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Dynamic response of piping system on rack structure with gaps and frictions
Nuclear Engineering and Design 111 (1989) 341-350 North-Holland, Amsterdam
341
DYNAMIC RESPONSE OF PIPING SYSTEM ON RACK STRUCTURE WITH GAPS AND FRICTIONS Hiroe KOBAYASHI
Nuclear Power Division, Ishikawafima-Harima Heavy Industries Co., Ltd., 1, Shin-Nakahara-cho Isogo-ku Yokohama, 235, Japan Misutoyo YOSHIDA
Storage Plant Division, Ishikawafima-Harima Heavy Industries Co., Ltd., 3-2-6 Toyosu Kohtoh-ku Tokyo, 135, Japan an d Yoshio O C H I
Research Institute, Ishikawafima-Harima Heavy Industries Co., Ltd, 1, Shin-Nakahara-cho Isogo ku Yokohama, 235, Japan Received 24 November 1987
In the seismic design of a piping system on a rack structure, the interaction between the piping system and the rack structure must be evaluated under the condition that the rack structure is not stiff and heavy enough compared with the piping system. Moreover, there are local nonlinearities due to the gap and friction between the piping system and the rack structure. This paper presents the influence of the interaction and the local nonlinearities upon the seismic response by numerical study and a vibration test using a shaking table. In the numerical study, the piping system and the rack structure were represented by the three degrees of freedom mass-spring model taking a vibration mode of the piping system into account. The nonlinearities due to gap and friction were defined as a function of motion and treated as the pseudo force vector (additional applied force) in an equation of motion. From the results of the numerical study and the vibration test, it was clarified that seismic response of both the rack structure and the piping system is reduced by gap and friction. Moreover, the piping system and rack structure can be represented by the three degrees of freedom mass spring model. And the local nonlinearities can be treated by the pseudo force in an equation of motion.
1. Introduction Some piping systems rest on rack structures. In a seismic design, the rack structure has been regarded as stiff and heavy enough compared with the piping system. And the interaction effect between the piping system and the rack structure (the piping-rack system) has been ignored. But this interaction effect must be evaluated when the rack structure is not stiff and heavy enough compared with the piping system. On the other hand, the gap in the piping-rack system is designed to allow for the thermal expansion and installation of a piping. Friction also exists here. The piping system may slide within the gap and cause an impact during a seismic event. The vibration energy may be dissipated
concurrently by the impact and friction within the gap [1-3]. So, the interaction and these nonlinearities strongly affect the seismic response of the piping-rack system. In order to clarify the influence of the interaction and these nonlinearities upon the seismic response, a numerical study and a vibration test were carried out. In the numerical study, the piping-rack system is represented by a three degrees of freedom mass-spring model taking a vibration mode of a piping system into account. The pseudo force method was used to analyze the nonlinear dynamic response of the piping-rack system. The nonlinearities due to the gap and friction were treated as the pseudo force vector in an equation of motion. The pseudo force is assumed to be constant for
0 0 2 9 - 5 4 9 3 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s io n )
H. Kobayashi et aL/ Dynamic response of piping system on rack structure
342
a small time increment in the direct integration of the equation of motion. The effect of the gap size, the mass ratio and a natural frequency of the piping system as well as the rack structure on the seismic response were surveyed. Moreover, nonlinear response spectra were generated [4]. A vibration test of the piping model using a shaking table was also carried out. A straight piping model of 3 / 4 inch in dia. and 6 m in length was set on two rack structures whose ends were fixed on the shaking table [4,5]. This model has gap and friction between the piping system and the stopper on the rack structure. The mass ratio of the piping system to the rack structure also varied in this test.
2. Numerical study
2.1. Modeling methodology The piping-rack system was represented by a massspring model in the numerical study. For convenience, consider one straight piping system on the rack structures placed with the same interval 1 as shown in fig. 1. The gap of size d and the friction of coefficient /~ exist between the piping system and the stopper on the rack structure. It is assumed that the oval deformation of the piping and higher vibration modes of the piping system are not caused by the impact between the piping and its stopper on the rack structure. Each rack structure is also supposed to have the same dynamic characteristics. So, each structure vibrates in phase as the seismic event as shown in fig. 1. The displacement of the piping system is composed of bending deformation and translational motion which is parallel to the resting position. In other words, it consists of elastic deformation and rigid body motion. Therefore, the mass of the piping system is divided into two parts. One mass Mpl represents the translational motion of the piping system. Mass Mp2 is for the
Xp2
Xs
Gap (d)
r ~ P i ping Stoppe Friction(~)
~Xb
K
Piping
/JJ///////////// Rack Structure
Fig. 1. Piping system on the rack structure.
~ XpI Kp Kg
~.Xp2
PK
~a Fig. 2. Mass-spring model.
bending deformation, mp2 is a generalized mass of the piping system corresponding to its vibration mode shown in fig. 1. MpI is the left mass of Mp2 from the total mass of the piping system. The spring Kp and the dashpot Cp represent the bending stiffness and damping characteristics of the piping system, respectively. The rack structure can be represented by one mass M~, one spring K s and one dashpot CS because each rack structure has the same dynamic characteristics and it is placed with sa ~ :e interval. Fig. 2 shows the mass-spring model of the piping-rack system of span l. In this figure, mass M b shows the ground on which the rack structures are fixed. Gap d and friction t~ are located between mass M s and Mpl.
2.2. Pseudoforce method In the pseudo force method for nonlinear structures, the system nonlinearities are treated as an additional applied force. The matrix is decomposed only once and iterative computation is not used as in a nonlinear analysis. [6] The equation of motion for the mass-spring system with friction and gap is as follows, [M]()(} +[C](X}
+ {R} = {F},
(1)
where [M] = mass matrix, [C] = damping matrix, { R } = force vector representing the nonlinearities due to gap and friction, ( F } = applied force vector. (.1~}, ()/'} and {X} are the acceleration, velocity and displacement vector, respectively. The nonlinear force vector is composed of the linearized force and nonlinear one due to the gap and friction. {R} = [ K ] { X }
R~Crkucture ~
~Xs
where [K ] ( R,i }
+ {Rn,},
= linearized stiffness matrix, = pseudo force vector.
(2)
H. Kobayashiet aL / Dynamicresponseofpipingsystemon rackstructure Substituting eq. (2) into eq. (1) gives
[ M I ( X } + [ C ] ( ) ( } + [ K ] ( X } = ( F } - {R.,}.
(3) Eq. (3) was solved using the Newmark-fl direct integration method [7]. The reasons why direct integration was chosen over modal superposition are as follows, (1) the pseudo force method is more easily used in the direct integration, (2) modal truncation is an additional factor when using modal superposition. Modal truncation could be a greater factor in the nonlinear problem than in a linear problem, since modes would have to be retained that not only sufficiently represent the linear structure, but also contain significant motion at the location where the nonlinear force was applied [8,9]. This could involve the retention of higher-order modes that would not be selected if the analysis involved no nonlinearities. If the time step size At in the direct integration method is small enough, the following assumption can be introduced. ( Rnl)0 = {Rnl)l"
(4)
In this method, the pseudo force is simply taken to be the one evaluated at the end of the previous time step. In other words, the time step size must be selected so small that the pseudo force vector ( R n l ) o is regarded to ( R n l } l . Suffix 0 and 1 means the pseudo force at time t o and q, respectively. Using eq. (4) and Newmark-fl direct integration method, eq. (3) expressed by,
= (F)I+[M]
(Rn,)0
{(+) -1
nonlinear elements are calculated. This information is used to calculate the pseudo force vector and its time derivative needed to integrate the equations during the next time step. In this method, the system is not truly in equilibrium at time t 1. In order to obtain an accurate solution, time step size must be so small as to satisfy eq. (4). However, computational advantages are that the mass and stiffness matrices are decomposed only once, as in a linear analysis, and the iterative computation is not required. Therefore, the computational cost was significantly reduced.
3. Experiment
3.1. Experimentalmodel The vibration test was carried out to clarify the effect of local nonlinearities and interaction on the piping-rack system. The modelling methodology and the pseudo force method were also verified for the solution of the seismic response of the piping-rack system with local nonlinearities. The straight piping model of 3 / 4 inch in diameter, 2.9 mm in thickness and 6 m in length is simply supported with a span of 4 m by the rack structures as shown in fig. 3. In order to realize the dynamic response characteristics of a serial piping system in a single span piping model, weights of 2.25 kg are attached to both free ends of the piping system. The rack structure in this experiment is the frame structure. Two types of rack structure were examined: the beam thickness of one type is 5.0 mm and of the other 6.5 mm. The piping system is constrained by elastic beam stoppers of 1.5 mm in thickness on the rack structure. Friction and gap between the piping system and the stopper on the rack structure are modeled. The gap size
1 ( J(}o + ~ - ~ ( 2 } o
+ B-~t~ (x}0} +[C](( ~ -
l){ 2}oat + ( ~---B-1){ 2}o
+ 2-~(x}0 ,
(5)
where At = t I -- t o. For a given time step, eq. (5) is integrated numerically and iterative computations are not performed. Then, the displacements and velocities of nodes associated with
343
_
< ~ Stopper~Excitation "-J ~Rack Structure Fig. 3. Experimental model.
344
H. Kobayashi et al. / Dynamic response of piping system on rack structure
4.0
Table 1 Properties of the experimental model
2.0
Weight Mass Natural Damping Beam (kg) ratio frequency (%) thickness (Hz) (mm) Piping Rack structure A
14.7 5.0 ~
-
9.6
0.4
-
0.6
8.9
0.4
5.0
].0
~" - 2 . 0 ;'q - 3 . 0
Rack 19.5 ~ structure B
0.3
6.9
0.6
-4.0
6.5
J 1.0
0
i 2.0
J 3.0
4.0
5.0
Time (sec)
a Per one unit
Fig. 5. Displacement at midpoint of piping.
was adjusted to 0.0 m m a n d 1.0 m m by sliding stoppers o n the rack structures. Friction was i n t r o d u c e d between piping a n d rack structure by a sliding interface of stainless steel. T h e friction coefficient is o b t a i n e d in the range from 0.2 to 0.3 by static loading test. T h e sinusoidal sweep test a n d r a n d o m wave excitation test of the white noise using the shaking table were p e r f o r m e d u n d e r the following s u p p o r t conditions: (1) linear s u p p o r t c o n d i t i o n (without gap a n d friction), (2) n o n l i n e a r s u p p o r t c o n d i t i o n (with gap a n d friction). In order to suppress the friction effect a n d to realize the linear s u p p o r t condition, the piping system was h u n g at the s u p p o r t p o i n t by two wires. Accelerations,
relative displacements a n d b e n d i n g stresses of b o t h the piping system a n d the rack structures were measured. 3.2. Test results
A t first, the natural frequency a n d the d a m p i n g ratio of each rack structure are o b t a i n e d b y the sinusoidal sweep test. Those of the piping system c o r r e s p o n d i n g to the vibration s h o w n in fig. 1 are also measured. These are summarized in table 1. T h e n a t u r a l frequency of the rack structure was adjusted by the thickness of the elastic b e a m of the frame structure a n d an additional weight to the rack structure.
180
\\x
90
S -180 °° F 200
150
- -
without
Gap and F r i c t i o n
....
w i t h Gap and F r i c t i o n
-----
w i t h Gap o n l y
100
o=
50
4
6
7
8
9
10
11
12
13
Frequency (Hz)
Fig. 4. Comparison of frequency response function at midpoint of piping.
t4
15
H. Kobayashi et al. / Dynamic response of piping system on rack structure
Support
Expt. Anal.
Linear
o
Gap, Friction
Mass Ratio:O.6
- - '
[]
345
Fig. 5 shows the typical time history of midpoint of the piping system by the random wave excitation test. The center of this vibratory amplitude moves within a gap size (1 mm) and repeats the stick and slip. This
2O u~ i
~I0 R
/
Support IExpt. Anal. j 0 --Gap, J _ I Friction °I !. . . .
Mass Ratio=O.6
4oi Linear
O
o. . . . . . I
o- . . . . I
c~]. . . . . . . .
-c
I
2 30
0//'/
/
]
I00 200 300 400 Input Acceleration (Gal)
2o
Fig.& Response displacementofpiping.
~= 1o []
Fig. 4 shows the variation of frequency response functions at the midpoint of the piping system due to support conditions. These frequency response functions were calculated from the results of the sinusoidal sweep test. The magnification factor of the piping system with gap only at the support point is a little smaller than that without gap and friction. The reason of the response reduction by gap are, (1) a part of the vibrational energy of the piping system of the fundamental vibration mode is reduced by the energy dissipation due to the collision of the piping and support, (2) a part of the vibrational energy of the fundamental vibration mode shifts to higher vibration modes excited by the impact. Moreover, response with both gap and friction becomes much smaller than that with gap only. It is indicated that the reduction effect by friction is greater than that by the gap because energy dissipation by friction is greater than that by impact.
2,0~ I Support
Expt.Ianal "I
/ILinear
o
100
20
15
t~
,7"
o.ol
0
1O0
,
200
,
300
,
400
Input Acceleration (Gal)
Fig. 7. Response acceleration of rack structure.
-~
Mass Ratio=0.3 / O
.~. . . . 9 ~ - ~ ' -
D
100 200 300 Input Acceleration (Gal)
400
Fig. 9. Response displacement of piping.
Support Expt. Anal. © 1.5 Linear
/
o ....... f~...~-----~ ......
Support Expt. Anal. Linear 0 Gap, m Friction ~ /
10
1.0
~o.st
400
o~
g
~
300 (Gal)
Fig. 8. Bendingstressofpiping.
Mass Ratio=Oo6
/--I
2~
Input A c c e l e r a t i o n
o.5
Mass Ratio=0.3 ,~
@~tion~ o / O _ / _ f ~ . . . . -~-/ . . . . []
0.0
i
L
I
100 200 300 Input Acceleration (Gal)
L
400
Fig. 10. Response acceleration of rack structure.
H. Kobayashi et al. / Dynamic response of piping system on rack structure
346
Sop__por ! Ept.[__all 60
Linear Gap,
4 -
i
Friction]
40
g c~
20
Ratio:O.3
0
~
o I
I
100 Input
200 Acceleration
300
I
400
(Ga])
Fig. 11. Bending stress of piping. phenomena indicates that vibration energy is dissipated by friction. Figs. 6-11 shows the relationship between the maximum input acceleration and the maximum response of the piping system and the rack structure excited by the random wave excitation test. The mass ratio 3' was calculated by the following equation:
T = M p / ( M v + M~),
(6)
where Mp = Mpl + Mp2. The response of both the piping system and the rack structure under linear support condition (without gap and friction) increases linearly with input acceleration. However, those having gap and friction do not increase linearly. A remarkable response reduction due to gap and friction is observed. The mass ratio hardly affects the response reduction by gap and friction. The solid and dotted line in these figures are the analytical results by the above mentioned pseudo force method. In this analysis, the friction coefficient was set to 0.25, which is the mean value of the test results, and the time step size in the direct integration was 0.001 s. Experimental results are found to nearly agree with analytical ones. From these comparisons, it is confirmed that the modeling methodology and the pseudo force method are adequate for the dynamic analysis of the piping-rack system with local nonlinearities such as gap and friction at support points.
4. Generation of nonlinear response spectra
parameters, which mainly affect the response of this system, were varied in this calculation: (1) natural frequency ratio f,/fo, (2) natural frequency of piping fp, (3) mass ratio T, and (4) gap size d, where fp = { K p / M p z / 2 7 r and fs = K ~ - ~ / 3 ~ / 2 ~r. Since other parameters hardly affect the response of the piping-rack system or are actually limited in a small range, they are fixed as follows: (5) damping ratio of piping h p = 0 . 0 5 , (6) damping ratio of rack structure h~ = 0.05, (7) friction coefficient # = 0.3, (8) stiffness of stopper Kg = 3.0 Kp, (9) max. ground acceleration 400 gal, where, hp = Cp/2V~-~pz.K p and h, = C , / ~ . ~ . K , .
4. 2. Linear response spectra Typical linear response spectra are shown in figs. 12-15. In case of the linear support condition, the acceleration response of both the piping system and the rack structure reduces with the increment of the mass ratio for any natural frequency of the piping system. And the response of the piping system is sensitive to the mass ratio.
4.3. Nonlinear response spectra Figs. 16-25 are the typical response spectra with gap and friction at the support point. In these figures, "gap
6.0
'
_i Mass ~
.........
5.0
0.1
. . . .
v
o T V
fp=2 Hz
g %.
-
4.0
3.0
z.o
1.0
4.1. Calculation parameter By applying the pseudo force method to the massspring system illustrated in fig. 2, the nonlinear response spectra were obtained. The following four
0.0 0.0
I
0.5
I
I
I
I
1.0 1.5 2,0 2,5 Frequency Ratio ( f s / f p ) Fig. 12. Linear response spectra of rack structure.
3.0
347
H. Kobayashi et aL / Dynamic response of piping system on rack structure 6.0
A
,
5.0
i
4.0
i %
l
,
i ,,'\
Mass Ratio:O.8 fp=2Hz
Mass Ratio 0.01 ........... 0.1 0.2 0.5 0.8 fp=2 Hz
2.0 %
i ," ",\
3.0
%
,
U
,,' I
i,",' //1
2.0
I','/,/"
.-.~-"
Gap Size ( m m ) 0.0 2.0 5.0 lO.O 20.0
•" - ~
A
-,................ ,~. . . . . .
~ /'"-\
f~.-.. /
"~Z --' ~ I ~
I
O.G
~.~:-..-=..
~
j
0.0
I
_
1.0
0.5
L
1.0 1.5 2.JO Frequency Ratio (fs/fp)
0.0
0.5
1.0 1.5 2.0 2.5 Frequency Ratio ( f s / f p )
Mass Ratio=O.8 fp=2Hz
3.0
Gap Size (~) 0.0 ..............
2.0
5.0 lO.O
Fig. 13. Linear response spectra of piping.
...... Mass Ratio 0.01 ........... 0.1 ....... 0.2 0.5 0.8 fp=10 Hz
3.0
,~ 2.0
v
20.0
2.0 1.0 0.0 ~ 0.0
i.o
-
0,5
-
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
3.0
Fig. 17. Nonlinear response spectra of piping.
0.0 0.0
0L
~0 ~
20 Frequency Ratio (fs/fp)
~ 2s
3.0
~0
II -
• ----L 3.0
Mass Ratio~ o o~
-- . . . . ---.
2.0
.
.
.
.
.
.
.
i ol 0.2 0.5 0.8
fp=10 Hz
2.0
Gap Size (mm) 0.0 2.0 5.0 lO.O 20.0 Mass Ratio=0.8
-
............ ----....
Fig. 14. Linear response spectra of rack structure.
Z
3.0
Fig. 16. Nonlinear response spectra of rack structure.
0.0
v
2.5
~ ~,i~~ I,~',.I
....
?!il
v
u~
I.C
fp:2Hz
"ii~,~
u
1.0 0.0 0.0
N I
0.5
I
I
l
I.O 1.5 2.0 Frequency Ratio (fs/fp)
l
2.5
Fig. 15. Linear response spectra of piping.
3.0
O.C 0.0
i 0.5
~
~
~~' 1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
Fig. 18. Nonlinear response spectra of rack structure.
3.0
348
H. Kobayashi et a L / Dynamic response of piping system on rack structure Mass Ratio=0.8 fp=10 Hz
_-•Gap
Mass Ratio=0.1 fs=2 Hz
Size (mm) 0.0 _ 2.0 _ _ 5.0
.... ~ - ........ . -----
Gap SioZ.~ ( m m ~ --
......
--2TC-
%
~o.o
%
2.o
u u
3.o
't 1.0
2.0
0.0 0.0
i
~
i
i
0.5
1.0
1.5
2.0
1.0
i
2.5
3.0
Frequency Ratio ( f s / f p )
Fig. 21. Nonlinear response of rack structure. 0.0 ~ 0.0
0.5
z 1.0 1.5 2.0 Frequency Ratio ( f s / f p )
2.5
3.0
Fig. 19. Nonlinear response spectra of rack structure. Gap Size (mm) 0.0 2.0 5.0 10.0 20.0
Mass Ratio=0.8
fp=10 Hz
size = 0 ram" represents the linear support condition. The response of the piping system is generally observed to be suppressed by gap and friction. In the case that the mass ratio y is relatively large, impulsive acceleration is generated by the impact between the piping system and the stopper on the rack structure. That is why some nonlinear acceleration response of the rack structure becomes larger than that of the linear model (fig. 16 and fig. 21). However, as shown in fig. 18, the relative displacement response between the rack structure and the foundation becomes smaller than that of the linear one. This means that both gap and friction
g
2.0 % o
1.0
0.0 0.0
i
I
0.5
i
i
I
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
3.0
Fig. 22. Nonlinear response spectra of piping. Mass Ratio=0.1 fp=2 Hz
I ~
Gap Size ( ~ ) 0.0 2.0 5.0 10.0 20.0
I
4.0
o=
3.0
~ Mass Ratio_
....
4.0
~U--L ........... ~ZT____ -
0.01 0.1 0.2 ......
~3.0 i I
2.0
2.0
,
1.0
0.0 0.0
, I
0.5
I
"---.
I
0.8
Gap=5.0 mm fp=2 Hz
1.0
I
1.0 1.5 2.0 Frequency Ratio ( f s / f p )
"'1
2.5
Fig. 20. Nonline~respon~spectraofpiping.
|/
3.0
0.0 0.0
L l
0.5
1.0
I
1.5
l
2.0
i
2.5
Frequency Ratio ( f s / f p )
Fig. 23. Nonlinear response spectra of rack structure.
3.0
H. Kobayashi et al. / Dynamic response of piping system on rack structure
4.0
Mass Ratio 0.01
[~,,
3.o
.
\
i
.
.
.
.
.
.
.
.
.
.
o.1
.
0.2
.......
/i
0.5 0.8
/' "~ I ,.'*', ~ '~,,~
2.0
Gap:5.0 mm fp:2 HZ
,.oI 0.0 ~ 0.0
__ 0.5
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2,5
3.0
Fig. 24. Nonlinear response spectra of piping.
absorb the vibration energy and consequently reduce the stress of the rack structure in spite of its impulsive acceleration by the impact. Both the acceleration response and the bending deformation of the piping system are also confirmed to be reduced by gap and friction for all mass ratios studied here as long as when the natural frequency ratio f s / f p is larger than 0.7. This reduction effect decreases with the natural frequency of the piping system. The gap size hardly affects the response reduction of both the piping system and the rack structure. The response of the piping system is more sensitive to the mass ratio than that of the rack structure. This characteristic is similar to that in case of linear support condition. For reference, the response spectra of the piping system with gap only are shown in fig. 25. No response is
349
smaller than that of the linear one (gap size = 0 mm). A comparison between fig. 20 and fig. 25 indicates that addition of friction to the gap contributes to reducing the system response. From a comparison between nonlinear response spectra and linear ones, the response of both the piping system and the rack structure is reduced for the following conditions: (1) the natural frequency ratio f=/fp is larger than 0.7, (2) the gap size is larger than 2 mm.
5. Conclusions From the results of the numerical study and the vibration test, the dynamic characteristics of the piping-rack system with nonlinearities (gap and friction) are clarified. In case that the piping-rack system has gap only, the response reduction is small. However, the introducing friction, the seismic response is greatly reduced for the following conditions: (1) the natural frequency ratio of the rack structure to the piping system ( f s / f p ) is larger than 0.7, (2) the gap size is larger than 2 mm. For the calculation of the seismic response, they can be modeled by the mass-spring system taking the vibration mode of the piping system into account. The pseudo force method is useful and adequate to analyze the dynamic response of the piping system with local nonhnearities such as gap and friction.
Acknowledgment MaSSfp:2Ratlo:O.IHz
~
The authors wish to express their appreciation to Prof. K. Suzuki and Dr. A. Sone, Tokyo Metropolitan University, for their precious advice.
Gap SiZeo.02.0(~) 5.0
4.O
20.0
v
3.0
.
..
,,,,
(,
References
,7 0.0
0.0
I
0.5
i
,
~
1.0 1.5 2.0 Frequency Ratio (fs/fp)
L
2.5
3.0
Fig. 25. Nonfinearr~ponsespec~aofpiping(~ponly).
[1] G. KiSnig,S. Aoyagi and J.-D. W6mer, Investigation of the influence of localized nonlinearities on the behavior of structures, 8th International Conference on Structural Mechanics in Reactor Technology, Paper K9/4, Brussels (Aug. 1985) pp. 421-426. [2] G. K~nig, and J.-D. WSrner, Nonlinear effects in component-structure-interaction and their influence on the dynamic behavior of the component, 8th International Conference on Structural Mechanics in reactor Technology, Paper K9/8, Brussels (Aug. 1985) pp. 449-454.
350
H. Kobayashi et al. / Dynamic response of piping system on rack structure
[3] G. K/Snig, and J.-D. WiSrner, Damping in piping systems due to local nonlinearities, 9th International Conference on Structural Mechanics in Reactor Technology, Div. K, Lausanne (Aug. 1987) pp. 963-968. [4] H. Kobayashi, T. Chiba, Y. Yoshida et al., Dynamic response of the piping system on the rack structure with gaps and friction, 9th International Conference on Structural Mechanics in Reactor Technology, Div. K, Lausanne (Aug. 1987) pp. 995-1000. [5] K. Suzuki and A. Sone, A load combination for aseismic design of multiple supported piping system, ASME PVP 127 (July 1987) 97-104.
[6] K.D. Balakely et al., Pipe damping studies and nonlinear pipe benchmarks from snapback tests at the Heissdampfreaktor, N U R E G / C R - 3 1 8 0 (Jul, 1983). [7] L.D. Hofmeister, Dynamic analysis of structures containing nonlinear springs, Computer & Structures 8 (1978) 609-614. [8] V.N. Shah and C.B. Gilmore, Dynamic analysis of a structure with Coulomb friction, J. of Press. Vessel Tecbnol. 105 (May 1983) 171-178. [9] V.N. Shah and A.J. Hartmann, Nonlinear dynamic analysis of a structure subjected to multiple support motions. J. Press. Vessel Technol. 108 (Feb. 1981) 27-32.
341
DYNAMIC RESPONSE OF PIPING SYSTEM ON RACK STRUCTURE WITH GAPS AND FRICTIONS Hiroe KOBAYASHI
Nuclear Power Division, Ishikawafima-Harima Heavy Industries Co., Ltd., 1, Shin-Nakahara-cho Isogo-ku Yokohama, 235, Japan Misutoyo YOSHIDA
Storage Plant Division, Ishikawafima-Harima Heavy Industries Co., Ltd., 3-2-6 Toyosu Kohtoh-ku Tokyo, 135, Japan an d Yoshio O C H I
Research Institute, Ishikawafima-Harima Heavy Industries Co., Ltd, 1, Shin-Nakahara-cho Isogo ku Yokohama, 235, Japan Received 24 November 1987
In the seismic design of a piping system on a rack structure, the interaction between the piping system and the rack structure must be evaluated under the condition that the rack structure is not stiff and heavy enough compared with the piping system. Moreover, there are local nonlinearities due to the gap and friction between the piping system and the rack structure. This paper presents the influence of the interaction and the local nonlinearities upon the seismic response by numerical study and a vibration test using a shaking table. In the numerical study, the piping system and the rack structure were represented by the three degrees of freedom mass-spring model taking a vibration mode of the piping system into account. The nonlinearities due to gap and friction were defined as a function of motion and treated as the pseudo force vector (additional applied force) in an equation of motion. From the results of the numerical study and the vibration test, it was clarified that seismic response of both the rack structure and the piping system is reduced by gap and friction. Moreover, the piping system and rack structure can be represented by the three degrees of freedom mass spring model. And the local nonlinearities can be treated by the pseudo force in an equation of motion.
1. Introduction Some piping systems rest on rack structures. In a seismic design, the rack structure has been regarded as stiff and heavy enough compared with the piping system. And the interaction effect between the piping system and the rack structure (the piping-rack system) has been ignored. But this interaction effect must be evaluated when the rack structure is not stiff and heavy enough compared with the piping system. On the other hand, the gap in the piping-rack system is designed to allow for the thermal expansion and installation of a piping. Friction also exists here. The piping system may slide within the gap and cause an impact during a seismic event. The vibration energy may be dissipated
concurrently by the impact and friction within the gap [1-3]. So, the interaction and these nonlinearities strongly affect the seismic response of the piping-rack system. In order to clarify the influence of the interaction and these nonlinearities upon the seismic response, a numerical study and a vibration test were carried out. In the numerical study, the piping-rack system is represented by a three degrees of freedom mass-spring model taking a vibration mode of a piping system into account. The pseudo force method was used to analyze the nonlinear dynamic response of the piping-rack system. The nonlinearities due to the gap and friction were treated as the pseudo force vector in an equation of motion. The pseudo force is assumed to be constant for
0 0 2 9 - 5 4 9 3 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s io n )
H. Kobayashi et aL/ Dynamic response of piping system on rack structure
342
a small time increment in the direct integration of the equation of motion. The effect of the gap size, the mass ratio and a natural frequency of the piping system as well as the rack structure on the seismic response were surveyed. Moreover, nonlinear response spectra were generated [4]. A vibration test of the piping model using a shaking table was also carried out. A straight piping model of 3 / 4 inch in dia. and 6 m in length was set on two rack structures whose ends were fixed on the shaking table [4,5]. This model has gap and friction between the piping system and the stopper on the rack structure. The mass ratio of the piping system to the rack structure also varied in this test.
2. Numerical study
2.1. Modeling methodology The piping-rack system was represented by a massspring model in the numerical study. For convenience, consider one straight piping system on the rack structures placed with the same interval 1 as shown in fig. 1. The gap of size d and the friction of coefficient /~ exist between the piping system and the stopper on the rack structure. It is assumed that the oval deformation of the piping and higher vibration modes of the piping system are not caused by the impact between the piping and its stopper on the rack structure. Each rack structure is also supposed to have the same dynamic characteristics. So, each structure vibrates in phase as the seismic event as shown in fig. 1. The displacement of the piping system is composed of bending deformation and translational motion which is parallel to the resting position. In other words, it consists of elastic deformation and rigid body motion. Therefore, the mass of the piping system is divided into two parts. One mass Mpl represents the translational motion of the piping system. Mass Mp2 is for the
Xp2
Xs
Gap (d)
r ~ P i ping Stoppe Friction(~)
~Xb
K
Piping
/JJ///////////// Rack Structure
Fig. 1. Piping system on the rack structure.
~ XpI Kp Kg
~.Xp2
PK
~a Fig. 2. Mass-spring model.
bending deformation, mp2 is a generalized mass of the piping system corresponding to its vibration mode shown in fig. 1. MpI is the left mass of Mp2 from the total mass of the piping system. The spring Kp and the dashpot Cp represent the bending stiffness and damping characteristics of the piping system, respectively. The rack structure can be represented by one mass M~, one spring K s and one dashpot CS because each rack structure has the same dynamic characteristics and it is placed with sa ~ :e interval. Fig. 2 shows the mass-spring model of the piping-rack system of span l. In this figure, mass M b shows the ground on which the rack structures are fixed. Gap d and friction t~ are located between mass M s and Mpl.
2.2. Pseudoforce method In the pseudo force method for nonlinear structures, the system nonlinearities are treated as an additional applied force. The matrix is decomposed only once and iterative computation is not used as in a nonlinear analysis. [6] The equation of motion for the mass-spring system with friction and gap is as follows, [M]()(} +[C](X}
+ {R} = {F},
(1)
where [M] = mass matrix, [C] = damping matrix, { R } = force vector representing the nonlinearities due to gap and friction, ( F } = applied force vector. (.1~}, ()/'} and {X} are the acceleration, velocity and displacement vector, respectively. The nonlinear force vector is composed of the linearized force and nonlinear one due to the gap and friction. {R} = [ K ] { X }
R~Crkucture ~
~Xs
where [K ] ( R,i }
+ {Rn,},
= linearized stiffness matrix, = pseudo force vector.
(2)
H. Kobayashiet aL / Dynamicresponseofpipingsystemon rackstructure Substituting eq. (2) into eq. (1) gives
[ M I ( X } + [ C ] ( ) ( } + [ K ] ( X } = ( F } - {R.,}.
(3) Eq. (3) was solved using the Newmark-fl direct integration method [7]. The reasons why direct integration was chosen over modal superposition are as follows, (1) the pseudo force method is more easily used in the direct integration, (2) modal truncation is an additional factor when using modal superposition. Modal truncation could be a greater factor in the nonlinear problem than in a linear problem, since modes would have to be retained that not only sufficiently represent the linear structure, but also contain significant motion at the location where the nonlinear force was applied [8,9]. This could involve the retention of higher-order modes that would not be selected if the analysis involved no nonlinearities. If the time step size At in the direct integration method is small enough, the following assumption can be introduced. ( Rnl)0 = {Rnl)l"
(4)
In this method, the pseudo force is simply taken to be the one evaluated at the end of the previous time step. In other words, the time step size must be selected so small that the pseudo force vector ( R n l ) o is regarded to ( R n l } l . Suffix 0 and 1 means the pseudo force at time t o and q, respectively. Using eq. (4) and Newmark-fl direct integration method, eq. (3) expressed by,
= (F)I+[M]
(Rn,)0
{(+) -1
nonlinear elements are calculated. This information is used to calculate the pseudo force vector and its time derivative needed to integrate the equations during the next time step. In this method, the system is not truly in equilibrium at time t 1. In order to obtain an accurate solution, time step size must be so small as to satisfy eq. (4). However, computational advantages are that the mass and stiffness matrices are decomposed only once, as in a linear analysis, and the iterative computation is not required. Therefore, the computational cost was significantly reduced.
3. Experiment
3.1. Experimentalmodel The vibration test was carried out to clarify the effect of local nonlinearities and interaction on the piping-rack system. The modelling methodology and the pseudo force method were also verified for the solution of the seismic response of the piping-rack system with local nonlinearities. The straight piping model of 3 / 4 inch in diameter, 2.9 mm in thickness and 6 m in length is simply supported with a span of 4 m by the rack structures as shown in fig. 3. In order to realize the dynamic response characteristics of a serial piping system in a single span piping model, weights of 2.25 kg are attached to both free ends of the piping system. The rack structure in this experiment is the frame structure. Two types of rack structure were examined: the beam thickness of one type is 5.0 mm and of the other 6.5 mm. The piping system is constrained by elastic beam stoppers of 1.5 mm in thickness on the rack structure. Friction and gap between the piping system and the stopper on the rack structure are modeled. The gap size
1 ( J(}o + ~ - ~ ( 2 } o
+ B-~t~ (x}0} +[C](( ~ -
l){ 2}oat + ( ~---B-1){ 2}o
+ 2-~(x}0 ,
(5)
where At = t I -- t o. For a given time step, eq. (5) is integrated numerically and iterative computations are not performed. Then, the displacements and velocities of nodes associated with
343
_
< ~ Stopper~Excitation "-J ~Rack Structure Fig. 3. Experimental model.
344
H. Kobayashi et al. / Dynamic response of piping system on rack structure
4.0
Table 1 Properties of the experimental model
2.0
Weight Mass Natural Damping Beam (kg) ratio frequency (%) thickness (Hz) (mm) Piping Rack structure A
14.7 5.0 ~
-
9.6
0.4
-
0.6
8.9
0.4
5.0
].0
~" - 2 . 0 ;'q - 3 . 0
Rack 19.5 ~ structure B
0.3
6.9
0.6
-4.0
6.5
J 1.0
0
i 2.0
J 3.0
4.0
5.0
Time (sec)
a Per one unit
Fig. 5. Displacement at midpoint of piping.
was adjusted to 0.0 m m a n d 1.0 m m by sliding stoppers o n the rack structures. Friction was i n t r o d u c e d between piping a n d rack structure by a sliding interface of stainless steel. T h e friction coefficient is o b t a i n e d in the range from 0.2 to 0.3 by static loading test. T h e sinusoidal sweep test a n d r a n d o m wave excitation test of the white noise using the shaking table were p e r f o r m e d u n d e r the following s u p p o r t conditions: (1) linear s u p p o r t c o n d i t i o n (without gap a n d friction), (2) n o n l i n e a r s u p p o r t c o n d i t i o n (with gap a n d friction). In order to suppress the friction effect a n d to realize the linear s u p p o r t condition, the piping system was h u n g at the s u p p o r t p o i n t by two wires. Accelerations,
relative displacements a n d b e n d i n g stresses of b o t h the piping system a n d the rack structures were measured. 3.2. Test results
A t first, the natural frequency a n d the d a m p i n g ratio of each rack structure are o b t a i n e d b y the sinusoidal sweep test. Those of the piping system c o r r e s p o n d i n g to the vibration s h o w n in fig. 1 are also measured. These are summarized in table 1. T h e n a t u r a l frequency of the rack structure was adjusted by the thickness of the elastic b e a m of the frame structure a n d an additional weight to the rack structure.
180
\\x
90
S -180 °° F 200
150
- -
without
Gap and F r i c t i o n
....
w i t h Gap and F r i c t i o n
-----
w i t h Gap o n l y
100
o=
50
4
6
7
8
9
10
11
12
13
Frequency (Hz)
Fig. 4. Comparison of frequency response function at midpoint of piping.
t4
15
H. Kobayashi et al. / Dynamic response of piping system on rack structure
Support
Expt. Anal.
Linear
o
Gap, Friction
Mass Ratio:O.6
- - '
[]
345
Fig. 5 shows the typical time history of midpoint of the piping system by the random wave excitation test. The center of this vibratory amplitude moves within a gap size (1 mm) and repeats the stick and slip. This
2O u~ i
~I0 R
/
Support IExpt. Anal. j 0 --Gap, J _ I Friction °I !. . . .
Mass Ratio=O.6
4oi Linear
O
o. . . . . . I
o- . . . . I
c~]. . . . . . . .
-c
I
2 30
0//'/
/
]
I00 200 300 400 Input Acceleration (Gal)
2o
Fig.& Response displacementofpiping.
~= 1o []
Fig. 4 shows the variation of frequency response functions at the midpoint of the piping system due to support conditions. These frequency response functions were calculated from the results of the sinusoidal sweep test. The magnification factor of the piping system with gap only at the support point is a little smaller than that without gap and friction. The reason of the response reduction by gap are, (1) a part of the vibrational energy of the piping system of the fundamental vibration mode is reduced by the energy dissipation due to the collision of the piping and support, (2) a part of the vibrational energy of the fundamental vibration mode shifts to higher vibration modes excited by the impact. Moreover, response with both gap and friction becomes much smaller than that with gap only. It is indicated that the reduction effect by friction is greater than that by the gap because energy dissipation by friction is greater than that by impact.
2,0~ I Support
Expt.Ianal "I
/ILinear
o
100
20
15
t~
,7"
o.ol
0
1O0
,
200
,
300
,
400
Input Acceleration (Gal)
Fig. 7. Response acceleration of rack structure.
-~
Mass Ratio=0.3 / O
.~. . . . 9 ~ - ~ ' -
D
100 200 300 Input Acceleration (Gal)
400
Fig. 9. Response displacement of piping.
Support Expt. Anal. © 1.5 Linear
/
o ....... f~...~-----~ ......
Support Expt. Anal. Linear 0 Gap, m Friction ~ /
10
1.0
~o.st
400
o~
g
~
300 (Gal)
Fig. 8. Bendingstressofpiping.
Mass Ratio=Oo6
/--I
2~
Input A c c e l e r a t i o n
o.5
Mass Ratio=0.3 ,~
@~tion~ o / O _ / _ f ~ . . . . -~-/ . . . . []
0.0
i
L
I
100 200 300 Input Acceleration (Gal)
L
400
Fig. 10. Response acceleration of rack structure.
H. Kobayashi et al. / Dynamic response of piping system on rack structure
346
Sop__por ! Ept.[__all 60
Linear Gap,
4 -
i
Friction]
40
g c~
20
Ratio:O.3
0
~
o I
I
100 Input
200 Acceleration
300
I
400
(Ga])
Fig. 11. Bending stress of piping. phenomena indicates that vibration energy is dissipated by friction. Figs. 6-11 shows the relationship between the maximum input acceleration and the maximum response of the piping system and the rack structure excited by the random wave excitation test. The mass ratio 3' was calculated by the following equation:
T = M p / ( M v + M~),
(6)
where Mp = Mpl + Mp2. The response of both the piping system and the rack structure under linear support condition (without gap and friction) increases linearly with input acceleration. However, those having gap and friction do not increase linearly. A remarkable response reduction due to gap and friction is observed. The mass ratio hardly affects the response reduction by gap and friction. The solid and dotted line in these figures are the analytical results by the above mentioned pseudo force method. In this analysis, the friction coefficient was set to 0.25, which is the mean value of the test results, and the time step size in the direct integration was 0.001 s. Experimental results are found to nearly agree with analytical ones. From these comparisons, it is confirmed that the modeling methodology and the pseudo force method are adequate for the dynamic analysis of the piping-rack system with local nonlinearities such as gap and friction at support points.
4. Generation of nonlinear response spectra
parameters, which mainly affect the response of this system, were varied in this calculation: (1) natural frequency ratio f,/fo, (2) natural frequency of piping fp, (3) mass ratio T, and (4) gap size d, where fp = { K p / M p z / 2 7 r and fs = K ~ - ~ / 3 ~ / 2 ~r. Since other parameters hardly affect the response of the piping-rack system or are actually limited in a small range, they are fixed as follows: (5) damping ratio of piping h p = 0 . 0 5 , (6) damping ratio of rack structure h~ = 0.05, (7) friction coefficient # = 0.3, (8) stiffness of stopper Kg = 3.0 Kp, (9) max. ground acceleration 400 gal, where, hp = Cp/2V~-~pz.K p and h, = C , / ~ . ~ . K , .
4. 2. Linear response spectra Typical linear response spectra are shown in figs. 12-15. In case of the linear support condition, the acceleration response of both the piping system and the rack structure reduces with the increment of the mass ratio for any natural frequency of the piping system. And the response of the piping system is sensitive to the mass ratio.
4.3. Nonlinear response spectra Figs. 16-25 are the typical response spectra with gap and friction at the support point. In these figures, "gap
6.0
'
_i Mass ~
.........
5.0
0.1
. . . .
v
o T V
fp=2 Hz
g %.
-
4.0
3.0
z.o
1.0
4.1. Calculation parameter By applying the pseudo force method to the massspring system illustrated in fig. 2, the nonlinear response spectra were obtained. The following four
0.0 0.0
I
0.5
I
I
I
I
1.0 1.5 2,0 2,5 Frequency Ratio ( f s / f p ) Fig. 12. Linear response spectra of rack structure.
3.0
347
H. Kobayashi et aL / Dynamic response of piping system on rack structure 6.0
A
,
5.0
i
4.0
i %
l
,
i ,,'\
Mass Ratio:O.8 fp=2Hz
Mass Ratio 0.01 ........... 0.1 0.2 0.5 0.8 fp=2 Hz
2.0 %
i ," ",\
3.0
%
,
U
,,' I
i,",' //1
2.0
I','/,/"
.-.~-"
Gap Size ( m m ) 0.0 2.0 5.0 lO.O 20.0
•" - ~
A
-,................ ,~. . . . . .
~ /'"-\
f~.-.. /
"~Z --' ~ I ~
I
O.G
~.~:-..-=..
~
j
0.0
I
_
1.0
0.5
L
1.0 1.5 2.JO Frequency Ratio (fs/fp)
0.0
0.5
1.0 1.5 2.0 2.5 Frequency Ratio ( f s / f p )
Mass Ratio=O.8 fp=2Hz
3.0
Gap Size (~) 0.0 ..............
2.0
5.0 lO.O
Fig. 13. Linear response spectra of piping.
...... Mass Ratio 0.01 ........... 0.1 ....... 0.2 0.5 0.8 fp=10 Hz
3.0
,~ 2.0
v
20.0
2.0 1.0 0.0 ~ 0.0
i.o
-
0,5
-
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
3.0
Fig. 17. Nonlinear response spectra of piping.
0.0 0.0
0L
~0 ~
20 Frequency Ratio (fs/fp)
~ 2s
3.0
~0
II -
• ----L 3.0
Mass Ratio~ o o~
-- . . . . ---.
2.0
.
.
.
.
.
.
.
i ol 0.2 0.5 0.8
fp=10 Hz
2.0
Gap Size (mm) 0.0 2.0 5.0 lO.O 20.0 Mass Ratio=0.8
-
............ ----....
Fig. 14. Linear response spectra of rack structure.
Z
3.0
Fig. 16. Nonlinear response spectra of rack structure.
0.0
v
2.5
~ ~,i~~ I,~',.I
....
?!il
v
u~
I.C
fp:2Hz
"ii~,~
u
1.0 0.0 0.0
N I
0.5
I
I
l
I.O 1.5 2.0 Frequency Ratio (fs/fp)
l
2.5
Fig. 15. Linear response spectra of piping.
3.0
O.C 0.0
i 0.5
~
~
~~' 1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
Fig. 18. Nonlinear response spectra of rack structure.
3.0
348
H. Kobayashi et a L / Dynamic response of piping system on rack structure Mass Ratio=0.8 fp=10 Hz
_-•Gap
Mass Ratio=0.1 fs=2 Hz
Size (mm) 0.0 _ 2.0 _ _ 5.0
.... ~ - ........ . -----
Gap SioZ.~ ( m m ~ --
......
--2TC-
%
~o.o
%
2.o
u u
3.o
't 1.0
2.0
0.0 0.0
i
~
i
i
0.5
1.0
1.5
2.0
1.0
i
2.5
3.0
Frequency Ratio ( f s / f p )
Fig. 21. Nonlinear response of rack structure. 0.0 ~ 0.0
0.5
z 1.0 1.5 2.0 Frequency Ratio ( f s / f p )
2.5
3.0
Fig. 19. Nonlinear response spectra of rack structure. Gap Size (mm) 0.0 2.0 5.0 10.0 20.0
Mass Ratio=0.8
fp=10 Hz
size = 0 ram" represents the linear support condition. The response of the piping system is generally observed to be suppressed by gap and friction. In the case that the mass ratio y is relatively large, impulsive acceleration is generated by the impact between the piping system and the stopper on the rack structure. That is why some nonlinear acceleration response of the rack structure becomes larger than that of the linear model (fig. 16 and fig. 21). However, as shown in fig. 18, the relative displacement response between the rack structure and the foundation becomes smaller than that of the linear one. This means that both gap and friction
g
2.0 % o
1.0
0.0 0.0
i
I
0.5
i
i
I
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2.5
3.0
Fig. 22. Nonlinear response spectra of piping. Mass Ratio=0.1 fp=2 Hz
I ~
Gap Size ( ~ ) 0.0 2.0 5.0 10.0 20.0
I
4.0
o=
3.0
~ Mass Ratio_
....
4.0
~U--L ........... ~ZT____ -
0.01 0.1 0.2 ......
~3.0 i I
2.0
2.0
,
1.0
0.0 0.0
, I
0.5
I
"---.
I
0.8
Gap=5.0 mm fp=2 Hz
1.0
I
1.0 1.5 2.0 Frequency Ratio ( f s / f p )
"'1
2.5
Fig. 20. Nonline~respon~spectraofpiping.
|/
3.0
0.0 0.0
L l
0.5
1.0
I
1.5
l
2.0
i
2.5
Frequency Ratio ( f s / f p )
Fig. 23. Nonlinear response spectra of rack structure.
3.0
H. Kobayashi et al. / Dynamic response of piping system on rack structure
4.0
Mass Ratio 0.01
[~,,
3.o
.
\
i
.
.
.
.
.
.
.
.
.
.
o.1
.
0.2
.......
/i
0.5 0.8
/' "~ I ,.'*', ~ '~,,~
2.0
Gap:5.0 mm fp:2 HZ
,.oI 0.0 ~ 0.0
__ 0.5
1.0 1.5 2.0 Frequency Ratio (fs/fp)
2,5
3.0
Fig. 24. Nonlinear response spectra of piping.
absorb the vibration energy and consequently reduce the stress of the rack structure in spite of its impulsive acceleration by the impact. Both the acceleration response and the bending deformation of the piping system are also confirmed to be reduced by gap and friction for all mass ratios studied here as long as when the natural frequency ratio f s / f p is larger than 0.7. This reduction effect decreases with the natural frequency of the piping system. The gap size hardly affects the response reduction of both the piping system and the rack structure. The response of the piping system is more sensitive to the mass ratio than that of the rack structure. This characteristic is similar to that in case of linear support condition. For reference, the response spectra of the piping system with gap only are shown in fig. 25. No response is
349
smaller than that of the linear one (gap size = 0 mm). A comparison between fig. 20 and fig. 25 indicates that addition of friction to the gap contributes to reducing the system response. From a comparison between nonlinear response spectra and linear ones, the response of both the piping system and the rack structure is reduced for the following conditions: (1) the natural frequency ratio f=/fp is larger than 0.7, (2) the gap size is larger than 2 mm.
5. Conclusions From the results of the numerical study and the vibration test, the dynamic characteristics of the piping-rack system with nonlinearities (gap and friction) are clarified. In case that the piping-rack system has gap only, the response reduction is small. However, the introducing friction, the seismic response is greatly reduced for the following conditions: (1) the natural frequency ratio of the rack structure to the piping system ( f s / f p ) is larger than 0.7, (2) the gap size is larger than 2 mm. For the calculation of the seismic response, they can be modeled by the mass-spring system taking the vibration mode of the piping system into account. The pseudo force method is useful and adequate to analyze the dynamic response of the piping system with local nonhnearities such as gap and friction.
Acknowledgment MaSSfp:2Ratlo:O.IHz
~
The authors wish to express their appreciation to Prof. K. Suzuki and Dr. A. Sone, Tokyo Metropolitan University, for their precious advice.
Gap SiZeo.02.0(~) 5.0
4.O
20.0
v
3.0
.
..
,,,,
(,
References
,7 0.0
0.0
I
0.5
i
,
~
1.0 1.5 2.0 Frequency Ratio (fs/fp)
L
2.5
3.0
Fig. 25. Nonfinearr~ponsespec~aofpiping(~ponly).
[1] G. KiSnig,S. Aoyagi and J.-D. W6mer, Investigation of the influence of localized nonlinearities on the behavior of structures, 8th International Conference on Structural Mechanics in Reactor Technology, Paper K9/4, Brussels (Aug. 1985) pp. 421-426. [2] G. K~nig, and J.-D. WSrner, Nonlinear effects in component-structure-interaction and their influence on the dynamic behavior of the component, 8th International Conference on Structural Mechanics in reactor Technology, Paper K9/8, Brussels (Aug. 1985) pp. 449-454.
350
H. Kobayashi et al. / Dynamic response of piping system on rack structure
[3] G. K/Snig, and J.-D. WiSrner, Damping in piping systems due to local nonlinearities, 9th International Conference on Structural Mechanics in Reactor Technology, Div. K, Lausanne (Aug. 1987) pp. 963-968. [4] H. Kobayashi, T. Chiba, Y. Yoshida et al., Dynamic response of the piping system on the rack structure with gaps and friction, 9th International Conference on Structural Mechanics in Reactor Technology, Div. K, Lausanne (Aug. 1987) pp. 995-1000. [5] K. Suzuki and A. Sone, A load combination for aseismic design of multiple supported piping system, ASME PVP 127 (July 1987) 97-104.
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