ML20247F897
| ML20247F897 | |
| Person / Time | |
|---|---|
| Site: | Byron, Braidwood, 05000000 |
| Issue date: | 09/07/1989 |
| From: | Chrzanowski R COMMONWEALTH EDISON CO. |
| To: | Murley T Office of Nuclear Reactor Regulation |
| References | |
| 0282T:1, 282T:1, NUDOCS 8909180307 | |
| Download: ML20247F897 (21) | |
Text
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'j~ Commonwealth Edison -
- 1f U. 72 West Adams Street, Chicago, Illinois
\\
- Address Reply to: Post OIlic~e Boi76F Chicago, Illinois 60690 - 0767 -
September 7, 1989 Dr. Thomas.E. Murley, Director Office.of Nuclear Reactor Regulation' U.S. Nuclear Regulatory Commission Nashington, DC 20555-
Subject:
Byron Station Units 1 and 2.
Braidwood Station Units'I and 2 l
Non-Linear Piping Analyses NRC Docket Nos. 50-454/455 and 50-456/451.
Reference:
(a) July.14,-1989, letter from R. A. Chrzanowski to T. E. Murley
Dear Dr. Murley:
Commonwealth Edison has submitted a non-linear pipe analyses in reference (a) that, once approved, will be used to justify removal of snubbers and pipe supports at Byron and Braidwood-Stations.
During discussions with-members of-your staff, it was felt that a theoretical discussion of the method used in the analyses would be helpful during the review of the submittal.
This letter, therefore, provides copies.of two technical papers that present the theoretical methodology used to justify the non-linear pipe analyses.
Attachments A and B contain.the technical papers.
Please direct and further questions regarding this matter to this office.
Very truly you s M
R.,. Chrzanowski y
Nuclear Licensing Administrator cc.:
Byron ~ Resident Inspector Braidwood Resident Inspector L. N. Olshan - NRR S. P. Sands - NRR M. Hartzmann - NRR Office of Nuclear Facility Safety - IDNS 0282T:1 8909180%O h.hb54 FDR AD FDC P
\\
l l
1 ATTACHMENT A J
I ANALYSIS OF PIPING SYSTEMS HITH GAPPED SUPPORTS USING THE RESPNSE SPECTRUM METHOD 0282T:2
e Analysis of Piping Systems with Gapped supports Using the Response Spectrum Method M.S. Yang and J.S.M. Leung Robert L. Cloud and Associates Berkeley, California Y.K. Tang Electric Power Research Institute Palo Alto, California ABSTRACT completely.
In order to make snubber reduction or elimination feasible, it is This paper presents a meth>dology that necessary to provide alternative pipe extends the capabilities of the conventional restraints to limit pipe displacements during linear response spectrum analysis (RSA) dynamic events.
method to the analysis of pipin; systems with gapped supports. By employing.2 minimization An innovative support design, which can i
procedure, a response dependent linearized overcome the problems escociated with snubber i
stiffness is calculated for each gapped has been conceived and developed by Robert L.
l support.
The mathematical models of piping Cloud Associates (RLCA),
Inc.
The systems, in which springs with the linearized development program was sponsored in part by stiffnesses representing the corresponding the Electric Power Research Institute (EPRI).
gapped supports, can then be analyzed by the This support design is called " Simplified conventional RSA method, owing to the Pipe Support System" (SPSS).
SPSS supports response dependent nature of the are also known as Seismic Stops.
linearization technique, an iterative procedure is used to achieve the solution.
Seismic Stops are passive seismic The theory on which the method is based and supports which do not employ potentially the computer implementation of the method are unreliable hydraulic or mechanical presented.
components.
They are compact, light-weight devices that allow pin-to-pin replacement for INTRODUCTION snuubers as illustrated in Figure 1.
They permit therpal expansion while preventing The desf.gu of nuclear power plant piping excessive dynamic pipe displacements through has IM t o the development and use of dynamic the use of internal or external gaps.
The I
restraints, known as snubbers. A snubber is gap size controls the maximum displacement intended tn permit pipe displacement due to permitted during dynamic events. The concept slow movemnt such as thermal expansion, but of using gaps to restrain dynamic pipe restrain rapid motion such as that induced by responses has been successfully used for many earthquakes or water hammer.
years in fossil fueled power plants (1) and l
other applications.
i While snubbers present convenient j
solutions in the design and analysis of Although the concept of using gapped nuclear piping systems, the use of snubbers supports to permit thermal growth but limit has also presented many problems from the dynamic response seems ideal, they are not view point of installation and maintenance.
currently used in nuclear piping support It is believed that considerable savings in design.
A major reason for this is because cost and significant improvement in piping there has been no analytical tool available reliability could be achieved by reducing the to economically calculate the dynamic number of snubbers or by eliminating them response of piping systems with gaps.
]
1
_ _ ___d
n.
9 5
4 s
RLA O -#- -
1 J
_. ~.. n.-...
...n.
.. ~.. - -.
Fig. 1 The Seizmic Stop Support The non-linear response of a piping' impleinented in the RLCA/EPRI computer program system with Seismic Stops may be computed bY GAPPIPE..This method co nbines the technique a time history analysis using one of the of the conventional response spectrum step-by-step time integration algorithms.
analysis and that of the equivalent However, this approach is extremely time linearization of non-linear gapped support consuming and would be excessively costly if properties.
used for a routine production piping analysis even if the project schedule permitted. One In the following section, the discussion fundamental difficulty associated with the first focuses on the equivalent linearization time history analysis is that the support technique adapted to analyze linear piping acceleration time historiers required are systems supported by non-linear Seismic usually not readily available.
Furthermore, Stops.
The response spectrum analysis the time history analysis using most procedure incorporating the linearizatier.
commercially available computer codes concept is then presented.
generally calculates erroneous impact forces unless a very detailed impact model and fine CONCEPT OF LINEARIZATION time steps are used.
The method implemented in GAPPIPE is For the Seismic Stop to become a viable based on the equivalent lineutization alternative to the snubber, there is a need technique. The concepts of linearization for to develop an analysis method that is capable non-line s.r dynamic systems are well of analyzing piping systems with gapped documented
[2,3,4).
The basic idea of supports.
The method should have the equivalent linearization is to determino a following characteristics:
linearized system which is " equivalent" to the actual non-linear system.
Equivalence
- Easy to use; may be defined in various ways and is usually
- Employing concepts familiar to defined in terms of tha minimization of some engineers and designerst measure of the difference between the
- Using the currently employed linearized and actual systems for an assumed forms of input, i.e.
support class (or pattern) of response, For a piping response spectras system with non-linear supports (e.g. Seismic
- Using the existing piping modelst Stops),
the method provides a set of
- Computationally efficient.
linearized support stiffnesses which may be used to model the non-linear supports in An analysis methodology that meets the order to obtain a solution for the system above requirements has been developed and response.
These linearized stif fnesses will
have properties which depend upon the Specifically the following assumptions response itself.
Therefore, an iterative are made:
procedure is generally required to obtain the
- response, o The system response may be uncoupled l
into mode-iike components which may be Strictly speaking, non-linear systems do analyzed separately.
not generally possess natural modes of vibration as do linear systems.
However, it o The response in a particular " mode" has been observed that most lightly damped is quasi-harmonic (sine wave-like) non-linear systems display a similar response with a slowly varM ng rendom character to linear systems in that the amplitude and phase.
- Hence, the frequency spectrum of the response exhibits a re spot.se in a
particular rode series of distinct peaks or " modes".
In such resembles a pure trigencmetric cases, the concept of uncoupling the response function over any one cycle of into different mode-like components is still oscillation.
very useful.
This approach has been used successfully for rigid multi-degree-of-LINEARIZED STIFFNESS FOR SYMMETRIC GAPPED freedom (MDOF) systems with gapped supports SUPPORT
[2] and is applied herein to the case of piping systems.
The equivalent linearized piping system is defined as that system which minimizes the Based on time history analysis and mean squared difference between the actual tests of piping systems with gapped linearized and actual systems for the class supports subjected to earthquake type of random like earthquake excitations.
For excitations [5-9], it is observed that the systems which consist of discrete masses response is strongly narrow-bound in nature.
Interconnected by non-linear spring elements, In other w0rds, there are only a few it may rigorously be shown that an equivalent predominant frequencies in the response linear system so defined can be obtained by associated with mode-like components and the replacing each individual non-linear element motion in each of these modes tends to be with a linear element whose stiffness is a nearly harmonic with a randomly modulated weighted average of the stiffness of the non-amplitude (see Figure 2).
This observation linear element [3).
motivates the special form of linearization which is employed in the computer program Consistent with the assumptions GAPPIPE.
described above, the pipe displacement at any 4
support for a particular mode-like component of response, x(t), is assumed to be quasi-harmonic with a modal frequency, w,
and slowly varying amplitude, A(t), and phase,
$(t). That is, x(t) = A(t) cos O(t)
(1)
E-where O(t) = wt-@(t).
N g.
The variable A(t) and c(t) are assumed j
to be random.
l Let D(x(t)) be the difference between the restoring force, F(x), for the gapped I
support and that for the equivalent iinear o
support. Then, I
C.tB
(. 8 8 d.t B 15.00 l'6.00 D(x(t)) = F(x(t))
- k x(t)
(2) y TIME (SEC) 1 where k is the equivalent linear support g
stiffness for quasi-harmonic response in a Fi g. 2 A Typical Pipe Displacement History particular mode.
at a Gapped Support k is determined by requiring that the g
mean squared value of D be a minimum for each l
l l
V m.
individual support over any one cycle.of
. oscillation of x(t). Since A(t) and $(t) are ai,sumed to"be slowly varying, they will roact remain nearly constant over any one cycle.
n Hence, the mean equared value of D(x) will ber
.J -
3 m
8 1
_ msmemn (D!x(t)))2 dt (3) 2.
D,,
=
'o where T=2r/w is the period of the modal response.
A necessary and sufficient condition for Fig. 3 Force-Displacement of Symmetric D,, to be a minimum with respect to kg Gapped Support is dD,,
= 0 (4)
Due to its irregular appearance and
'dk1 broad frequency band, earthquake excitation is often modeled as a random process. For a random excitation process, the response will where d's denote the differentials.
also be random.
For the assumed case of narrow-banded response within mode-like Performing the differentiation indicated components, the amplitude of quaskharmonic yields an' algebraic equation for kg which may response.
A, must therefore be a random be written as a function of "A" in the form variable.
For a singir-Jegree-of-freedom (SDOT) 2r system, a direct linearization analysis may e performed cons W ing A to be alwly k (A) =
F(x)cosede (5) y varying random variable.
In this case, the f'
ir A minimization criterion used is the O
minimization of the expected value of D (see where e is defined in Eqn. (1).
Eqn.(3)) (10).
This leads to the following expression for the effective linear support Figure 3 shows the typical force.
stiffness, displacement relationship of a symmetric 2
Seismic Stop. The support stiffness is equal E(A k (A))
y to zero until the gap is closed, after that, KLin (8) 2 the stiffness is K Let 6 denote the E[A )
g.
dimension of the gap.
Then, the support restoring force may be written as where E(
)
denotes the probabilistic operation of expectation and kg(A) is the
-0 when lxl g g linearized stif fness quasi-harmo'nic response (6) defined by Eqn.(7).
F(x)
=
9(lx(t)l-6)
K Equation (8) states that the final when lxl > d linearized stiffness of the support is a where l l denotes the absolute value.
weighted average of the harmonic linearized k (A) over all possible response stiffness y
amplitudes, A.
From a purely theoretical Substituting from Eqns. (1) and (6) into Egn.(5) and integrating yield point of view, an equation of the form of Eqn.(8) would logically be applied to each individual mode of the piping system.
This
0, for A56 would result in a different set of values k (A)
(7) y (2K /r)(cos~1(6/A)
K in for each mode of the system.
Such a
=
g s tuation could be dealt with numerically but
-(6/A) (1-(d/A)2)), for A>g not without considerable computational effort.
l' D
An alternative approach would be to where 6 @~
and 6 are the effective r
egg,1 define a single set of values KLin which is right and 'l' eft gaps, r%pectively.
the same f or all of the active modes of the system.
This could be accomplished by An average unbalanced force can be specifying a value for A in Eqn.(8) which defined as represents the total response of the system at each Seismic Stop.
This leads to a F(x )
= F (x,A,) <rI*o*A I Ill) 1 o m
o substantial reduction in computational effort over the separate mode approach since all of where F and F are average impact forces 1
r the modes could be determined at one time from the left and right
- impacts, using a standard eigen-solver.
Based on respectively.
Note that they are functions available experimental data, the alternative of the offset x,
and the maximum o
approach provides results which are as displacement response, A,.
accurate as those obtained from the separate mode approach.
In addition, an unbalanced force _F has sign.
A positive unbalanced force pushes In the linearization analysis, the the pipe toward the right and vice versa.
highly localized secondary damping effect due The offset x also has sign.
A positive to impacts between pipe and supports is offset implies,that the pipe moves toward the ignored for simplicity and inherent right.
In general, a gapped support has the conservatism.
The conservatism means that same stiffness K for either the right or the g
higher response may be calculated because left gaps.
In tnis case, the net effect of this impact damping is neglected.
impact, in terms of unbalanced force, shall push the pipe toward the larger gap.
LINEARIZED STIFFNESS FOR ASYMMETRIC CAPPED SUPPORT The average impact force from the right impacts is estimated using the following The derivation delineated above is for equation:
symmetric gap configurations only.
In cases of non-symmetric and one-sided gap A,
configurations, the assumption of a symmetric response prescribed by Eqn. (1) is no longer pr(*o'A I "
fr(A'*olp(A)dA (12) valid.
To account for the response m
asymmetry, an offset is introduced into the
'Eeff,r response as illustrated in the following equations where 6,gf is the effective size of the right gap clefined above in Eqn. (10); p(A) is y(t) = x, + x(t)
(9) the probability density function of displacement amplitudes.
fr(A**o) is the where average right impact force over one cycle j
y(t) = the assumed pipe displacement in with an amplitude equal to A.
j the direction of the asyn. metric i
gapped support; It may be expressed as x
= the of fset cf the rerponte; xit)=a rymmetric displacement about f (A**o) " IKgr/r) ( Asina-6,g g,7o)
(13) r the of" set.
where The physical meaning of offset is that a
= cos-1(6m g A);
the pipe assumes a new position during dynamic impacts.
The offset is caused by geff,r = effective right gap bize non-symmetric impact forces. In other wottis, the unbalanced impact forces from both sides K
= Seismic Stop stiffness after generates a net effect to push the pipe to a 9#
the right gap closure.
new position from the original stationary position. And the pipe is assumed to vibrate Similarly, the equation for the average symmetrically about the offset position.
impact force from the left may be derived.
The only differences are that Kai replaces To accommodate the new average pipe F
in Eqn.(13) and that the effective gap w
position, the gap sizes are redefined as follows:
size becomes 6,fg,1 A linear relationship is assumed to 8 eff,r " E ~*o exist between this average unbalanced force r
(10) and pipe offset.
In fact, this has been
'eff,1 * # **o 1
shown to be the case from the test data [11).
That is determining the amount of change in (KLin]
between the.two consecutive iterations.
_I(*o). =Kx (14) po In general, the procedure begins where K represents the. global pipe stiffness assuming that all linearized stiffnesses are p
in the support' direction. Since the ' average zero as if the gapped Seismic Stops are not unbalanced impact force F is also a function present.
The pipe displacement responses at of offset x an iterative numerical o,
Stop locations are then calculated using the procedure is used to solve Eqn.(10) for the conventional response spectrum method. Based offset value.
on these responses, a new set of linearized stiffnesses are calculated,using the The theory on KLin dolineated in the linearization procedure described above.
previous. section for the symmetric gap with this new set of linearized stif fnesses configuration may also be applied to the added to the piping system, the response i
estimation of KLin for the asymmetric gap spectrum analysis procedure repeats.
The configuration. The method of application is iteration continues until the changes in the described below.
linearized stiffnesses for all gapped supports are within the preperly prescribed The main. difference in calculating the tolerance, c.
linearized stiffness between symmetric and asymmetric Seismic Stops is that we need to Other than the iteration flow chart, determine the offset for every asymmetric Figure 4 also illustrates that the converged Stop before stiffness calculation.
The pipe response is that associated with the offset is computed by solving Eqn.(14).
cross point of two KLin vs. pipe displacement curves.
One of the. curves is - the predicted The KLin for the asymmetric Stop is KLin-curve, and the other - is the calculated estimated by taking the average of the KLin-cuive.
Note that the pipe displacement linearized stiffnesses ' of two hypothetical represented ty the horizontal axis is symmetric Seismic Stops.
One of the evaluated in the gapped support direction.
hypothetical symmetric Stops has d as Around the neighborhood of the solut. ion, the eff,r its symmetric gap size while the other has response is inversely proportional, as u
6,gga for its symmetric gaps.
These two.
indicated by.the predicted KLin-curve, to the hypotfietical Stops take, respectively, the value of the predicted linearized stiffness stiffnesses, Eg and ' K of the original used in the calculation.
However, based on g
g, seismicStopfortheirst[ffnessesbeyondgap the line e riza tion. theory, the calculated cloture.
linearized stiffness, as indicated by the calculated KLin-curve, increases as the The methodology described above, for th9 response does.
In the process of iteration, calculation of a linearized stiffness for a the two E g3 values approach the croEs point n
symmetric Stop, is used to calculate the simultaneously along the two curves.
When linearis,ed stiffnesses for these two they reach within the small crea representing hypothetical symmetric Stops.
the allowed tiolerance, the linearization solution is achieved.
In summary., the lintuarized stiffness for an asymmetric Stop can be expressed as The iterative procedure depicted uy the.
follows flow chart is outlined step-by* step as follows:
KLin = (Kg M y2 (15) r (1) Assume a null (FLinI*
where Kg and K are the linearised r
stiffnesses of the two hypothetical (2) Add (KLin) to {K) piping.
symmetrical stops (12).
(3) Perform the RSA to determine the LINEARIZATIO 4 S0WTION PROCEDURE maximum displa :ement amplitudes at Stops.
Because the equivalent linearized stiffness is response dependent, an iterative (4) Use the maximum displacement procedure is required to obtained the amplitudes to calculate a new solution.
The iterative procedure is (KLin), together with the offsets illustrated by the flow chart in Figure 4.
for asymmetric Stops.
In the flow chart, (K] pining is the global stiffness matrix of the piping system (5) Compare the old and new (KLin]'s to excluding the gapped supports. [KLin) is the see if the difference is within the linearized gapped support stiffness matrix.
prescribed tolerance for every stop.
Variable "b"
is the convergence factor If all differences are within the
tolerance, the solution is (6) If the tolerance is exceeded by at 1
converged.
least one Stop, a new updated (KLin3 is calculated for use in the next iteration using the following start formula:
[KLin3 Updated "
(1-b)[KLin3 ld+b[Knin)New O
"'"IM where "b" is the convergence factor.
(7) Go to Step ' (2) and the process o
repeats.
- N*
.The whole solution process is a repetition of the response spectrum analysis procedore.
The non-linearity of gapped g
supports is accounted for in the Responee SpecWu linearization procedure while the interaction D
between gapped supports is also inherently accounted for through the step-by-step iterative process.
n N IKm],
NUMERICAL EXAMPLE
[Km]. e[Km] + b[Km ),
Shown in Figure 5 is a simply supported a+b=1 3sch40 pipe with a symmetric Seismic Stop u
j support at the center.
The gap size is 0.5 inch and the length of the pipe is 20 feet.
The 4 valus is 21 kips / inch.
The piping IM
-I systel is subjected to a seismic input m
e within Tolerance?
depicted by the reponse spectrum shown in Figure 6 in the z-direction.
The analysis of this simple span piping "3
takes nine linearization iterations to convarge.
The cpu time required is. 34.5 Done seconds on the DEC VAX 11/750 computer using VMS operating system.
The convergence
\\
process is pict+ red in Figure 7, of which the ho rizontal axis is the pipe displacement at l K'. 3l < e convere=d the gapped support location and the vertical 1
axis is tae linearired stiffnese.
The curve g (1-c)*Ky + } ec with squares represents the predicted A
ctherwise K
=
g linearized stiffness used to calculate the
,, M response.
The other curve with crosses D
PED 7crto represents the calculated linearized CAfruLATED E
stiffness based on the ralculated resperse.
D The intersection point of the two curves U
represents the solution.
A pair of square g
and cross symbols stands fcr one iteration.
I g.
The calculated pipe displacement in the a
direction of the seismic Stop is 0.66 inch.
3 The impact force is 1.81 kips. The measured 6
PIPE DISPMCEMENT displacement and impact force are 0.62 inch and 1.31 kips, respectively.
CONCLUSION Fig. 4 Solution Flow Chart and Convergence The linearization method and the Criterion iterative response spectrum analysis procedure presented in this paper show piping systems with gapped restraints can be
~.
analyzed economically using the existing model and input spectra.
This analysis 7.
" Reduction and Examination of the Data from the Response of a Large Piping methodology together with the Seismic Stop hardware (see Piqure 1) provide a viable System with Alternative Support solution to the problem of eliminating Configurations," EEBI Proiect R e n t, r t fPR-3 troublesome snubbers.
1444-7, December, 1988 (to be published).
]
REFERENCES 8.
- Malcher, L.,
Steinhilber, H.,
and Schrammel, D.,
" DESIGN REPORT
- 1. Cloud, R.
L., " Seismic Performance of Servohydraulic Excitation of Mechanical Piping in Past Earthquakes," presented at Equipment, HDR Test Group SHAM, VS. No. T41,"
PMDR Work Reoort No.
4.338/88, ASCE Specialty Conference, Knoxville, TN, j
September 1980.
Kernforschungszentrum Karlsruhe, Project HDR
]
Safety Program, Commissioned by the 2.
- Iwan, W.
D.,
"The Earthquake Design Federal Minister for Research and Technology, and Analysis of Equipment Isolation Systems,.
West Germany, March 1988.
Earthauake Enaineerina te nd Structural Rymardq, Vol.
6, pp. 523-534 (1978).
9.
- Leung, J.
M.
S.,
- Yang, M.
S.,
and
- Singh, A., " Shaking Table Testing of a Piping 3.
- Iwan, W.
D.,
" Application of Non-System Supported by Seismic Stops,"
)
Linear Analysis Techniques,"
Anelled EIDgeedinas of the loth SMIRT Conference, liechanics in Earthauake Encineerina, AMD-Vol.
August 1989 (in Preparation).
3, ASME, N.Y.,
pp. 135-162 (1974).
10.
- Caughey, T.
K.,
" Equivalent
- " 9 ***"
4.
- Spanos, P-T.
D.
and Iwan, W.
D.,
" Harmonic Analysis of Dynamic Systems with
^"*
pp. 1706-1711, Nov. 1963.
Nonsymmetric Nonlinearities," Journal of Dynamic Systems. Measurement. and Control, 11.
" Calculation of offsets from Test Vol. 101, pp. 31-36, March (1979).
Data," RLCA Cale. Packsoe, Record No. P94-0, hojed No. N4, 1987.
5.
- Cloud, R.
L.,
- Anderson, P.
H.,
and Leung, J. S.
M.,
" Seismic Stops vs. Snubbers, 12.
"Linearized Stiffness of a Gapped A Reliable Alternative," Nuclear Encineerina S
M Pd% hd h P96 and Desian, 107, pp.205-213, 1988.
4/29, Project No. P94-4, 1987.
6.
Wu, K.
K.
and Loey, H.,
" Dynamic Testing of Seismic Stop Pipe Support Device,"
(1987 Shaking Table Test) Proceedings of the MUL_PvP Conferga, Pittsburgh, PA, June 1968.
a e
d=
GAP SIZE a 0.5 INCH Kg = 21 KIP / INCH GAPPED SUPPORT (BETWEEN NODES 4 AND 11)
MODE FREQUENCY 1
10.8 HZ 2
24.2 HZ 3
68.8 HZ 11 Q
2 6
c 3
1 3
4 5
10 7
A 8
g Fig. 5 Simple Single Span Piping Model
a, INPUT RESPONSE SPEC'.;UM ss
's4 -
I3 -
12 -
18 -
10 -
9..
8-o 7-6-
5-4-
3-2-
/
8-0 c
10 20 30 4D HZ Fig. 6 Input Flocr Spectrum o.s y
D.4 -
\\
m
{
CAICUIATED U-I
/
s
/
l 0.2 -
~
I c.s -
PREDICTED 1
0 O
C4 0.8 1.2 1.6 P
2.4 2.8 PIPE DISPLACEMENT / CAP SIZE l
Fig. 7 Convergence Process of Simple Span Piping Analysis l
l l
t l
ATTACHMENT B RESPONSE SPECTRUM ANALYSIS OF MULTIPLE SUPPORT EXCITATION-GN PIPING SYSTEMS HITH GAPPED SUPPORTS l
- . 4,
.4 o
.g y
RESPONSE SPECTRUM ANALYSIS OF NULTIPLE SUPPORT EXCITATION ON PIPING SYSTEMS WITE GAPPED SUPPORTS E.-C. Tsai and C.-W. Lia Robert L. Cloud and Associates, Inc., Berkeley, California Y. E. Taag Electric Power Research Institute, Palo Alto, California ABSTRACT important task by the nuclear industry for improving plant safety and reducing snubber reduction has been considered by operational cost.
the nuclear industry as an important means of improving plant safety and reliability. In An inmovative support design known - as lieu of or. in conjunction. with snubber seismic stop, developed by Robert-L.
Cloud reduction, gapped pipe supports, such as Associates, Inc..and partially sponsored by seismic stops, could also be used. This the Electric power Research Institute, is-approach would allow free-unrestrained intended to be a practical alternative to thermal expansion while preventing excessive snubbers (1). The seismic stop, a persive seismic motion.
seismic support requiring little maintenance and minimal inspection, has pre-s(t gaps to-
-i Secause of the constraint of gapped allow free. thermal expansion bat is capable I
supports, the dyramic response of a piping of limiting the large seismic movement.
system becomes non-linear, which can only be because of the presence of the gaps, the analysed using an equivalent linearisation dynamic response of a piping system with technique when the response sp.e method meismic stops becomes non-linear, which can is used. presented in this paper is an be directly solved by the step-by-step time iterativt algorithm oc using the multiple integration approach if the complete time support response spectrum technique to history of excitation is known. However, this compute the piping systou response. In is neither practical-nor cost effective. In addition, some numerfcal results are also addition, it has become a common practice in presented for piping with gapped supports.
earthquake engineering design to specify earthquake excitations in terms of design INTRODUCTION response spectra. When the response spectrum method is used, the. non-linear gapped On piping system design, various loading supports must be modeled by an equivalent conditions need to be evaluated. He piping linearisation technique and an iterative layout must keep the thermal expansion procedure is required to solve the piping stresses within the code limits and at the response.
Such methodology has been same time maintain the seismic stresses at an incorporated into a piping analysis computer acceptable level. Most strut-type of supports program GhPPIPE for uniform response spectrum are not well suited for thermal expansion analysis [2].
because they tend to stiffen the system. To satisfy the design, es=hhars are widely used However, most piping systems of nuclear in the nuclear power plants, which permit power plants are attached to the structures free movement of the pipe for slow motions at many support points subjected to quite such as thermal expansion but restrain rapid different earthquake motions. The multiple motions such as that induced by an support excitation analysis would provide earthquake.
However, snubbers have been more realistic results than the uniform found to be unreliable. As a result, excitation analysis. Thus, an iterative
. excessive inspection and maintenance are algorithm of using the multiple support required for snubbers which has caused response spectrum technique to compute the snubber reduction to be regarded as an response of piping systems with non-linear
a
,4 gapped supports has been implemented in the GAPPIPE program and will be presented in this The dynamic displacement yd is due to the paper. In addition, some numerical results inertia effect. The pseudostatic displacement are also presented to illustrate the Y,
is caused by the different relative influence of gapped supports on the piping movements among the constrained degrees of response
- freedom and equal to GAPPED SUPPORTS 3
The force-displacement relationship of a where I is the influence vectors defined by gapped support is shown in Figure 1.
The support has a gap sire, d, to allow the pipe g. _ g-1 r p) to move freely. When the deformation of the "9
pipe at the support is larger than the gap Assuming C I + c., = 0, the upper portion of size, the pipe will touch the support and its Equation (1) can be rearranged in terms cf 7
further movement will be resisted by a dynamic displacement as follows:
constraint with stiffness, k,
from the
' support. In general, this support stiffness EId+EId + E Id "
- E E I"g 0)
,is much higher than the piping stiffness so that the support can prevent exoessive piping By the modal-superposition techniques, displacement. Due to the opening and closing the dynamic displacement can be expressed as of the gap between the pipe and support, the the lineer combination of the n mode shapeu gapped support acts on the piping system as a as non-linear spring.
An equivalent linearization technique [3] is adapted to find a linearized stiffness such that the E di Yg
@)
Id " I"I mean difference between the support force acting on the piping system by the gapped support and that by the equivalent linearised spring is minimized.
The equivalent where di is the l'th mode shape and Yg is the linearization technique makes it possible to corresponding amplitude. Equation (5) can be analyze the piping system with gepped decoupled to the following n independent supports using the response spectrum method.
equations But, because the linearized stiffness varies 2
(7)
Yi+2#1 wi Yi + v1 yi..yi yg with the corresponding pipe displacement, an iterative procedure is required and will be discussed in the remaining part of the paper.
where #1 and wi are the damping ratio and modal frequency of the l'th mode. The MULTIPLE SUFFORT EXCITATION participation factors 21 are defined as t
t The aquations of motion for a multiply-pg-gg g g j g,i g gi) g)
supported piping system with lumped-mass assumption can be expressed as which are corresmnding to the motions on the constrained degrees of freedom 29 H2 Ye C
C Yt g
+
+
The support points of the piping system Eg
$g can be divided into several statistically t
2E Y
E gg g
g g
independent input groups as recommended in E
E Ig 2
Reference (5). All the input response spectra g
(1) of Support points belongirvJ to the sama inpat E
E I
A group are the same or proportional. The t
g gg g
responses due to the antions of support where 5' k
and
.Yt are the total pointo in a
same input group are t
t acceleration, velocity and displacement of algebraically summed (5). In other words, the theunconstraineddegreesoffreedom,k,f input motions f on the constrained degrees g
g and 2 are the prescribed input motion at the of freedom can be expressed as the, linear 9
combination of the group excitations U.
g constrained degrees of freedom pertaining to the support points. E, C and E are the mass, Ig"EEg desping and stiffness matrices of the unconstrained degrees of freedom. E g
g, gg where 3 is a linear transfor; ration matrix.
and Egg are those of the constrained degrees The right-hand side of Equation (7) becomes of freedom, g and E are the damping and
-gg U in which the new participation factor g
g g
stiffness coupling terms between the gg is equal to constrained and unconstrained degrees of
- freedom, gg - Eg 3 00 The total response of the unconstrained If there are a statistically. independent degrees of freedom consists of two components input groups and 'd (j,k) represents the k'th g
[4]
directional component of the j 'th group,I*
Yt " Yo + Ya (2) is a row vector with 3a elements of og
representing the participation factor of the accelerations and stresses, component t)g og.esponding l'th mode to the group The whole solution process is a
repetition of the response spectrum analysis Denoted the spectral displacement of procedure. The number of iterations to Sg(j,k)
,, D
- U at the i'th mode as the converge depends on the accuracy desired, the U
g maximum response of dynamic displacements can number of gapped supports and the response be calculated from the following equation:
spectra.
PARTICIPATION FACTORS n
3 m
Max (yd) =MCB(dgDCB[CCB Mg SgII*N))) (11)
U'U i=1 k=1 j=1 New participation factors defined in Equation (8) will be calculated in each where MCB is the modal combination rule that iteration. According to Equation (4), the combines the maximum responses of individual influence vectorc r requires to be resolved modes, DCD is the direction combination rule in every iteration because the stiffness
'that, for each individual mode, combines the matrix and stiffness coupling term are-
. maximum responses of each direction and CCB changed. To avoid inverting the stiffness is the group combination rule that combines matrix and to afford an efficient way to calculate participation factors in the non-the maximum responses of each group.
Combinational tvles, such as square-root-of-linear iteration analysis, Equation (8) can be further simplified [6] by considering the sum-of-the-squares (SRSS), absolute sum, etc., can be selectively applied by the user.
eigenvalue problem, ITERATIVE ALGORITEM FOR RESPONSE SPECTRUM E di " wi Mdi (12)
ANALYSIS which can be rewritten as Because the linearized stiffness of gapped supports is response dependent, the (1/Wibd t g g-1 g333 i "di stiffness matrix of a piping system E contains two parts. One part is the response.
Post-multiplying by E and applying Equation g
independent stiffness matrix without the (4)*
gapped support stiffness, which is constant 2
t t
in the iterution. Thu other part corresponds (1/W1 ) di 19 " "di MI (14) to the gapped support stiffness which requires to update for each iteration.
Then, substituting into Equation (8), the participd lon factors becom.e Similarly, the stiffness coupling term E can g
be divided - Jnto the constant part and the t
2
- t. E di)
(15)
El " ~#1 Ig / I *i II gapped support part.
Through this equation, the participation The iterative algorithm to determine the facWs can be dhedy calmlated from the final response of piping systems with gapped s e p te s E without solving supports by the response spectrum nethod is g
yg summarized as follows:
N RICAL REC m s (1) Form the constant parts of the stiffness matrix and the stiffness coupling The iterative algnritha described above term without the gapped support stiffness.
has been implemented in tts piping analysis (2) Use the equivalent linearized pyp stiffness to form the element s,tiffness with gapped supports have been analyzed and uatrix of gapped supports.
their results are presented to demonstrate (3) Form the complete stiffness matrix the influence of gapped supports on the and stiffness coupling terla by adding the response of piping systems subjected to the stiffness of gepped supports to the multiple support excitation.
corresponding constant parts.
(4) Calculate the modal frequencies and The first example is a simple three-dimen8ional piping bend shown in Figure 2.
C 1culate the participation factors.
Both the transnational and rotational degrees (6) Employ the group, directional and of freedom at anchor point 1 are constrained.
modal combination rules to find the only the transnational degrees of freedom at deformation in the direction. of gapped anchor point 31 are constrained so that it acts as a moment-released node. At piping Evaluate the equivalent linearized n de 12, there is a spring support in the Z stiffness of gapped supports from the direction. Two gapped supports at piping calculated deformation.
nodes 13 and 22 have a gap size of 0.5 inch (8)
Check if the quivalent stiffness (12.7 mm). The support points were subjected is converged. The convergence criteria could
~
to two statistically independent group be based on maximum per centage of change in excitations in the XY plan. Anchor point 31 the stiffness. If not converged, go to step belongs to the first group and the other (2) to start the next iteration.
supports belong to the second group. The (9) Employ the combination rules to response spectra of two group excitations in calculate
~the piping displacements,
+
the X and Y directions, for simplicity, are and has lower modal frequencies when it is assumed to be the same and are shown in analyzed by the multiple support motion Figure 3. As a result, only phase differences method. As shown in Table 3,
the maximum between support motions have been considered piping moment calculated by the envelope in the present example. The piping responses spectrum method is 31.69 kip-in (3580 kN-mm}
have been calculated under different and decreases to 23.21 kip-in (2620 kN-mm) by excitation intensity. The intensity scales of the absolute sum method and 27.56 kip-in input excitation vary from one to ten times (1980 kN-mm) by the SRSS method. The number of the response spectrum shown in Figure 3.
of iteration required to converge is 33 for When the scale equals to one, gapped supports the envelope spectrum method, 35 for the do not close so that the piping response is absolute sum method and 30 for the SRSS linear. As input intensity is increased, the method. This shows that the multiple support non-linear behaviors begin to show up. Figure excitation method does not increase more 4 shows the variation of the first and second computational effort but will provide more modal frequencies under the different scales realistic results in the response spectrum of input. For the larger scale, the acdal analysis of piping systems with gapped
. frequencies become higher because gapped supports.
supports provide more constraint forces and
'the piping system becomes stiffer. The CONCLUSION distributions of moments in the Z direction under several different scales are plotted in An iterative algorithm of using the Figure 5 along the pipeline. It reveals that multiple support response spectrum technique the moment distributions along the pipeline to compute the response of piping systems vary with intensity scales. At node 1 which with non-linear gapped supports has been is far from the gapped supports, the presented. The whole solution process is a increasing of moment is smaller than those at repetition of the response spectrum analysis nodes 13 and 22 where the gapped supports are procedure in which the stiffness of gapped located. The variations of moments at nodes supports is determined by the equivalent 1,
13 and 22 with different scales are linearization technique and updated in each plotted in Figure 6.
This shows that the iteration. The numerical results reveal that moments are not increased in proportion to the gapped supports can prevent the the excitation intensity. The moment increase proportional increase of piping stresses when becomes very small when the pipe deformation the intensity of excitation is increased. It is greater than the gap size except at node is also shown that, using the presented 22 where the moment increase in the 2 iterative algorithm, the multiple support direction is larger than the linear case excitation analysis which can provide more without gapped suppets. This is because node accurate results requires almost the same 22 h&s a,
very su.all moment ir. the linear computation effort as that in the uniform excitation analysis.
case.
The second example is piping system with REFERENCES l
a three branch configuration an shown in l
Figure 7. The su.> ports were divided into four (1) R. L. Cloud, P. H. Anderson and J 1Aung, 1
groups corresponding to four distinct input
" Seismic Stops vs. Snubbers, A Reliable excitation spectra sets. Group one consists Alternative", Nuclear Engineering and only of unchor point 1. Two hangers at nodes Design, Vol. 107, pp 205-213, 1988.
11 and 13 and three gapped supports at nodes (2)
J.
Leung, et al.,
" Development of A 7,
9 and 15 are in group two. Group three Simplified Piping Support System",
consists only of anchor point 17 and group Transaction of the 9th 8KiRT Conference, a
four only anchor point 21. These excitation pp 923-928, 1987.
spectra correspond to actual spectra (3)
W.
D.
Iwan, "The Earthquake Design and developed for a real res,ctor structura (7)
Analysis of Equipment Isolation Systems",
j and are again used in this example. The Earthquake Engineering and Structural
)
responses of different support group Dynamics, Vol. 6, pp 523-534, 1978.
contributions were combined by ' the absolute (4) R.
W. Clough and J.
Penzien, Dynamics of sum method and the SRSS method. In addition structures, McGraw-Hill, New York, NY, to the analysis of multiple suppott motions, 1975.
this example was also solved by the envelope (5)
C.-W. Lin and F.
- Inceff, "A New Approach spectrum method where all supports are to Compute Response with Multiple Support uniformly excited by a spectrum that envelops
Response
Spectra Input",
Nuclear all group spectra. The original gap sizes of Engineering and Design, Vol. 60, pp 347-three gapped supports and their maximum 352, 1980.
displacements solved by three methods are (6)
C.-W. Lin and W.
H. Guilinger, " System shown in Table 1. Also shown in this table Response to Multiple Support Response are the equivalent linearized stiffnesses of Spectra Input". Transaction of the 6th gapped supports. The results of the SRSS SMiRT Confersnoe, Paper No. K10/4, 1981.
i method has the smallest displacements at (7) p.
Berler, M.
Subudhi and M.
- Hartzman, i
gapped supports and, thus, has the lowest
" Piping Benchmark probleme, Dynamic I
equivalent linearized stiffnesses. he first Analysis Independent Support Motion three modal frequencies computed by three Response Spectrum Method", NUREG/CR-1577, methods are compared in Table 2, which shows 70 L. II, 1985.
that the piping system becomes more flexible
Table 1 Sizes and Displacements of Capped Supports (1 in = 25.4 mm, 1 kip /in = 0.175 kN/mm)
Node Size Displacement (in)
Stiffness (kip /in) l l
No.
tin)
Envelop Abs Sun SRSS Envelop Abs Sun SRSS I
l 7
0.120 0.169 0.161 0.155 7.184 5.353 3.945 9
0.045 0.064 0.058 0.054 8.114 4.397 2.124 i
15 0.240 0.275 0.268 0.262 1.015 0.659 0.384' Table 2 The First Three Modal Frequencies Mode Envelop Abs sua sKSS Number (Hz)
(Hz)
(Hz) 1 7.157 6.528 5.943 2
9.943 9.133 8.403 3
11.94 10.77
$.737 Table 3 Maximum Moments and Iteration Numbers (1 kip-in = 113 kN-am)
Envelop Abs Sun SRSS Maximum Moment (kip-in) 31.69 23.21 17.56 Iteration Number 33 35 30 l
i.
l l
s
' \\
so" #
se as
\\
ase,,
as na Og*
k 1
1 n
-d d-m 7
k S"
/
rigure 1 c Displa nt R.lation of Figure 2 Piping Systen of First Numerical Example (1 in = 25.4 son) 5 DAMPING RATIO = 2 %
E l.
w l
e 40 s
0 FREQUENCY (Es) rior. 3 R..,oo... ct o... i. rir.,.
ric.1....p1 j
i 1
l
\\
r
.e l
l 16 MODE 2
.E U
MODE 1 4
0 10 INTENSITY SCALE Figure 4 variation of Modal Frequencies with Excitation Intensity a i 40 moes is sons a 6
m.
g
==n g
m w
E moes as E
scau>.1 ee 6
l NODE 25 0
O DISTANCE McIOC FIPELINE (IN)
Figure 5 Mounent Distribution along Pipeline under Different Intensit/
(1 in = 25.4 an, 1 kip-in = 113 kN-an)
I l
9 40 x,.c mod. 1 M at Mode 22 g
M at mode 13 y
E M at Mode 13 g
M at mode 1 y
~
M at Nede 22 s
0 10 O
INTENSITY SCALE Figure 6 Variation of Moments with Excitation Intensity (1 kip-in = 113 kN-am)
N Z/
4 4
g e-s e
p&
- s
,e gm h
se q
Figure 7 Piping System of Second Numerical Example i
___.______________.__________m
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