ML20094N068
| ML20094N068 | |
| Person / Time | |
|---|---|
| Site: | Mcguire, Catawba, McGuire  |
| Issue date: | 07/27/1995 |
| From: | Aradhya P, Gupta A, Gupta A North Carolina State University, RALEIGH, NC |
| To: | |
| Shared Package | |
| ML20094N047 | List: |
| References | |
| NUDOCS 9511270349 | |
| Download: ML20094N068 (183) | |
Text
{{#Wiki_filter:_ _.._.___ _..-_.__ _.. _ .__..___..____..__ _._ _ _ _..__ -_ _____._ I Attachment 4 ; i j ! l ! l ! I j Presentation To l United States Nuclear 1 l Regulatory Commission and l Brookhaven National Laboratory i j For Duke Power Company i i Ajaya Kumar Gupta j Professor and Director i Abhinav Gupta Research Engineer .l l Pradeep Aradhya i Research Assistant i July 27,1995 l Center for Nuclear Power Plant l Structures, Equipment and Piping ' North Carolino State University O TBA227.88M o'ias!!6, Raleigh, NC 27695-7908
~
P. _ _ ______PDR - - __
Presentation To United States Nuclear l Regulatory Commission and l Brookhaven National Laboratory { For Duke Power Company l Ajaya Kumar Gupta j Professor and Director Abhinav Gupta Research Engineer { Pradeep Aradhya Research Assistant { { July 27,1995 { { Center for Nuclear Power Plant Structures, Equipment and Piping A 9 North Carolino State University I Raleigh, NC 27695-7908 .
; , i
l { f I CO:NTE:NTS Presentation { Reference Publication j Verification Problems l l i { l I ( l i
1 T f i l l 4 I l i
! 1 1
J s { f i 1
- 1 1
l 4 l
- 1 1
l 1 1 j e Response of Secondary Systems i i t 1 i i i i i 1 1 4 i i I 1 l t i i 1 i I, 1 1 4 5 1 l i 1 a I.,
I i l l Primary Systems: Building that receives the motion directly from the ground Secondary Systems: Piping systems and equipments that are attached to the building 4 1
Conventional Analysis Method I j{ f Building Design Spectrum Compatible Time History
^
Ab b[ An. . l VV l V" i o 1 o Building Model i na b { jV y I l i Aa l l 'u' l V " V Floor Time History Floor Response Spectrum l I
l l I l l Problems
- 1. Evaluation of the compatible time history is not unique. (Several time histories can be developed that are
" compatible.")
~
- 2. The method does not account for mass-interaction between secondary and primary systems - response is overestimated.
. . . . . . . . . . . - . ~ .. ... - ..- .....- -.. . .-... ... _ ..- ..~. - . - .- -.-. . ~. - --. - . ... . ..-~ ... .-. . .
i t i 1 l 1 t l Multiply Supported Secondary l System (Piping Systems) i 1. Floor spectra are enveloped - ! overestimation of response. 4 l 2. For relative displacements between the j supports - worst case analysis is l l performed - overestimation of response. l l l Overall, calculated piping stresses may ! be an order of magnitude too high. i i 1 .I i i l
l . l i 1 i i i l i i l l } i 1 i 1 ! New Methods l i 3
- 1. Coupled Response Spectrum Analysis l l
i
- 2. Floor Response Spectrum Analysis -
approximating the coupled analysis l
l I 1
)
i These new methods are rational, rigorous l and elegant. In the overall scheme of things, the new methods do not significantly increase the cost of analysis. i 1
l l i I New methods give significantly reduced ! seismic stresses. They are useful for: l l
- 1. Requalification of existing systems for higher .than the original design seismic forces.
i 2. Reduction of piping supports (snubbers). l
- 3. Margin evaluation.
d 4
T Free Vibration Equation [M] ; 6: + [C] ; 0; + [.K]( U = l0: 3
~
- y'
=
- o3 -
x'- x' T = xp-'.T:'s-x 'T_ i
' o, O' ; X, : T = [X,, X,2 . . . ]
[$1 = ys2 0 $s. ' Xs.'.T = [ xsi -) [&,] = [$,2 4,2 l 1 i i [&,] = [o,, 4,2 1 i 1 1 l 1
- --- - - - - - - _ - _\
1 1 1 ) i i i l i 1 u , l r g,/2 = [ y,,] 0,, ; i
- SDOF Secondary System l
) I [yCG] = mS i i
- l
- SDOF Primary System l
- . l l , ,
1 1 = 4ct - . i m l P i l l 1 Ms i r.is/2 =
- \ M
! 1 P d
The definitions of f and [6 are as accurate as they can be. ] For moderately light secondary systems, these
- expressions can be used to obtain approximate l but accurate eigenvalues and eigenvectors.
I
1 ) 4 i ! Eigenvalue Problem i l [K*] :X = ;0; i ;
- 1 i l 2
l [K*] = 1 [1] + 1[C] + :K] i l A, = coupled complex eigenvalue [ = - co + icof1 -(2 co = coupled frequency ( = coupled damping
Primary System Secondary System Coupled System m R m k @ R m @
/////
m ma no b R R @ m @ m k, m k
'//// @ m k @
'/////// '///////
m Node Number m Each Story Mass (mo ) and stiffness (ko) Mass m:1 Kip s /in2 are varied to obtain a range @ Element Number n Stiffness k=5000 Kips /in of r,,,and r, values
Comparison of Coupled Frequencies, Damping Ratios Case 2 Frequency, (Hz) Damping Ratio (%) Percent Error Percent Error Mode Exact Gupta-Jaw IDK Exact Gupta-Jaw IDK - 1 2.5 0.05 10.21 5.7 0.19 22.38 2 4.2 0.17 -9.12 3.2 0.52 -37.49 3 6.5 0.02 1.61 2.2 0.25 -10.94 4 8.2 0.05 -2.48 6.7 -0.15 4.69 5 12.8 -0.01 -0.42 7.0 -0.35 0.25 6 16.9 -0.01 -0.33 7.0 -1.65 -0.50 7 19.9 0.00 -0.05 7.0 -0.28 -0.09 8 21.9 0.00 -0.16 7.0 -0.43 -0.06 t l
1 l I 1 i l J Example: Case 9, El Centro (NS,1940) Damping 7 %,2 % ; - Forces in Kips Element 1 Element 2 Element 3 Time History 88.6 6.18 89.4 Our Method 83.9 5.61 86.5 Conventional 243 16.8 243 i s 1
i l l l l High Frequency Mode Effect l on Coupled Analysis i l e i d
- Equation of motion of the coupled system M' {U3 } s,
.M' {@} + C {d} + 'K' {U} =
{U} = , 5 P ,
.{U,}.
{U,} ,{U,} are the primary and secondary system displacements, respectively, relative to the base of the primary system. l f
I l Secondary system displacement vector relative to the primary systems connecting DOF {&,} = {U,} - ;U,c {Uc}
;U,c. contains one secondary system displacement vector for each connecting DOF.
Each such vector represents the displacement vector when the corresponding connecting DOF undergoes a unit displacement. l i i
-+.
Transformation
-{U} =
SU},= F k / fO } ', P
. {Us } , .UaP.
I .,{0,}, {Up} s {&p} 'U,,. is obtained from the matrix ;U,c. by adding ) zeros for the non-connecting DOF of the primary l l system. l i
;5f {&} + [C {U} + ;f {&} = - ;ff {C6} dy l
i 1 I l l
1 i 1 Primary system residual mode vector l lK,. {Uo} = - lM,: {Us,} 9; scated out l
)
\
np {U6,} = {U3,} - I' {py } 7,eg l l i=1 Normalize {U,} to get { ,} such that i {do}' M {do} = 1 l I
\
I { 1 I l i Define a fictitious frequency ; w$ = {4o} K p {do} ! l l Treat { ,} as an extra uncoupled primary system mode just like any other mode.
1 Evaluation of Secondary System Residual Mode Vectors
'M,; {d,} + :C.. {d,} + 'K, {D,} = - M,U,c.{5}
{&'} = Total acceleration at the primary system connecting DOF.
M, fd,)+ 'C. fN + lK, {0,} = - E lM,. {d a} E %s Q e 1
% = Participation factor for the secondary system mode a at connecting DOF c, an element of TJ j
?c, = $, ,
'M, U,c l
l 1 i i i I 0"'
'K, {0,,} = - lM, E {U,c,} Yl
'd \
R, {U,,,} = {U,,} - E { ,,} % Q=1 Calculate nc values of {U,c,} vectors l
;K,; {&,,} = 'M,; -{U,c,} ;
Normalize each {&,,} to give { u} such that {dR}# 'M,. {dR} = 1
-Each of the nc { u} vectors is orthogonal to the given ns modal vectors { ,,}.
However, the { n} vectors themselves are not ; orthogonal to each other. ; i I l l l l l
l l 1 l 1 1 Coupled eigenvalue problem
'M' {&} + c {&} + ;K' {U} = - ;M' {U3} n y l IUF } = 4{X}
{U} = 4
. {U.} .
;0' ;0 S#F } ' i
= 4>F ' ~f#' } ' ;
.0 '4>,. '4> 3 -
,{Xa},
if {i} + ;d' {i} + fr {X} = - ;&r y. {y,) g,
- . ._ . -. _ . - . . - .. . . . - ._ . - - . . . - . . .~ _
4 l UF unn uninnnnnn onn NP ES Ks m, IAS 1 mp KP ms ES EP K' Kg g, m, KP Ks
'////////////////////////////
m, K, K,
////////////////////////////,
Secondary System Primary System m s= 0.1 Kip-s */in m,= 1.0 Kip-s */in Ks= 1000 Kips /in K,= 3000 Kips /in l Primary and Secondary Systems for Case 1 l I l l j l I 4 I f l
} 4 3 1 4 i d I a i 6 4 5 I 3 $ 4 4 2 $ ; 1 i 1 b i 4 2 @ i ,i 1 1 i i i i
'////////////////////////////,
i 4 i J Coupled System for Case 1 i ( } I I i l a l 1 i i l J i A i l
1 Frequencies and Damping Ratios for Case 1 6-DOF Primary System 4-DOF Secondary System l Mode Freq. Damping Mode Freq. Damping No. (Hz.) Ratio No. (Hz.) Ratio l 1 2.10148 0.07 1 9.83625 0.02 2 6.18237 0.07 2 18.71025 0.02 3 9.90389 0.07 3 25.75100 0.02 4 13.04991 0.07 4 30.27286 0.02 5 15.43755 0.07 6 16.92772 0.07 i I { l I i
( l Comparison of Nodal Displacements (inch) for ) Secondary System DOF CREST / PIPESTRESS Node Including all Truncated modes; TIME HISTORY ! no. modes Including missing mass 1 1.281 1.277 1.289 l 2 1.498 1.493 1.498 3 1.688 1.682 1.682 I 4 1.849 1.842 1.838 1 l l
Comparison of Spring Forces (kip) for Secondary System Elements CREST / PIPESTRESS Element Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 240.4 239.7 239.9 2 217.6 217.0 214.8 3 191.0 190.3 185.9 4 161.3 160.6 155.6 5 128.8 128.2 124.1
Description ofInput Parameters Description Notation I. Primary System Modal frequencies eip Modal dampings (pi Modal participation factors yi p Modal vectors for connecting DOF &ci
l ! Description of Input Parameters i i i l II. Secondary System h 4 Modal frequencies . w,i Modal dampings (,i Modal vectors poi Mass matrix M, Stiffness matrix K, Connecting stiffness matrix K8 Support influence matrix U,c Base influence vector U3, III. Primary System Base Response Spectra Spectral accelerations SI(wi ,(i) Corresponding frequencies wi Corresponding dampings (i Rigid frequency wr i
MASTER PROGRAM l PtPESTRESS ControlData Primary SecondaryConnecdyty , input BaseSpectra Mass and ~ Stiffness Matrices Primary System Eigenvalue Problem ModalProperties CREST
^
l
~
CoupledModal PIPESTRESS Displacements MemberForces : Stresses CREST Support Reactions ModeCombination Generalized Flow Chart for Interaction of CREST and PIPESTRESS
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Research on Coupled Seismic Response of Secondary Systems Ajaya Kumar Gupta l Hussein Ahmad Megahed2 l Abstract Coupled seismic analysis of primary-secondary systems may reduce msponse of a secondary system by one or more orders of magnitude over that calcula*ed from a l conventional uncoupled secondary system analysis. When individually classically damped primary and secondary systems are coupled, the combined system becomes nonclassically damped if the modal damping values of the two systems are different. Various topics related to such systems have been studied to further the state of the art, and thereby improve the accuracy of the analysis. Two methods of analyzing nonclassically damped systems, an
" original" and a new canonical, are shown to be mathematically identical. The former is preferred because it represents response in terms of physical quantities, relative displacements and velocities. 'Ihe response spectrum method for nonclassically damped systems requires the conventional relative displacement-based response spectrum and a new relative velocity-based spectrum. For combination of modal responses, three sets of correlation coefficients are needed. Evaluation of the response of a single-degree-of-freedom (SDOF) system is studied to be able to evaluate the response spectra and correlation coefficients. It is shown that the commonly used time-domain analysis gives incorrect relative velocity values for high frequency SDOF systems. An altemative frequency-domain approach is proposed. New expressions for correlation coefficients showing better agreement with the numerical values i
than the existing. expressions are developed. Impact of damping values on rigid response coefficients is studied and is incorporated in new equations. Empirical values for the key frequencies that define the rigid response coefficients are derived. To account for uncalculated high frequency modes of the uncoupled primary and secondary systems, a residual mode method is developed. Introduction Various topics related to analyzing nonclassically damped coupled primary-secondary systems have been investigated to further the state of the art, and thereby improve the I L accuracy of the analysis. Coupled response of a secondary system may be one or more orders L of magnitude less than that calculated from the conventional uncoupled analysis. Dynamic ; 4 analysis of multi-degree-of-freedom (MDOF) systems is often performed by the modal superposition-time history method, or the equivalent response spectrum method. Mode shapes 1 Professor and Director, Research Program on Nuclear Power Plant Structures, Equipment and Piping, Department of Civil Engineering, North Carolina State University, Raleigh, NC 27695-7908. 2 Assistant Professor, Depanment of Civil Engineering, Cairo University, Caim, Egypt. Former graduate student, Department of Civil Engineering, Nonh Carolina State University, Raleigh, NC 27695 7908. III/5-1
, ,-- -+---e :-
and fmquencies of the undamped system are calculated. It is assumed that when the equation of motion (including the damping term) is transformed using the mode shapes and normal coordinates, a set of uncoupled equations in terms of the normal coordinates are obtained. These equations are similar to the equations of motions of single-degree-of freedom (SDOF) systems, thus greatly simplifying the solution process, In the above process, the off-diagonal terms in the modally transformed damping matrix are assumed to be zero. The systems, in which this assumption is valid, are called classically damped. MDOF systems in which the off-diagonal terms in the transformed damping matrix cannot be ignored are called nonclassically damped. A coupled primary-secondary system is an example of such a , system. A detailed state of the art of analyzing nonclassically damped coupled systems is presented by Gupta (1992). A summary of recent research performed by us (Megahed and Gupta,1992) at North Carolina State University to further the state of the art is presented here. Methods of Analysis The equation of motion of an N-DOF coupled nonclassically damped system is given by: , 1 MC+C0+KU=-MUQ, (1) s 1 where M, C and K denote mass, damping and stiffness matrices, respectively; U is the relative displacement vector, U3is a displacement vector obtained by statistically displacing the support by. unity in the direction of the input motion; u, is the ground displacement; and l the super dot (-) represents a derivative with respect to the ume variable. In the Foss approach (Foss,1958), Equation (1) is cast into a 2N-dimensional matrix equation, and a i complex eigenvalue problem is solv:d. 'Ihere are N pairs of complex eigenvalues and eigenvectors. Each pair consists of eigenvalues and eigenvectors that are conjugates of each other. The complex eigenvalue pairs give the values of modal frequencies mg (rad /sec) and damping ratios (;, in which (here and elsewhere in the paper) the subscript denotes the mode number. Each complex eigenvector pair gives two real vectors d and M of the rank Nx1. j The modal superposition equation of the nonclassically damped system is given by: ! N N N U = E U, = E Ul - U;' = E yfx; - y,'x; (2) i=1 i=1 i=1 in which xg is the relative displacement of an equivalent SDOF system and can be calculated j from: f, + 2 mj (,x; + mfx; = -4, (3) A new " canonical" method was recently proposed by Yang, Sarkani and Long (1987). They define sine and cosine responses of a SDOF system, denoted here by s, and cg, respectively. The modal superposition equation is written in terms of these responses and new modal vectors M and M as: III/5-2
N N U, = E U{ + U ' = E f s; + f c; g (4) i=1 i=1 We have shown that: N"O I N Di
+5;W V[l i Vs'"V[
s; = -m xg, c, = .t;-(ge,x, g (5) in which mo;is the damped frequency of the ith mode. Equations (4) and (5) give a vector U, that is identical to the vector U of Equation (2). In the time domain, therefore, the new canonical method would give responses that are identical to those given by the " original" method.~ The corresponding response spectrum methods should give comparable results. The-original method deals with physically meaningful quantities, relative displacement (x;) and velocity (1), and the sine and cosine responses (sg and eg) used in the canonical method are physically not as meaningful. Therefore, we propose to continue to use the original method. In the response spectrum method of nonclassically damped systems, two response spectra need to be defined (Gupta,1992; Gupta and Jaw,1986). 1 l 1 i Sj = max lx(t)l, SE = max lk(t)l (6) Both the spectra can be defined in alternate units. SI = mS[ = m2 Sj, 2 SJ = mSE = m Sf (7) in which the small letter superscripts d and v denote the source of the spectral value (relative displacement and velocity, respectively), and the capital letter subscripts A, V and D denote the units, acceleration, velocity, and displacement, respectively. The maximum value of a response R in mode i obtained from the relative displacement spectra is denoted by RI and that from the relative velocity spectra by RI. Since these maximum values do not occur at the same time, the following modal combination equation is used: III/5-3
1 R2 = EE(efRlRj +el,R[Rl-2p;;R Rj) g (8) IJ l in which efj, tij, and p;j are appropriate correlation coefficients. In the analysis of classically damped systems, only relative displacement based design spectra are needed and 6: fined. Analysis of a system as nonclassically damped is uncommon, and therefore, relative velocity-based design spectra are not readily available. A method for estimating a velocity spectrum from the corresponding displacement spectrum is proposed by Gupta and Jaw (1986) that will be further investigated here. Analytical values of the correlation coefficients Eh Eip and gj based on the assumption that the ground motion is a white noise are available m hterature. Empirical studies based on real earthquake ground motions are also available in literature for tyj. No such studies have been performed, however, for cij and p;,. It is proposed to empirically investigate all three coefficients in the present work. In all these numerical investigations on the response spectra and the correlation coefficients, we need to evaluate the response of SDOF systems subjected to earthquake motions in all frequency ranges. Problems with the calculations of the responses were , investigated and are summarized (along with the solutions to the problems) in the next section. 1 Response of an SDOF System j Response of an SDOF system subjected to earthquake ground motions is usually calculated in time domain using an " exact" Nigam and Jennings method (1969). The processed earthquake data has nonzero initial values of the acceleration, velocity and displacement. That is due to the instrument triggering threshold and the process of data filtering. When these nonzero initial conditions are not explicitly accounted for, the calculated response becomes erroneous for the low-frequency SDOF oscillators. This problem has been long recognized and can be solved by introducing a fictitious, prefixed pulse proposed by Pecknold and Riddell (1978). The ground motion is defined in terms of acceleration values at discrete time intervals. In the response evaluation it is assumed that the ground acceleration varies linearly between the data points. The assumption of linear acceleration variation between the data points introduces a practically limitless frequency content in the ground motion that is well beyond the Nyquist frequency,f, (Hz) = 1/2 At, the . upper frequency limit of the accurate representation, in which At is the record interval in ! seconds. The spurious high frequency content of the motion does not significantly affect the calculation of the relative displacement values that have been calculated in the past for the l 1 classically damped systems. The same is, however, not true of the relative velocity calculations. For illustration, let us consider a 100 Hz oscillator with 2% damping subjected to the El Centro earthquake (SOOE,1940). The earthquake record is digitized at 0.02 i intervals, and is capable of representing a maximum frequency of 25 Hz. It is not expected l l to generate a significant 100 Hz response in the oscillator. Figure I shows a 0.5 second , l segment of the relative velocity history of the oscillator, that clearly shows a major 100 Hz ! content (for any 0.1 second duration in the figure, we can count approximately 10 sinusoidal i III/5-4 , I
4 i j l . waves). We were unable to develop a suitable time-domain technique to suppress this j spunous response. t 1 1 i b ! 0.05- -
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-0.10 - -
-0.15 2.0 2.1 2.2 2.3 2.4 2.5 Time (sec)
Figure 1 0.5 Second Segment of Relative Velocity History of an SDOF System,100 Hz, 1 Damping Ratio = 0.02, El Centro (SOOE,1940) To calculate the response of the oscillator accurately in all frequency ranges, therefore, ; we proceeded to perform a frequency domain analysis. The method consists of obtaining the ! Fourier transform of both the input motion and the unit impulse function over an appropriate i range of frequencies, multiplying the two transforms, and performing an inverse Fourier j transform on the product resulting in the response time history. The nonzero initial conditions are handled the same way as in the time-domain analysis. When the unit impulse function is also discretized in the time domain (the input motion is already discretized), two new problems appear in the solution. The Fourier analysis implicitly introduces periodicity in the input and output, the period being equal to the length of the record being processed. 'Ihis results in an "end effect," due to nonzero output at the end of any " period" interfering with the outputs of the subsequent periods. The error is cumulative from period to period and can give very erroneous results. The problem is solved by adding enough zeroes at the end of both the earthquake and the unit impulse data, assuring a zero output at the end of the period, thus avoiding'the end effect. Another problem is " aliasing" that occurs in the high frequency range. When the oscillator frequency is beyond the earthquake record's Nyquist frequency, 111/5-5
the true high frequency content of the unit impulse function cannot be accurately represented. The mathematical process interprets the unit impulse function records in terms of the lower
" alias" frequencies within the Nyquist frequency, thus giving incorrect response values.
To avoid the aliasing problem, we used an alternate procedure in which the unit impulse function is not discretized to evaluate its Fourier transform. Instead, we use the analytical form of the Fourier transform of the unit impulse function, thus avoiding the aliasing problem. To overcome end effect problems in this case, a correction proposed by Veletsos et al (Veletsos and Ventura,1985) is used. Figure 2 shows the velocity response of the 100 Hz oscillator (the seme as the one used before) from the frequency-domain analysis along with that from the time-domain analysis. The frequency-domain analysis has clearly eliminated the spurious 100 Hz output. . IT f a Frequency Domain Time Domain
.g 0.05- -
o i 3 !}
> 11 3
} 0.00
~~
"- /\jm "
fi m
/
JC -
\
[ 1 I I 1
-0.05 - -
-0.10 - -
l
-0.15 !
2.0 2.1 2.2 2.3 2A 2.5 Time (sec) Figure 2 Comparison of 0.5 Second Segment of Relative Velocity History of an SDOF System from Time and Frequency Domain Analyses,100 Hz, Damping Ratio = 0.02, El Centm (SOOE,1940) III/5-6 I J
1 4 j l Correlation Coemeients l The cormlation coefficients EIj, eij, and pgj are used in Equation (8) to obtain the combined response R from the corresponding maximum modal responses RI and RI. These I coefficients depend upon the closeness of the modal frequencies, except in the high frequency i d range, where the correlations e j and eij become practically equal to unity irrespective of the (lack of the) closeness of modal frequencies. Rosenblueth and Elorduy (1969) have proposed an expression for eyj (called the Rosenblueth equation) assuming earthquake motion to be a finite segment of white noise, and 1 d Igusa, Der Kiureghian and Sackman (1984) have given expressions for e E}j, and p,j (called the Der Kiureghian equation) for white noise input of infinite duration. th sets of equations are applicable only when the high frequency effects are not present. We found that Rosenblueth and Der Kiureghian equations gave values of Eyj that were numerically very close when the finite segment length (duration) of the earthquake is assumed to be infinity and the two modal damping values are identical. When a practical earthquake duration,10 - 20 seconds, is substituted in the Rosenblueth equation,it gives correlation values that are too high. Also, the Rosenblueth equation gives efj values that do not agree with the numerical results when the two modal damping values are significantly different. A modified Rosenblueth equation, given below, can be developed by setting the earthquake duration to infinity and by incorporating an expression to account for different modal damping values based on the Der Kiureghian equation: i l l 1 l r v d W~O i j 1 NiNJ (9) c'.' . = bgj , b'.' . = a;j 1+ , a'.' . = m;
- Cjw;s - Ci+C;
( Ci l l According to Igusa and Der Kiureghian, cij - cyj, and: l 2 G-S i i (10) p'. . - {;G m; + m; b'. . l l Equations (9) and (10) are in reasonable agreement with the average numerical data from twelve earthquakes for low modal damping values. Further modification of the equations is needed for the modal damping values that are on the order of 10%. We propose to modify the above equations as follows: III/5 7
I g Q Egj = . O (b-cf),gj g E"j = Q'I (bjj-cj"), j Oj~Cj i i Gj~C]i i f > 1 G-W i j p = b C? al \ NiUi
- NjU j)
/ 5 i d Ni*Nj Ni*Nj v Ni
- Nj ejj = 2.5 5 ,
egj = 0.005 + 0.7 , mj +m s j; 2 2 cj = 0.35 + 0.653, m j2 m; g Ui
= 0.35 + 0.653, m 2 mj ; g I s e,j s 3 (II)
*j I
Figure 3 shows one set of comparisons between the average numerically obtained correlation coefficients and those predicted by Equation (11). 1.0 , , , , , , , ,
'7
, 0.9 - - ,
~
fi = 0.5 Hz = 0.10 f 0.8 - - fj = 0.5-5.0 Hz = 0.20 g 0.7- - 0.6 - -
}0 L. l 5- -
. . Numerical h 04- -
.\J, Modified Rosenblueth i
j \ Proposed i g 0.3 - -
*\ - .
6 0.2 - -
* 'h., -.
0.1 - -
*W~ --
*I ' ' ' ' ' ' '
0.0
~
1 2 3 4 5 6 7 8 910 f/f j i Figure 3a Comparison of Displacement Correladon Coefficients in the Frequency Range 0.5 5.0 Hz, Numerical, Modified Rosenblueth, Proposed. (=0.10, (c=0.20 i 111/5 - 8 l
I l ! l d e i l j f.0 , , , , , , , , } 'c 0.9 - -. i o,g .-y f, = 1 Hz ( = 0.10 _. fj = 1-10 Hz ( = 0.20
- 0.7 - - - .
j g 0.s. 1 ! 0.5 - - . . Numerical -. l ! Der KiuregNan 0.4 - - _ , l j0.s-y Proposed l C 0.2 - - 1 0.1 - - - . 0.0
' m ----- - - T -
' '~'+--'
1 2 3 4 5 6 7 8 910 l f/f 3 i Figure 3b Comparison of Velocity Cormlation Coefficients in the Fmquency Range 1-10 Hz, Numerical, Der Kiureghian, Proposed. (,=0.10, (,=0.20 6 i i i i i i i E 0.0 . . . . . . . . . . . j -0.1 A
^^
g, _ p.- _,
-0.s - , . - -[
0,4 _ '. - . . . Numerical - - D*r KiuregNan 5 -0.5 - - Proposed j .0.6 - - _ , f 0.7-0.8 - - fi = 1 Hz fJ = 1-10 Hz ( = 0.05 0,g - - Q = 0.20 -. 1.0 1 2 3 4 5 6 7 8 910 f/f j i Figure 3c Comparison of Cross Correlation Coefficients in the Fmquency Range 1-10 Hz, Numerical, Der Kiureghian, Proposed, (,-0.05, (,=0.20 111/5-9
I High Frequency Effect - Rigid Response Coefficients To account for the high frequency effect on correlation, modal responses, RI and RJ, can be decomposed into two parts each: the rigid parts, R7 and R7, and the damped periodic parts, R7 and R}P. The following relationships have been proposed by Gupta et al (Gupta, 1992; Gupta and Chen,1984): R[' = afRf, R[P , } _(gf)2 R[ R[' = a[R[, R[F , } _(g )2 R[ (12) in which af and ai are rigid response coefficients, both of which are assumed to vary between 0 and 1. The rigid parts from various modes are combined algebraically, and the damped periodic parts are combined in accordance with Equation (8). R#=ERi #, R " = E R[' i i (RP)2 , EE(gl,glP glP ,g;gjPg]P _ gy;;gfPg]Py i) Finally, the two rigid parts (R7 and R}') and the combined damped periodic part (RP) are assumed to be statistically independent. Therefore, R2 , (g e)2 + (R )2 + (RF)2 (14) Equations (12) through (14) give: R = E E(iifjR fRf + e[jR [RJ -2j;;Rf Rj) (15)
*1 in which the modified correlation coefficients are defined as follows:
Efj = f1-(af)2' 1 -(af)2' efj + afaf E[j = f1-(a[)2' 1 -(aj)2' c[j+a[aj Fij = f1-(af)2' 1 -(aj)2' pgj (16) 1 l l III/5-10 l
Note, in the high frequency range, af, af, a}, aj = 1, and ejj, Qj = 1; and in the low frequency range af, ay, ai, aJ = 0, and eyj = eyj, Qj = eij, gj = pgj. f Gupta and Chen (1984) originally pmposed an approximate equation for ad that can be represented by a straight line on a semi log chart. Such an equation did not account for i the effect of damping on the coefficient. We have modified the original equation for aI and j 1 proposed a similar equation for al. The two sets of modified equations are given below. , l d af = afg + af,(1 -a ) Aaf, g a[ = a", , + al;(1 -ul,) Aa[, d IU([i[1/ ) d Goi " i[1 [i S[2 In(f2/ /[ } v In(f;/fi ) Ooi " *lt,$$i $2$y in(f2"//t" ) (17) d # " " l in whichfj is the modal frequency,dandff ,f2 .ft /2 are key frequencies (all in Hz) yet to be defined. The expressions for a,g and OL define the coefficients without taking into account the effect of damping that is introduced in Equation (17) by including AaI and Ani. The key frequencies,ff d,/2 >ft"> /2"'were studied by performing numerical analysis on twelve , earthquake ground motions. Empirical expressions of these frequencies, and Any and Aa} based on the averages of the twelve earthquakes are given below:
/t = 0.86fj, /2 = 0.84f,# ,
i fi = 1.86fj, f2 #= 1.20f# , d ' f = SA *" 2 x Sy,, Aaf = 2.1 + 0.6(g , Aal = 1.2 + 0.5(; (18) in which f,d is the rigid or the ZPA frequency defined as the lowest frequency at which the (relative displacement-based) spectral acceleration becomes almost equal to the zero period III/5-11
b acceleration (ASCE,1986). We found that the definition is not very precise since it can lead
- to the use of differentf/ values by different engineers for the same earthquake motion.
Therefore, we propose to add the requirement that the spectral acceleration become practically independent the numerically calculated of thecx'damping and cQ values [ values with thoseat the rigid predicted frequency. by Equations (14) and Figure (15) 4 sho for three damping values. Figure 5 shows a comparison of numerically calculated correlation coefficients with those calculated using Equation (16) in the high frequency range.
) i i ,
,,0 - ElCentre
~
0 q a0 - .e. , -. A.*..=
- L as
- /
- e /s'..=,...- .
.<' ya' *
, + .
+
- 1+-
a . $ ,.,, jp .., p,
/.x,., *** .* + *.
as -
' *. .+ . . .c.coiew -.
*. c.e ei m
+g Ec., .. / .+. *.+ . = c.o os w 4
a0 -.
.+* ., . ., .- + . ..* - -
. - - . c.o os % w
. c.omown
'=_.e e t, . . - -
, , , ,c.am ,, , , ,
g . , , f.0 140 Frequency (Hz) Figure da Comparison Between the Numerical of Values with those from the Proposed Model, El Centro (SOOE,1940) b f.0 - Taft ....
$**,*l*
- 5; '
j 8' 0.0 j Os f
/l/ ,
*.sphan ,
,e +
- c. -
p
;[ ,i .. .
..s g" 0.4 -- 4,* -.
,y4,,. * ,. * .
+
OJ -
*p*Ge ,/ * +
. + c o ot Num.new -.
(.4 of Pres d g0
* +* *('** s
. (.o os Num.ne.1 c.c os proo e -.
+
. . c.c a sun.new
' ,,* .* *. ..].. =:,.,..f - . . - c.am p, e u w_ , ,
10.0 100.0 f.0 Frequency (Hz) Figure 4b Comparison Between the Numerical & Values with those from the Proposed Model, , Taft (S69E,1952) J i 111/5 12 1
1 f.0 , , , , , , , , g's 0.9 - -. 18 0.8 - - -. j 0. 7 - - ^^^^^^^^^^^^^^^^^^^^^^^^^^^^--^^^^ _ al-l 0.6 - - f, = 10 Hz = 0.05 _. O f3 100 Hz = 0.20 0.5 - - -. 0.4 - - -. 0.3 - - . . Numerical -. 0.2 - - .. Proposed _, 0.1 - - - . 0.0 1 2 3 4 5 6 7 8 910 f/f j i Figure Sa Comparison of Displacement Correlation Coefficients including the Effect of ad in the Fmquency Range 10-100 Hz. Numerical, Proposed, (,=0.05, (,=0.20 l l l 1.0 , , , , , , , , l l
\c 0.9 - - -.
c 0.8 fi = 10 Hz Q = 0.05 _.
- f3 = 10100 Hz (g = 0.20 I!
==
- 0. 7 - - -.
g 0.6 - _. U 0.5 - - . . . Numerical -. h5 0.4, -
. , * . . . . . . . . . . . . . . . . Proposed
' O.3 - - -.
h 0.2- - -. O. , . - -. 0.0 1 2 3 4 5 6 7 8 9 10 f/f 3 i Figure Sb Comparison of Velocity Cormlation Coefficients including the Effect of n' in the Frequency Range 10100 Hz, Numerical, Proposed, (,=0.05, (,=0.20 III/513
l I Velocity Response Spectrum j As was discussed earlier, two types of response spectra are needed in the analysis of d nonclassically damped systems: the relative displacement spectrum, S , and the relative j velocity spectrum, S". Conventionally, most structures and equipments are assumed to be l i classically dan-d for which only the relative displacement spectrum is needed. Herefore, as a rule, only relative displacement-based design spectra are available. To analyze nonclassically damped systems, we need to be able to estimate a relative velocity spectrum from a relative displacement spectrum. A method proposed by Gupta and Jaw (1986) is , investigated here. i l In the intermediate frequency range, it can be shown that the relative displacement and velocity-bred spectra are almost equal when represented in the same units. In the low and high frequency ranges, the following relationships hold: ft - A = 85" , fy . . b % 2xSj 2mu sma 2xSj 2 m G,,, _ l If the key frequencies (Hz)ft andfy are known, the relative velocity-based spectrum can be calculated from a relative displacement spectrum in the low and high frequency ranges using Equation (19). Gupta and Jaw hr.d given empirical values of these frequencies based on twelve earthquake ground motions. Rey used time-domain analysis that does not give accurate relative velocity values in the high frequency ranges. Therefore, we repeated their work with our frequenci-domain analysis. The new relationships forft andfy are: , ft = fy$, fy$ = 2xSp , fy = 2.68fj = 0.22f , d dd Sg ,, <
/av " d (20) 2xSym ,
There is a small transition zone between the intermediate and the high frequency ranges, the relative velocity spectrum for which can be evaluated using the procedure presented in the references (Gupta,1992; Megahed and Gupta,1992; Gupta and Jaw,1986). Figuit 6 shows a comparison between the estimated and the directly calculated velocity spectrum for the El Centro earthquake (1940, SOOE), which is one of the twelve given in the original report (Megahed and Gupta,1992). In the estimation of the velocity spectrum in Figure 6, actual values of ft andfy for the carthquake were used. III/5-14 I i
l 3 I I 6I6l 6 6 6 6 6 863l 4 6 6 i 6 66Il 3 I e ;_" ~ 1000 r El Centro w > l!
- !l g =u - .
100 r= - r: . 2 Estimated
~
. . Actual - '
10; r j 9 f f fffl t f f f ffffl f I f f ffffl f I"' 1.000 10.000 100.000 Frequency (Hz) Figure 6 Comparison of Estimated and Actual Velocity Spectra, Damping Ratio = 0.05. El Centro (SOOE,1940) Secondary Systems with High Frequency Modes In recent years, several modal synthesis methods have been developed to perform coupled analysis of primary and secondary systems. In these methods, it is assumed that all uncoupled modal properties of the primary and secondary systems are known, which is not practical for systems with large degrees of freedom. Typically, several (often, much less than all) modes each of both the primary and the secondary systems are calculated. It is well s known that in certain complex problems, the effect of higher uncalculated modes can be quite significant. In the analysis of (uncoupled) single systems, this effect is represented in tenns of a residual response vector obtained through a pseudo-static analysis of the system subjected to the missing mass inertia forces. We have developed a method in which we use this type of residual response vector as a fictitious mode, called the residual mode, to represent the response of all the uncalculated higher medes both for primary and secondary systems. In the existing coupled analysis methods, both the primary and secondary degree of freedom are represented relative to the fixed base of the primary system. Thus, the secondary system degrees of freedom include a rigid body motion component that the residual mode vector cannot model - since, by definition, it represents a deformed shape. Therefore, we reformulated the coupled problem such that the secondary system degrees of freedom are defined relative to the primary system degrees of freedom at which the secondary system is connected to the primary system. The new formulation, that at the present is applicable to only singly connected primary-secondary systems, has been implemented in a new version of the CREST program III/5-15
originally developed by Gupta and Jaw (1985). The modified program was used to analyze , five primary.secoAary coupled systems of the type shown in Figure 7. The five coupled systems han identical primary systems, and have secondary systems with identical story masses and with five different story stiffnesses. The frequencies (Hz) of the uncoupled primary system modes are: 8.139,23.944,38.358,50.542,59.789 and 65.561. Uncoupled secondary system frequencies (Hz) are: Case 1,8.155,22.849,33.018; Case 2,12.495, 35.009,50.589; Case 3,13.690,38.358,55.429; Case 4,23.398,65.562,94.758; Case 5, 38.328, 107.302, 155.186. The uncoupled primary system modal damping is 2%, and all the uncoupled secondary systems have 7% modal damping. The frequencies and damping ratios 1 for the five coupled system are given in Table 1. All the coupled systems were subjected to the El Centro canhquake (1940, SOOE). The coupled analysis was performed two ways: considering all the uncoupled modes, and considering only the uncoupled modes up to the rigid frequency and accounting for the remaining modes with the residual mode vector. The two sets of nodal displacements and spring forces given in Table 2 are practically the same. ISI lsI
@ T
@i m @ T C i.
O ls b 7 7 Node Number g @ Element Number i gwwnwerMMMMMMMMe&M Figure 7 Example Coupled System 111/5 - 1 6
P 1 Tr.ble 1 Coupled Frequencies and Damping Ratios for Vanous Cases i l Case Frequendes (Hz) / Damping Ratios (Ye) 1 2 3 4 5 6 7 8 9 l 1 8.132 8.155 22.833 23.966 33.019 38.360 50.548 59.790 65.567 ! 5.338 3.663 2.071 6.932 2.002 6.999 6.997 7.000 6.998 ! 2 8.123 12.506 23.953 35.006 38.367 50.558 50.590 59.790 65.576
- 6.970 2.029 6.994 2.015 6.991 6.973 2.028 7.000 6.996 3 8.124 13.696 23.958 38.356 38.361 50.557 55.434 59.791 65.580 l 6.976 2.024 6.994 2.053 6.955 6.991 2.010 7.000 6.997 4 8.122 23.133 24.210 38.367 50.547 59.787 65.587 65.618 94.771 l 6.993 3.179 5.827 6.998 6.990 6.997 6.788 2.229 2.002' !
5 8.129 23.874 38.292 38.371 50.676 59.794 65.562 107.480 155.200 6.990 6.959 2.707 6.378 6.959 6.999 6.990 2.008 2.000 1 Table 2 Comparison of Nodal Displacements and Spring Forces from the Present Method j l I Nodal Displacements Spring Forces Case Node (inch) Element (kips) All Residual All Residual Modes Modes Modes Modes 1 1 0.4328 0.4322 1 30.180 30.270 1 2 0.7296 0.7299 2 23.850 23.940 3 0.8936 0.8948 3 13.140 13.200 1 0.1113 0.1112 1 5.812 5.809
.2 2 0.1322 0.1321 2 4.230 4.120 3 0.1431 0.1431 3 2.141 2.144 1 0.1051 0.1051 1 5.323 5.322 3 2 0.1231 0.1212 2 3.702 3.699 3 0.1295 0.1295 3 1.898 1.900 1 0.0912 0.0912 1 6.546 6.559 !
4 2 0.0977 0.0977 2 4.673 4.703 3 0.1010 0.1011 3 2.437 2.461 1 0.0843 0.0843 1 4.360 4.353 5 2 0.0860 0.0860 2 2.932 2.922 3 0.0868 0.0868 3 1.474 1.468 III/5-17
f Summary and Conclusions , , The conventional uncoupled analysis of secondary system gives resonse values that . may be one or more orders of magnitude higher than the actual coupled resonse. In recent years, several efficient methc<is have been developed that treat the primary and secondary systems as a single coupled mutli-degree-of-freedom (MDOF) system. It is shown that the original method of analysis in which the modal responses are represented in terms of the relative displacements and velocities of the equivalent SDOF oscillators is mathematically
- identical to a newly pmposed canonical method in which the sine and cosine responses are q used. Since the relative displacements and velocities are physical quantities, it is proposed to continue to use the original method of analysis. 'Ihe corresponding response spectrum method requires definition of relative displacement-based response spectrum, which is also used in the conventionally performed analysis of the classically damped systems and is commonly available, and of a new relative velocity-based spectrum. The modal responses are calculated in terms of these two spectral values. Combination of modal responses requires definition of correlation coefficients efj, eig, and pgj. Investigations related to the new relative
- velocity spectrum and the correlation cocmcients are reported in the paper.
1 To numerically evaluate the spectrum and the correlation coefficients for a given ' eamquake, we need to calculate the response of SDOF systems subjected to ground motion. . It is shown that the time-domain analysis introduces spurious inputs in the calculation of the i relative velocity of an SDOF system having frequencies beyond the Nyquist frequency of the earthquake record. An ' appropriate frequency-domain analysis eliminates the problem. i The Rosenblueth equation for EIj is first modified to be in better agreement with the Der Kiureghian equation by eliminating the earthquake duration term and by including a term l to account for different modal damping values, thus obtaining a better agreement with the average numerical values from the twelve actual earthquakes. These equations are further 4 modified to improve agreement between the equation and numerical values, especially for l' systems with relatively higher damping values, in the order of 10%. New equations are also proposed for Eij and pgj. The high frequency effects in modal combination are incorporated using the rigid ! response coefficients af and ai. A previously proposed equation for aI, that gives a straight l line variation with respect to the modal frequencyfg on a semi-log chart and is independent of , the modal damping value, has been modified to include a damping dependent term j introducing a parabolic representation of the coefficient. A similar equation is proposed for I
- ai. Empirical equations are obtained for the key frequenciesf/,f/,f/ andf/ that are needed in the evaluation of the rigid response coefficients.
In the present seismic analysis practice, only relative displacement-based design spectra are given. For the analysis of the nonclassically damped systems, therefore, one i needs to estimate the relative velocity spectrum from a given relative displacement spectrum. I An existing method of making such estimations that is based on two key frequencies ft andfu is reviewed. The method gives estimates of the relative velocity spectra that are in good . l 1 III/5-18 , l
1 f agreement with the actual spectra, provided that the key frequency,fy, is recalculated based on the new frequency-domain analysis. ; ~ Fmally, results of the coupled analysis from a new formulation are presented. It is shown that the effect of uncalculated uncoupled primary and secondary system modes having frequencies beyond the rigid frequency can be accounted for by introducing residual mode vectors both for prunary and secondary systems. Acknowledgments This research was partially supported under the Research Program on Nuclear Power Plants, Structures, Equipment and Piping in the Civil Engineering Department of North Carolina State University. Present contributing companies are: AECL Technologies, Baltimore Gas and Electric Company, Carolina Power and Light Company, Consumers Power l and Light Company, Duke Power Company, Florida Power and Light Company, Niagara j Mohawk Power Corporation, South Carolina Electric and Gas Company, and Virginia Power. l
- Authors (and not the contributing companies) are solely responsible for the material presented l in the paper.
1 References !
- 1. American Society of Civil Engineers, Standardfor the Seismic Analysis of Sqfety- ;
Related Nuclear Structures, September 1986. , 1 i
- 2. K.A. Foss, " Coordinates Which Uncouple the Equations of Motion and Damped Linear Systems," Journal of Applied Mechanics, 25:361-364, September 1958.
- 3. A.K. Gupta, Response Spectrum Method in Seismic Analysis and Design of Structures, Blackwell Scientific Publications, Boston, MA,1990; CRC Press, Inc., Boca Raton, FL, 1992.
- 4. A.K. Gupta and D.C. Chen, "A Simple Method of Combining Modal Responses,"
Transactions, Seventh International Conference on Structural Mechanics in Reactor Technology, Paper No. K3/10, Chicago, August 1983.
- 5. A.K. Gupta and J.W. Jaw, " CREST, A Computer Program for Coupled Response Spectrum Analysis of Secondary System, User's Manual, Department of Civil Engineering, North Carolina State University, Raleigh, NC, June 1985.
- 6. A.K. Gupta and J.W. Jaw, " Response Spectrum Method for Nonclassically Damped
- Systems," Nuclear Engineering and Design,91:161-169,1986.
- 7. T. Igusa, A. Der Kiureghian and J.L. Sackman, " Modal Decomposition Method for Stationary Response of Nonclassically Damped Systems," Eanhquake Engineering and Structural Dynamics, 12:121-136, 1984.
III/5-19
- 8. H.A. Megahed and A.K. Gupta, " Topics in Seismic Response of Nonclassically Damped Systems," Report, Research Program on Nuclear Power Plant Structun:s, Equipment and Piping, Department of Civil Engineering, North Carolina State University, August 1992.
- 9. N.C. Nigam and P.C. Jennings, " Calculations of Response Spectra from Strong Motion Earthquake Records," Bulletin of the Seismological Society of America,104(EM2), April 1969.
- 10. D.A. Pecknold and R. Riddell, "Effect of Initial Base Motion on Response Spectra,"
I wnal of the Engineering Mechanics, ASCE,104(EM2):485-491, April 1978. I 1. d. Rosenblueth and J. Elorduy, " Response of Linear Systems in Certain Transient Disturbances," Proceedings,4th World Conference on Earthquake Engineering, Volume A-1, Santiago, Chile,1969.
- 12. A.S. Veletsos and C.E. Ventura, " Dynamic Analysis of Structures by the DFT," Journal of Engineering Mechanics, ASCE, 111:2625-2642, 1985.
- 13. J.N. Yang, S. Sarkani, and F.X. Ieng, " Modal Analysis of Nonclassically Damped Structural Systems using Canonical Transformation," Technical Report NCEER-87-0019, George Washington University, September 1987.
111/5 - 2 0
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/ //////////////////////////fo Coupled System Figure 1: Primary, Secondary and Coupled Systems, Case 1 2
l i Table 1: Frequencies and Damping Ratios - Case 1 ; 6-DOF Primary System 4-DOF Secondary System Mode Freq. Damping Mode Freq. Damping i No. (Hz.) Ratio No. (Hz.) Ratio 1 2.10148 0.07 1 9.83625 0.02 2 6.18237 0.07 2 18.71025 0.02 3 9.90389 0.07 3 25.75100 0.02 4 13.04991 0.07 4 30.27286 0.02 5 15.43755 0.07 6 16.92772 0.07 Table 2: Comparison of Nodal Displacements (inches) for Secondary System - Case 1 CREST / PIPESTRESS Node Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 1.281 1.277 1.289 2 1.498 1.493 1.498 3 1.688 1.682 1.682 4 1.849 1.842 1.838 l l 1 Table 3: Comparison of Spring Forces (kips) for Secondary System - Case 1 CREST / PIPESTRESS Element Including all Truncated modes; TIME HISTORY no, modes Including missing mass 1 240.4 239.7 239.9 2 217.6 217.0 214.8 i 3 191.0 190.3 185.9 ! 161.3 160.6 155.6 I 4 5 128.8 128.2 124.1 l l
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01.dat 1/1 IDEN JBa1111 1U=1'00=1 PLs/PIPESTRESS INPUT FILE / TITL SU=1 CV=2 T2=/ CASE-1, 4-DOF PROBLEM / FRTQ FR=33 LOm1 MX=4 tim / INCLUDING ALL MODES / RCAS CA=1 EV=1 TY=1 SU=3 LO=1 FX=1 FY=1 FZs1 RSEC CA=2 EV=1 SU=1 FX=1 FY=1 FZal SPEC EV=1 NE=1 FP=0 SH=0 ' LVal DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0. DI=Z 1.0/1.0 50.0/1.0 LV=2 DX=1 DY=1 DZ=1 ! i DI=X 1.0/1.0 50.0/1.0. DI=Y 1.0/1.0 -50.0/1.0 l dis 2 1.0/1.0 50.0/1.0 LVm3 DX=1 DY=1 DZ=1 ) DIsX l 1.0/1.0 50.0/1.0 DI=Y l 1.0/1.0 50.0/1.0 l DI=2 1.0/1.0 50.0/1.0 LV=4 DX=1 DY=1 DZul DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 I DI=2 1.0/1.0 50.0/1.0 LV=5 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y
-1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=6 DX=1 DY=1 DZ=1 DIsX 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 MATL CD=3 BC=28.0 SC=75 SH=75 KL=1 ANCH PT=1 LV=1 SPRS PT=2 DX=1.0 AZ=1000.0 LUMP PT=2 MA=38.64 RSUP PT=2 DY=1 LV=2 RSUP PT=2 DZ=1 LV=2 ROTR PT=2 RX=1 ROTR PT=2 RY=1 ROTR PT=2 RZs1 i
SPRS PT=3 DX=1.0 AZ=1000.0 LUMP PT=3 MA=38.64 RSUP PT=3 DY=1 LVs3 RSUP PT=3 DZal LVs3 ROTR PT=3 RX=1 ROTR PT=3 RY=1 ROTR PT=3 RZs1 SPRS PT=4 DX=1.0 AZ=1000,0 LUMP PT=4 MA=38.64 RSUP PT=4 DY=1 LV=4 RSUP PT=4 DZ=1 LVs4 ROTR PT=4 RX=1 ROTR PT=4 RY=1 ROTR PT=4 RZ=1 SPRS PT=5 DX=1.0 AI=1000.0 LUMP PT=5 MA=38.64 RSUP PT=5 DY=1 LV=5 RSUP PT=5 DZal LVab ROTR PT=5 RX=1 ROTR PT=5 RY=1 ROTR PT=5 RZ=1 SPRS PT=6 DX=1.0 AZs1000.0 ANCH PT=6 LV=6 ENDP
031.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-1, WITH ALL THE S.S. MODES 6 2 2 6 4 1 0 11 1 1 0 0 1 900 10 10 1.0E-6 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 0.07 0.02 0.02 0.02 0.02 2 6 1 16 100000000. 0.0 0.0 100000000. 16 2.10148200 6.18237000 9.9038940 13.0499100 15.4375500 16.9277200
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cir.dat 1/1 IDEN JB=1111 IUm1 00=1 PL=/PIPESTRESS INPUT FILE / TITL SU=1 CV=2 TI=/ CASE-1. 4-DOF PROBLEM / FREQ FR=33 Lo=1 MX=2 TI=/ TRUNCATED MODES / RCAS CA=1 Eval TY=1 SU=3 LO=0 FX=1 TY=1 FZal RSEC CA=2 Eval sus 1 FXal TYm1 FZal SPLC RV=1 ME=1 FP=0 SH=0 LVal DX=1 DY=1 DZs1 DI=X 1.0/1.0 50.0/1.0' DImY 1.0/1.0 50.0/1.0 dim 2
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1 I c:alr.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-1, TRUNCATED MODES OF S.S. 6 2 2 6 2 1 0 11 1 1 0 1 1 900 10 10 1.0E-6 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 0.07 0.02 0.02 , 2 6 1 16. 1 100000000. 0.0 0.0- 100000000. 16 2.10148200 6.18237000 9.9038940 13.0499100 15.4375500 16.9277200
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Coupled System Figure 2: Primary, Secondary and Coupled Systems, Case 2 I
1 l Table 4: Frequencies and Damping Ratios - Case 2 l 6-DOF Primary System 5-DOF Secondary System Mode Freq. Damping Mode Freq. Damping No. (Hz.) Ratio No. (Hz.) Ratio 1 5.83144 0.07 1 5.82555 0.02 2 17.15531 0.07 2 11.25400 0.02 1 l 3 27.48288 0.07 3 15.91549 0.02 - 4 36.21252 0.07 4 19.49171 0.02 l 5 42.83814 0.07 5 21.74056 0.02 ) 6 46.97300 0.07 1 Table 5: Comparison of Nodal Displacements (inches) for Secondary System - Case 2 CREST / PIPESTRESS l Node Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 0.3875 0.3875 0.3874 i 1 2 0.6098 0.6096 0.6097 3 0.7038 0.7035 0.7037 4 0.6497 0.6497 0.6496 5 0.4659 0.4657 0.4658 ) Table 6: Comparison of Spring Forces (kips) for Secondary System - Case 2 CREST / PIPESTRESS Element Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 154.0 154.0 154.0 2 112.9 112.9 112.9 3 48.04 47.98 48.02 4 33.6 33.61 33.61 5 100.2 100.1 100.2 6 146.2 146.1 146.2
I i c2.dat 1/2 I IDEN JB=1111 IU=1.OU=1 PL=/PIPESTRESS INPUT FILE / I TITL sus 1 CVa2 TI=/ CASE-2, 5-DOF PROBLEM / I FRZQ FR=33 LO=1 MX=5 TI=/ INCLUDING ALL MODES / RCAS CA=1 EV=1 TY=1 SU=3 Lo=1 FX=1 FY=1 FZ=1 RSEC CA=2 EV=1 SU=1 FX=1 FY=1 FZ=1 i SPEC Eval HE=1 FP=0 SH=0 LV=1 DX=1 DY=1 DZ=1 l DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 ) DI=Z 1.0/1.0 50.0/1.0 LV=2 DX=1 DY=1 DZ=1 1 DI=X 1 1.0/1.0 50.0/1.0 i DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=3 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0- 50.0/1.0 LV=4 DX=1 DY=1 DZ=1 1 DI=X ' l 1.0/1.0 50.0/1.0 ' DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 1 LV=5 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 ) DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 ! LV=6 DX=1 DY=1 DZul. I DI=X l 1.0/1.0 50.0/1.0 DI=Y ! 1.0/1.0 50.0/1.0 l DI=Z 1.0/1.0 50.0/1.0 LV=7 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 I DI=Y. 1.0/1.0 50.0/1.0 DIsZ 1.0/1.0 50.0/1.0
- MATL CD=3 EC=28.0 SC=75 SH=75 KL=1 ANCH PT=1 LV=1 !
SPRS PT=2 DX=1.0 AZ=500.0 l LUMP PT=2 MA=38.64 RSUP PT=2 DY=1 LV=2 RSUP PT=2 DZal LV=2 ROTR PT=2 RX=1 ROTR PT=2 RY=1 ROTR PT=2 RZ=1 SPRS PT=3 DX=1.0 AZ=500.0 LUMP PT=3 MA=38.64 RSUP PT=3 DY=1 LV=3 RSUP PT=3 DZ=1 LV=3 RCTR PT=3 RX=1 ROTR PT=3 RY=1 ROTR PT=3 RZ=1 I SPRS PT=4 DX=1.0 AZ=500.0 LUMP PT=4 MA=38.64 RSUP PT=4 DY=1 LV=4 RSUP PT=4 DZ=1 LV=4 ROTR PT=4 RX=1 ROTR PT=4 RY=1 1
c2.dat 2/2 i i' ROTR PT=4 RZal l SPRS PT=5 DX=1.0 AZ=500.0 ! LUMP PT=5 MA=38.64 l RSUP PT=5 DY=1 LV=5 l RSUP PT=5 DZ=1 LV=5 i ROTR PT=5 RX=1 l l ROTR PT=5 RY=1 i ROTR PT=5 RZ=1 1 SPRS PT=6 DX=1.0 AZ=500.0 l l LUMP PT=6 MA=38.64 l RSUP PT=6 DY=1 LV=6 RSUP PT=6 DZal LVm6 RCfrR PT=6 RX=1 ROTR PT=6 RY=1 ROTR PT=6 RZ=1 l SPRS PT=7 DX=1.0 AZ=500.0 .I ANCH PT=7 LV=7 . ENDP l l i l l l l l f i
c32.dat 1/1 l CREST / PIPESTRESS RUN FOR CASE-2, WITH ALL THE S.S. MODES 6 2 2 6 5 1 0 11 1 0 0 0 1 1 900 11 11 1.0E-6 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 0.07 0.02 0.02 0.02 0.02 0.02 2 6 1 19 1 100000000. 0.0 0.0 100000000. 17 5.83144000 17.1553100 27.4828800 36.2125200 42.8381400 46.9730000 l -0.25778 -0.55066 0.36783 0.13275 0.51865 0.45651 -0.55066 0.51865 0.45651 0.36783 -0.25778 0.13275
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c2r.dat 1/2 IDEN JB=1111 IUm1 0U=1 PL=/PIPESTRESS INPUT FILE / TITL sus 1 CV=2 TI=/ CASE-2, 5-DOF PROBLEM / FREQ FR=33 LO=1 MX=3 TI=/ TRUNCATED MODES / RCAS CA=1 EV=1 TY=1 SU=3 LO=0 FX=1 FY=1 FZ=1 RSEC CA=2 EV=1 SU=1 FX=1 FY=1 FZal SPEC Eval ME=1 FP=0 SH=0 LV=1 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=2 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0
- i LV=3 DX=1 DY=1 DZ=1 I DI=X l 1.0/1.0 50.0/1.0 '
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LUMP PT=5 MA=38.64 ' RSUP PT=5 DY=1 LV=5-RSUP PT=5 DZ-1 LV=5 ROTR PT=5 RX=1 ROTR PT=5 RY=1 ROTR PT=5 RZ=1 SPRS PT=6 DX=1.0 AZ=500.0 LUMP PT=6 MA=38.64 RSUP PT=6 DYsi LV=6 ! RSUP PT=6 DZal LV=6 ROTR PT=6 RX=1 ROTR Pr=6 RY=1 - , ROTR PT=6 RZal . SPRS PT=7 DX=1.0 AZ=500.0 ANCH PT=7 LV=7 ENDP l l 4 1 4 l i i 1 I
c22r.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-2, TRUNCATED MODES OF S.S. 6 2 2 6 3 1 0 11 1 0 0. 1 1 900 1.0E-6 11 11 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 C.07 0.02 0.02 0.02 3 6 1 19-100000000. 0.0 0.0 10C000000. 1-7 5.83144000=17.1553100 27.4828800 36.2125200 42.8381400 46.9730000
-0.25778- -0.55066 0.36783 0.13275 0.51865 0.45651
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l Table 7: Frequencies and Damping Ratios - Case 3 6-DOF Primary System 6-DOF Secondary System Mode Freq. Damping Mode Freq. Damping No. (Hz.) Ratio No. (Hz.) Ratio .) 1 9.85694 0.07 1 14.16606 0.02 -l 2 28.99800 0.07 2 27.62134 0.02 ) 3 46.45145 0.07 3 39.69324 0.02 4 61.20940 0.07 4 49.77253 0.02 i 5 72.40913 0.07 5 57.35785 0.02 6 79.39922 0.07 6 62.06565 0.02 1 Table 8: Comparison of Nodal Displacements (inches) for Secondary System - Case 3
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CREST./ PIPESTRESS Node Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 0.07188 0.07182 0.07186 2 0.0944 0.0944 0.09435 3 0.1097 0.1097 0.1097 4 0.1168 0.1167 0.1167 , 5 0.1152 0.1151 0.1152 ! 6 0.1053 0.1052 0.1053 l l 1 l Table 9: Comparison of Spring Forces (kips) for Secondary System Case - 3 CREST / PIPESTRESS i Element Including all Truncated modes; TIME HISTORY l no. modes Including missing mass ( 1 112.9 112.9 112.9 ) 2 90.40 90.46 90.59 l 3 61.62 61.58 61.63 l 4 28.92 28.86 28.92 5 8.58 8.51 8.58 6 40.50 40.67 40.50 l 7 71.17 71.26 71.17 I
03.dat- 1/1 IDEN JB=1111 IU=100=1 PL=/PIPESTRESS INPUT FILE / TITL.SU=1 CVs2 tis / CASE-3. 6-DOF PROBLEM / FREQ FR=65 LO=1 MX=6 tim / INCLUDING ALL MODES / RCAS CA=1 Eval TY=1 SU=3 Losi FX=1 FY=1 FZe1 RSEC CA=2 Eval SU=1 FX=1 FY=1 FZ=1 SPEC Eval NE=1 FP=0 SHs0 LV=1 DX=1 DY=1 DZs1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 dis 1 1.0/1.0 50.0/1.0 LV=2 DX=1 DY=1 DZal DIsX - 1.0/1.0 50.0/1.0
.DIsY 1.0/1.0 50.0/1.0 DIsZ 1.0/1.0 50.0/1.0 LV=3 DX=1 DY=1 DZ=1 DI=X 1.0/1.0-50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=3 1.0/1.0 50.0/1.0 LV=4 DX=1 DY=1 DZs1 DIsX 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=5 DX=1 DY=1 DZ=1 DIsX 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 dim 2 1.0/1.0 50.0/1.0 LVs6 DX=1 DY=1 DZ=1 DIsX 1.0/1.0 50.0/1.0 DI=Y
. 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LVs7 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 1.0/1.0 50.0/1.0 DIsZ 1.0/1.0 50.0/1.0 LV=8 DX=1 DY=1 DZe1 DI=X-1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0
'DI=2 1.0/1.0 50.0/1.0 ,
MATL CD=3 EC=28.0 SC=75 SH=75 KL=1 I ANCH PT=1 LV=1 SPRS PT=2 DX=1.0 AZ=4000.0 1 LUMP PT=2 MA=38.64 I MULR PT=2 DY=1 DZe1 RX=1 RY=1 RZ=1 LVs2 SPRS PT=3 DX=1.0 AZ=4000,0 s LUMP PT=3 MA=38.64 MULR PT=3 DYa1 DZ=1 RX=1 RYsi RZal LV=3 SPRS PT=4 DX=1.0 AZ=4000,0 LUMP PT=4 MA=38.64 NULR PT=4 DY=1 DZs1 RX=1 RY=1 RZ=1 LVs4 SPRS PT=$ DX=1.0 AZa4000.0 LUMP PT=5 MA=38.64 MULR PT=5 DY=1 DZul RX=1 RY=1 RZ=1 LV=5 SPRS PT=6 DX=1.0 AZ=4000.0 LUMP PT=6 MA=38.64 i MULR PT=6 Dv=1 DZ=1 RX=1 RY=1 RZ=1 LV=6 l SPRS PT=7 DX=1.0 AZ=4000.0 i LUMP PT=7 MA=38.64 KULR PT=7 DY=1 DZ=1 RX=1 RY=1 RZe1 LV=7 SPRS PT=8 DX=1.0 AZ=4000.0 ANCH PT=8 LVs8 i ENDP l 1
C33.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-3, WITH ALL THE S.S. MODES 6 2 2 6 6 1 0 11 1 1 0 0 1 900 11 10 1.0E-6 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 0.07 0.02 0.02 0.02 0.02 0.02 0.02 I' 2 6 1 22 l 100000000. 0.0 1 0.0 100000000, i 18 ' 9.85694300 28.9980000 46.4514500 61.2094000 72.4091330 79.3992200 0.36456 -0.77874 0.52020 -0.18773 -0.73349 -0.64560 1 0.77874 0.73349 0.64560 -0.52020 0.36456 -0.18773 ' O.1615E+01 -0.5171E+00 0.2841E+00 0.1737E+00 0.1029E+00 0.4834E-01 l 10 0.020151 ' 8.4050.712511.6820.682818.4690.517425.8030.443029.4140.390330.4910.4236 46.6630.348861.2760.348272.5310.348679.7720.3498 10 0.021093 l 8.4050.710011.6820.680818.4690.512125.8030.439929.4140.388830.4910.4216 i 46.6630.348861.2760.348372.5310.348779.7720.3498 10 0.029183 8.4050.680111.6820.656518.4690.464425.8030.409029.4140.374930.4910.4024 46.6630.348561.2760.348572.5310.349079.7720.3498 10 0.037682 i 8.4050.649511.6820.629618.4690.431225.8030.384429.4140.364330.4910.3875 l 46.6630.348561.2760.348572.5310.349179.7720.3498 1 10 0.047921 8.4050.626511.6820.598718.4690.415225.8030.363829.4140.358130.4910.3749 46.6630.348661.2760.348672.5310.349279.7720.3497 1 10 0.058319 1 8.4050.603711.6820.570318.4690.406325.8030.360629.4140.354530.4910.3660 ' 46.6630.348761.2760.348772.5310.349279.7720.3496 l 10 0.069036 I 8.4050.580211.6820.544418.4690.402125.8030.357629.4140.353230.4910.3594 46.6630.348861.2760.348772.5310.349279.7720.3495 10 0.069702 8.4050.578811.6820.543018.4690.402225.8030.357429.4140.353230.4910.3590 46.6630.348861.2760.348772.5310.349279.7720.3495 10 0.069721 . 8.4050.578811.6820.542918.4690.402225.8030.357429.4140.353230.4910.3590 l 46.6630.348861.2760.348772.5310.349279.7720.3495 , 10 0.069726 ' 8.4050.578711.6820.542918.4690.402225.8030.357429.4140.353230.4910.3590 l 46.6630.348861.2760.348772.5310.349279.7720.3495 l 10 0.070000 8.4050.578211.6820.542318.4690.402225.8030.357429.4140.353130.4910.3589 46.6630.348861.2760.348772.5310.349279.7720.3495 l l l o
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03r.dat 1/1 IDEN JB=1111 IU=1 00=1 PLs/PIPESTRESS INPUT FILE / TITL SU=1 CV=2 TI=/ CASE-3, 6-DOF PROBLEM /. FREQ FR=65 LO=1 MX=4 TI=/ INCLUDING ALL MODES / RCAS CA=1 EV=1 Ty=1 sum 3 LO=0 FX=1 FY=1 FZal RSEC CA=2 EV=1 sus 1 FX=1 FY=1 FZ=1 SPEC Eval ME=1 FP=0 SH=0 LV=1 DX=1 DY=1 DZs1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 - 50.0/1.0 DIsZ-1.0/1.0 50.0/1.0 LVs2 DX=1 DYa1 DZal DIsX 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=3 DX=1 DY=1 DZs1 DIaX 1.0/1 0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0
- LV=4 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=5 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 -50.0/1.0 i DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=6 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 dis 2 1.0/1.0 50.0/1.0 LVs7 DX=1 DY=1 DZal DI=X 1.0/1.0 50.0/1.0 DI=Y 1 1.0/1.0 50.0/1.0 l DI=Z I 1.0/1.0 50.0/1.0 I LVm8 DX=1 DY=1 DZal '
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- 10. 0.029183 8.4050.680111.6820.656518.4690.464425.8030.409029.4140.374930.4910.4024 46.6630.348561.2760.348572.5310.349079.7720.3498 10 0.037682 8.4050.649511.6820.629618.4690.431225.8030.384429.4140.364330.4910.3875 46.6630.348561.2760.348572.5310.349179.7720.3498 10 0.047921 8.4050.626511.6820.598718,4690.415225.8030.363829.4140.358130.4910.3749 46.6630.348661.2760.348672.5310.349279.7720.3497 10 0.058319 8.4050.603711.6820.570318.4690.406325.8030.360629.4140.354530.4910.3660 46.6630.348761.2760.348772.5310.349279.7720.3496 10 0.069036
' 8.4050.580211.6820.544418.4690.402125.8030.357629.4140.353230.4910.3594 46.6630.348861.2760.348772.5310.349279.7720.3495 10 0.069702 ! 8.4050.578811.6820.543018.4690.402225.8030.357429.4140.353230.4910.3590 l 46.6630.348861.2760.348772.5310.349279.7720.3495 t 10 0.069721 8.4050.578811.6820.542918.4690.402225.8030.357429.4140.353230.4910.3590 46.6630.348861.2760.348772.5310.349279.7720.3495 I 10 0.069726
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Table 10: Frequencies and Damping Ratios - Case 4 6-DOF Primary System 4-DOF Secondary System Mode Freq. Damping Mode Freq. Damping No. (Hz.) Ratio No. (Hz.) Ratio 1 7.25094 0.07 1 9.83625 0.02 2 21.33154 0.07 2 18.71025 0.02 3 34.17216 0.07 3 25.75100 0.02 4 45.02652 0.07 4- 30.27286 0.02 5 53.31372 0.07 6 58.40668 0.07 i Table 11: Comparison of Nodal Displacements (inches) for Secondary System - Case 4 CREST / PIPESTRESS Node Including all Truncated modes; TIME HISTORY , no. modes Including missing mass 1 0.1380 0.1377 0.1380 2 0.1878 0.1878 0.1878 3 0.2032 0.2031 0.2032 4 0.1821 0.1816 0.1822 Table 12: Comparison of Spring Forces (kips) for Secondary System - Case 4 CREST / PIPESTRESS Element including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 76.41 76.41 76.42 2 50.75 51.03 50.81 3 16.55 16.51 16.51 4 24.35 24.79 24.52 5 56.86 56.92 56.78 I
l. C4.dat 1/1 IDDI Ca1111 IU=1'OU=1 PL=/PIPESTRESS INPUT FILE / TITL SU21 CV=2 TI=/ CASE-4, 4-DOF PROBLEM /. I FREQ FRs33 L0=1 MX=4 tis / INCLUDING ALL MODES / l RCAS Caul EV=1 TY=1 sus 3 L0=1 FX=1 FY=1 FZ=1 l ! RSEC CAa2 EV=1 SU=1 FX=1 FY=1 FZ=1 ' SPEC Eval ME=1 FP=0 SH=0 l LVal DX=1 DY=1 DZ=1 DI=X 4 1.0/1.0 50.0/1.0 l DI=Y 1.0/1.0 50.0/1.0
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LUMP PT=3 MA=38.64 ' RSUP PT=3 DY=1 LV=3 RSUP PT=3 DZ=1 LV=3 ROTR PT=3 RX=1 ROTR PT=3 RY=1 ROTR PT=3 RZ=1 SPRS PT=4 DX=1.0 AZ=1000.0 LUMP PT=4 MA=38.64 RSUP PT=4 DYa1 LV=4 RSUP PT=4 DZs1 LV=4 ROTR PT=4 RXal ROTR PT=4 RY=1 ROTR PT=4 RZal SPRS PT=5 DX=1.0 AZ=1000.0 LUMP PT=5 MA=38.64 RSUP PT=5 DY=1 LV=5 RSUP PT=5 DZ=1 LV=5 ROTR PT=5 RX=1 ROTR PT=5 RY=1 ROTR PT=5 RZ=1 SPRS PT=6 DX=1.0 AZ=1000.0 ANCH PT=6 LV=6 ENDP
t CC4.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-4, WITH ALL THE S.S.' MODES 6 2 2 6 4 1 0 11 1 1 0 0 1 900 11 10 1.0E-6 . 20.5 386.4 0.10
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Coupled System Figure 5: Primary, Secondary and Coupled Systems, Case 5
Table 13: Frequencies and Damping Ratios - Case 5 6-DOf Primary System 6-DOF Secondary System Mode Freq. Damping Mode Freq. Damping No. (Hz.) Ratio No. (Hz.) Ratio 1 9.85694 0.07 1 9.84007 0.02 2 28.99800 0.07 2 19.18613 0.02 3 46.451455 0.07 3 27.57201 0.02 4 61.20940 0.07 4 34.57323 0.02 5 72.40913 0.07 5 39.84126 0.02 6 79.39922 0.07 6 43.11189 0.02 Table 14: Comparison of Nodal Displacements (inches) for Secondary System - Case 5 CREST / PIPESTRESS Node Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 0.09631 0.09636 0.0963 2 0.1435 0.1436 0.1435 3 0.1729 0.1729 0.1729 4 0.1803 0.1804 0.1803 5 0.1651 0.1652 0.1651 6 0.1303 0.1304 0.1303 Table 15: Comparison of Spring Forces (kips) for Secondary System - Case 5 CREST / PIPESTRESS Element Including all Truncated modes; TIME HISTORY no. modes Including missing mass 1 117.3 117.4 117.3 2 93.56 93.56 93.56 3 57.57 57.57 57.58 4 16.18 16.18 16.18 5 35.15 35.14 35.15 6 75.41 75.51 75.41 7 106.0 106.0 106.0
c5.dat 1/1 IDEN JB=1111 IUs1 00=1 PL=/PIPESTRESS INPUT FILE / TITL SU=1 CV=2 TI=/ CASE-5 6-DOF PROBLEM / FREQ FR=45 LO=1'MX=6 tis / INCLUDING ALL MODES / RCAS CA=1 EV=1 TY=1 sus 3 LO=1 FX=1 FY=1 FZ=1 RSEC CA=2 EV=1 SU=1 FX=1 FY=1 FZul SPEC Eval ME=1 FP=0 SH=0 LVal DX=1 DY=1 DZel-DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=2 1.0/1.0 50.0/1.0 LV=2 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=3 DX=1 DY=1 DZal DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=2 1.0/1.0 50.0/1.0 LV=4 DX=1 DY=1 DZal DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 dis 2 1.0/1.0 50.0/1.0 LV=5 DX=1 DY=1 DZal DI=X i 1.0/1.0 50.0/1.0 i DI=Y 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=6 DX=1 DY=1 DZ=1 DI=X , 1.0/1.0 50.0/1.0 DIsY 1.0/1.0 50.0/1.0 DI=Z 1.0/1.0 50.0/1.0 LV=7 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y 1.0/1.0 50.0/1.0 DI=2 1.0/1.0 50.0/1.0 LV=8 DX=1 DY=1 DZ=1 DI=X 1.0/1.0 50.0/1.0 DI=Y l 1.0/1.0 50.0/1.0 l DI=2 1.0/1.0 50.0/1.0 MATL CD=3 EC=28.0 SC=75 SH=75 KL=1 ANCH PT=1 LV=1 CPRS PT=2 DX=1.0 AZ=1930.0 LUMP PT=2 MA=38.64 MULR PT=2 DY=1 DZ=1 RX=1 RY=1 RZ=1 LV=2 SPRS PT=3 DX=1.0 AZs1930.0 LUMP PT=3 MA=38.64 MULR PT=3 DY=1 DZ=1 RX=1 RY=1 KZal LV=3 SPRS PT=4 DX=1.0 AZm1930.0 LUMP PT=4 MA=38.64 MULR PT=4 DY=1 DZal RX=1 RY=1 RZs1 LV=4 l SPRS PT=5 DX=1.0 AZ=1930.0 LUMP PT=5 MA=38.64 l MULR PT=5 DY=1 DZs1 RX=1 RY=1 RZs1 LV=5 ( SPRS PT=6 DX=1.0 AZ=1930.0 LUMP PT=6 MA=38.64 MULR PT=6 DY=1 DZ=1 RX=1 RY=1 RZ=1 LVs6 SPRS PT=7 DX=1.0 AZ=1930.0 LUMP PT=7 MA=38.64 MULR PT=1 DY=1 DZ=1 RX=1 RY=1 RZ=1 LV=7 SPRS PT=8 DX=1.0 AZ=1930.0 ANCK PT=8 LV=8 ENDP
cc5.dat 1/1 CREST / PIPESTRESS RUN FOR CASE-5, WITH ALL THE S.S. MODES 6 2 2 6 6 1 0 11 1 1 0 0 1 900 11 10 1.0E-6 20.5 386.4 0.10 0.07 0.07 0.07 0.07 0.07 0.07 0.02 0.02 0.02 0.02 0.02 0.02 2 6 1 22 100000000. 0.0 0.0 100000000. 18 9.85694300 28.9980000 46.4514500 61.2094000 72.4091330 79.!492200 0.36456 -0.77874 0.52020 -0.18773 -0.73349 -0.64560 0.77874 0.73349 0.64560 -0.52020 0.36456 -0.18773 0.1615E+01 -0.5171E+00 0.2841E+00 0.1737E+00 0.1029E+00 0.4834E-01 10 0.019999 8.4050.712511.6820.682818.4690.517425.8030.443029.4140.390330.4910.4236 46.6630.348861.2760.348272.5310.348679'.5720.3498 10 0.021093 - 8.4050.710011.6820.680818.4690.512125.8030.439929.4140.388830.4910.4216 46.6630.348861.2760.348372.5310.348779.5720.3498 10 0.029183 8.4050.680111.6820.656518.4690.464425.8030.409029.4140.374930.4910.4024 46.6630.348561.2760.348572.5310.349079.5720.3498 10 0.037682 8.4050.649511.6820.629618.4690.431225.8030.384429.4140.364330.4910.3875 46.6630.348561.2760.348572.5310.349179.5720.3498 10 0.047921 8.4050.626511.6820.598718.4690.415225.8030.363829.4140.358130.4910.3749 46.6630.348661.2760.348672.5310.349279.5720.3497 10 0.058319 8.4050.603711.6820.570318.4690.406325.8030.360629.4140.354530.4910.3660 46.6630.348761.2760.348772.5310.349279.5720.3496 10 0.069036 8.4050.580211.6820.544418.4690.402125.8030.357629.4140.353230.4910.3594 46.6630.348861.2760.348772.5310.349279.5720.3495 10 0.069702 8.4050.578811.6820.543018.4690.402225.8030.357429.4140.353230.4910.3590 46.6630.348861.2760.348772.5310.349279.5720.3495 10 0.069721 8.4050.578811.6820.542918.4690.402225.8030.357429.4140.353230.4910.3590 46.6630.348861.2760.348772.5310.349279.5720.3495 10 0.069726 8.4050.578711.6820.542918.4690.402225.8030.357429.4140.353230.4910.3590 46.6630.348861.2760.348772.5310.349279.5720.3495 10 0.070000 8.4050.578211.6820.542318.4690.402225.8030.357429.4140.353130.4910.3589 46.6630.348861.2760.348772.5310.349279.5720.3495
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