 EDASP PSD analysis with y excitationSTAR0 3.2.8 PS0s from STAROYNE analysis

}

3.2.9 3.2.10 EDASP PSD analysis with z excitation 3.2.1* EDASP PSD :nalysis with x and y excitations 3.2.12 Cochination of x iand y excitations c k

s

.j 3.2.14 Combination of hyckand z excitations ch tions l 3.2.16 Combination of x and 2 excitations c e k 3.2.17 EDASP PSD analysis (1) with x, y, and 3.2.19 E0 ASP PSD analysis for envelope check (2) k l

3.2.10 EDASP P50 analysis for envelope ch l

123

l modification case 1 STARDYNE eigenvalue analysis for structura 4.0.1 EDASP structural modificationforanalysis case 1l modification case 2**

4.0.2 STARDYNE eigenvalue analysis for structura foranalysis case 2l modification case 3**

4.0.3 EDASP structural modification 4.0.4 STARDYNE eigenvalue analysis for structura foranalysis case 3l modification case 4**

4.0.5 EDASP structural modification se 4 4.0.6 STARDYNE eigenvalue analysis for structura 4.0.7 EDASP structural modification analysis for cal mod 4.0.8 STARDYNE eigenvalue analysis for structuralysis for case 5 4.0.9 4.0.10 EDASPofstructural Verification modification program modification anacorrection #2 for cfor for correction #3 5 .1.1 5.2.1 Verification of program modificationVerification 5.3.1 un & Associates offices.

• • STARDYNE outputs are stored in Stevens i

1 ,

124 i  !

i i