Response of piping system on friction support to bi-directional excitation

S.V. Bakre a, R.S. Jangid a,∗, G.R. Reddy b

a Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, Indiab Reactor Safety Division, Bhabha Atomic Research Center, Anushaktinagar, Mumbai 400 058, India

Abstract

Installation of friction devices between a piping system and its supporting medium is an effective way of energy dissipation in the piping systems.In this paper, seismic effectiveness of friction type support for a piping system subjected to two horizontal components of earthquake motion isinvestigated. The interaction between the mobilized restoring forces of the friction support is duly considered. The non-linear behavior of therestoring forces of the support is modeled as an elastic-perfectly plastic system with a very high value of initial stiffness. Such an idealizationavoids keeping track of transitional rules (as required in conventional modeling of friction systems) under arbitrary dynamic loading. The frictionalforces mobilized at the friction support are assumed to be dependent on the sliding velocity and instantaneous normal force acting on the support.A detailed systematic procedure for analysis of piping systems supported on friction support considering the effects of bi-directional interactionof the frictional forces is presented. The proposed procedure is validated by comparing the analytical seismic responses of a spatial piping systemsupported on a friction support with the corresponding experimental results. The responses of the piping system and the frictional forces of thesupport are observed to be in close agreement with the experimental results validating the proposed analysis procedure. It was also observed thatthe friction supports are very effective in reducing the seismic response of piping systems. In order to investigate the effects of bi-directionalinteraction of the frictional forces, the seismic responses of the piping system are compared by considering and ignoring the interaction under fewnarrow-band and broad-band (real earthquake) ground motions. The bi-directional interaction of the frictional forces has significant effects on the

response of piping system and should be included in the analysis of piping systems supported on friction supports. Further, it was also observedthat the velocity dependence of the friction coefficient does not have noticeable effects on the peak responses of the piping system.

1

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. Introduction

The technology of passive control of structures has nownitialized its applications in the lifeline structures like pip-ng systems in industrial installations and utilities like nuclearower plants (Kunieda et al., 1987). Owing to its advantagesver its sister branches i.e. active control or semi-active controlechnologies, this technology finds a wide range of applica-ions varying from its use in upcoming piping installations tohe retrofitting of existing facilities without significantly affect-ng the production/operation and the structural characteristics.he primary objective of the control technology is to satisfy the

otion related design requirements wherein strength is viewed

s a constraint and not as a primary requirement (like assumedn conventional designs). The passive control technology is now

sepwca

aving a reputation of over five decades since its emergence andad been successfully used in structures like buildings, bridgesnd water tanks in the form of supplemental devices. Several pas-ive control devices are reported in the literature namely friction,iscous or visco-elastic dampers, yielding elements or auxiliaryscillators like tuned mass dampers and multiple tuned massampers (Mahmoodi, 1972; Hanson and Soong, 2001; Oliveto,002). These passive control devices act as energy sinks or fusesnd control the motion of the system by significantly increasinghe damping due to energy dissipation in the structural sys-em. These devices can be easily replaced after an earthquake ifequired due to occurred damage.

There had been an attempt to prove seismic effectiveness ofupplemental devices for piping systems in the past (Kuneidat al.,1987). However, the proposed devices were very com-

lex in design, fabrication and implementation, as a result, thereas an urgent need to develop simple control devices that are

apable to dissipate the earthquake forces and at the same timellow free expansion of the piping systems for thermal load-

ispfiKsLesoBi1dtcf(idoatfn(ttpt

lfprie

wpfowfbaiselvo

2

s

ngs. Using friction in the supports was the best choice as itatisfies the above-mentioned requirements without altering theiping characteristics. Anderson and Singh (1976) were therst to propose the modeling of friction at the piping support.obayashi et al. (1989) modeled friction between the piping

ystem and its gap support to establish its seismic effectiveness.ater on, Suzuki et al. (1992), Chiba et al. (1992) and Shimuzut al. (1996) performed extensive shake table tests on a large-cale piping system resting on a friction support in the formf Teflon, which were further studied by Yokoi et al. (1993).ased on the results obtained from above experimental stud-

es (Suzuki et al., 1992; Chiba et al., 1992; Shimuzu et al.,996; Yokoi et al., 1993), Reddy et al. (1999) presented theesigner’s methods to analyze piping systems resting on fric-ion supports. Recently, Bakre et al. (2004) have presented theomparison of time-history analysis of the piping system onriction support, with the methods presented by Reddy et al.1999) and the corresponding experimental results, by model-ng the frictional forces independently in the two orthogonalirections (two-dimensional planar) and ignoring the effectsf bi-directional interaction. Few observations from all of thebove-mentioned studies to be noted are: (i) the non-linear fric-ion support is referred as equivalent linear, (ii) the frictionalorces are assumed to be acting independently in two orthogo-al directions without any effects of bi-directional interaction,iii) the limiting friction force of the friction support is assumed

o remain constant throughout the analysis, and (iv) the effects ofhe velocity dependence of friction coefficient on the controllediping system were not considered. Mokha et al. (1993) reportedhe interaction effects of restoring forces for the friction type iso-

(iYA

Fig. 1. Model of piping system

125

ation system on the seismic responses of buildings, which areurther confirmed by Jangid (1996). The seismic behavior of theiping systems is far different than the building structures, as aesult, there is a need to conduct separate study on the behav-or of piping systems with friction supports under bi-directionalxcitation.

In this paper, seismic response of a spatial piping systemith friction support is investigated for two horizontal com-onents of four narrow-band (frequency range of 3–9 Hz) andour real earthquake ground motions. The specific objectivesf the present study are summarized as: (i) to present a methodhich incorporates the interaction effects of the frictional forces

or dynamic analysis of piping systems with friction support toi-directional ground excitations; (ii) to validate the proposednalytical procedure by comparing the analytical and exper-mental results; (iii) to study the seismic response of pipingystem under variation of the friction coefficient; (iv) to study theffects of bi-directional interaction of the frictional forces mobi-ized at the friction support; and (v) to investigate the effects ofelocity dependence of the friction coefficient at friction supportn the seismic response of piping system.

. Structural model

Fig. 1(a) shows a schematic layout of the piping system con-idered in the present study which was tested by Shimuzu et al.

1996) and further studied by Reddy et al. (1999). The pip-ng system is made of pipe elements of carbon steel havingoung’s modulus (E) of 192.2 GN/m2, poisons ratio = 0.214.ll the bends in the piping system are 90◦ with bend radius

with friction support.

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f 225 mm. The piping system is supported on a friction sup-ort and on rod restraints with its ends rigidly anchored to thehake table, as shown in Fig. 1(a). The damping in the pipingystem without friction support obtained from the experimen-al test model is equal to 1.3% (Reddy et al., 1999). Fig. 1(b)hows an enlarged view of the friction support illustrating theeflon coated mild steel plate that was used to dissipate the

nput dynamic energy. The plate was placed below a steel sliderhat was welded to the piping system. The coefficient of fric-ion between the two surfaces (Teflon and steel) is adopted as

max = 0.15 (at large velocity). The energy gets dissipated whenelative sliding between the two interfaces occurs during shakingf the piping system.

The experimental results of the selected piping system modelith friction support are available for comparison to validate

he proposed analysis procedure and to establish the need ofonsidering bi-directional interaction of the frictional forces.hese experimental tests were performed at the lab facilities ofational research Institute for Earth Science and Disaster pre-ention (NIED), Tsukuba, Japan, which houses a shake tablef area 15 m × 15 m with 500 T capacity. From the series ofests performed on the full-scale piping system, experimentalesults of the piping system without internal water pressurend subjected to only narrow-band random ground motions areonsidered for comparison between analytical and experimen-al response. To study the effect of broad-band random motionsnalytically, few components of real earthquake ground motionsre also considered. It is to be noted that the piping system waslaced diagonally (in plan) for performing the tests to producehe effects in the two orthogonal directions of the piping system.he piping system was subjected to few components of narrow-and random ground motions, which were again recorded onhe shake table. The recorded ground motions are having peakround acceleration (PGA) of 0.2, 0.27, 035 and 0.53 g, and.22, 0.29, 0.35 and 0.56 g, along X- and Y- directions, respec-ively. In addition to this, the piping responses and the frictionalorces mobilized at the friction support in the two orthogonalirections were also recorded.

. Finite element model of piping system

Fig. 1(c) shows finite element (FE) model of the piping sys-em with a friction support. Following assumptions are madeor dynamic analysis of piping systems supported on frictionupport subjected to bi-directional excitation.

1) The straight members in the piping system are modeled asthree dimensional (3D) Beam elements and the bends aremodeled as 3D curved beams (considering the flexibilityfactors) having 6 degrees-of-freedom at each node. Beforestep-by step integration, the degrees-of-freedom with zeromass are statically condensed out.

2) The mass of each member is assumed to be distributed

between its two nodes as a point mass. In addition to themass of piping system the externally lumped masses areassumed to be effective in the three translational degrees-of-freedom.

Xfbt

3) Only the friction support behaves non-linearly and the pip-ing system is assumed to behave linearly throughout thedynamic analysis.

4) The non-linear force-deformation relationship at the frictionsupport is considered similar to a elastic-perfectly plasticsystem (with a very high value of initial stiffness), basedon the non-linear model proposed by Park et al. (1986), inwhich there is no need to keep track of transitional rules forsliding and non-sliding phases.

Solving above FE model for its eigenvalues and eigenvectorsields the fundamental natural frequency of the piping systemqual to 4.38 Hz, which is in close agreement with the cor-esponding experimental frequency of 4.74 Hz (Reddy et al.,999).

. Modeling of the friction support

Conventionally, the non-linear friction support is representedy its transitions between the stick and slip conditions, where,t is required to keep track of the transitional rules. In this paper,nstead of using conventional method, a non-linear model pro-osed by Park et al. (1986) is used to represent the frictionalorces of the support. The characteristic of this type of model-ng is that the friction support behaves like a fictitious springith very high stiffness during non-slip mode and zero stiffnessuring slip mode. The following relation expresses the restoringorces at the friction support in the X and Y-directions

Fx

Fy

}= Fs

{Zx

Zy

}(1)

here Fx and Fy are the restoring or frictional forces in X- and-directions of the friction support, respectively; Zx and Zy are

he non-dimensional hysteretic components (value varies from1 to +1); and Fs is the limiting friction force at the friction

upport.The hysteretic components, Zx and Zy, satisfies the following

oupled non-linear first order differential equations presented inPark et al., 1986){

Zx

Zy

}= [G]

{xp

yp

}(2)

=[A−βsgn(xp)|Zx|Zx − τZ2

x −βsgn(yp)|Zy|Zx−τZxZy

− βsgn(xp)|Zx|Zy − τZxZy A − βsgn(yp)|Zy|Zy−τZ2y

]

(3)

here β, � and A are the parameters, which controls the shapend size of the hysteretic loop of the friction support; q ishe yield displacement; sgn denotes the signum function; and˙p and yp represent the relative velocity of piping system in

- and Y-directions of the friction support, respectively. Theorce-deformation behavior of friction support can be modeledy properly selecting the parameters β, � and A. However, ashe behavior of friction support is modeled as elastic-perfectly

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s

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sqf

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Fig. 2. Force-deformation characteristics of the friction support.

lastic, the yield displacement should be taken equal to zero,owever, to avoid computational error a very small value has toe adopted.

The off-diagonal terms of the matrix [G] expressed by Eq.3) provide the coupling or interaction between the restoringorces at the friction support. If these terms are ignored (i.e.ssumed equal to zero), the frictional forces behave indepen-ently in the two orthogonal directions. It will be interesting tonvestigate the effects of this interaction (i.e. ignoring and con-idering the off-diagonal terms of matrix [G]) on the seismicesponse of piping system. Two typical hysteresis loops of theriction support using Eqs. (1)–(3) with and without interactionnder sinusoidal displacement of amplitude 0.1 m and frequencyf 3 Hz in X-direction are shown in Fig. 2. The parameterselected are: Fs = 250 N, q = 0.0012 m, β = τ = 0.5 and A = 1. Theoop for no interaction case represents the independent unidi-ectional model of friction support in two orthogonal directionseported in (Wen, 1989). This is obtained by neglecting the effectf force interaction i.e. by replacing the off-diagonal element ofhe matrix, [G] in Eq. (3), by zero. The difference between thewo hysteresis loops is that when the force interaction is takennto account the friction support starts sliding at the lower valuef sliding force i.e. as soon as the resultant of the frictional forcesttains the limiting frictional force (i.e. F2

x + F2y = F2

s ).The limiting friction force at the friction support is generally

onsidered to be invariant (i.e. assumed as a constant) during thexcitation by earthquake ground motions. However, it is well-nown that the normal reaction at the friction support will bearying with time, consequently, the limiting friction force willlso be affected. This change in the limiting friction force isncluded in the present study and is expressed by

s(t) = μFz(t) (4)

here Fz(t) is the time dependent vertical support reaction at theriction support; and μ is the coefficient of friction.

Constantinou et al. (1990) performed several tests on Teflon-teel interfaces and observed that the coefficient of friction isependent on the relative velocity between the contact inter-aces and provided an expression to track this effect. Using the

e

[

127

xpression, the coefficient of sliding friction, μ, at a resultant

liding velocity u =√

x2p + y2

p is approximated as

= μmax − (�μ) exp(−a|u|) (5)

here μmax is the coefficient of friction at large velocity of slid-ng (after leveling off); �μ, is the difference between the frictionoefficient at large and zero velocity of the system; and a ishe constant dependent on the bearing pressure and conditionf interface. The value of the constant a is taken as 0.2 s/cmConstantinou et al., 1990).

. Governing equations of motion

The equations of motion for the piping system model ashown in Fig. 1(a) under the two horizontal components of earth-uake ground motions are expressed in the following matrixorm

M]{u} + [C]{u} + [K]{u} + [D]{F } = −[M][r]{ug} (6)

u} = {x1, y1, z1, x2, y2, z2, x3, y3, z3, xi, yi, zi . . .}T (7)

ug} ={

xg

yg

}(8)

here [M], [C] and [K] represents the mass, damping and stiff-ess matrix, respectively of the piping system; {u}, {u} andu} represent structural acceleration, velocity and displacementectors, respectively; [D] is the location matrix for the frictionalorces at the friction support; {F} is the vector containing therictional forces governed by Eq. (1); [r] is the influence coeffi-ient matrix; {ug} is the earthquake ground acceleration vectorxpressed by Eq. (8); xg and yg represents the earthquake groundccelerations in the X- and Y-directions of the piping system,espectively; and xi, yi and zi are the displacements of the nodef the piping system in X-, Y- and Z-directions, respectively.he mass matrix has a diagonal form with the lumped mass atach node. The stiffness matrix of the piping system with frictionupport is constructed separately and then static condensation isarried out to eliminate the rotational degrees-of-freedom. Theamping matrix of the system is constructed using Rayleigh’sethod by considering the first two natural frequencies of the

iping system and the damping ratio obtained from the experi-ental test results.

. Incremental solution technique

Since the force-deformation behaviour of the friction supports non-linear, the governing equations of motion are to be solvedn the incremental form using Newmark’s time-stepping methodssuming linear variation of acceleration over small time inter-al �t. The equations of motion in the incremental form are

xpressed as

M]{�u} + [C]{�u} + [K]{�u}+[D]{�F }= − [M][r]{�ug}(9)

128

Table 1Peak ground acceleration of various ground motions

Earthquake Recording station Applied in X-direction Applied in Y-direction

Component PGA (g) Component PGA (g)

Imperial Valley, 1940 El-Centro N90E 0.214 N00E 0.348N 90EL 90EK 90E

wf

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orthridge, 1994 Sylmar Converter Station Noma Prieta, 1989 Los Gatos Presentation Center Nobe, 1995 JMA N

here {�F} is the incremental restoring force vector of theriction support.

Following the assumption of linear variation of accelerationver the small time interval, �t, {�u} and {�u} are given as

�u} = a0{�u} + a1{ut} + a2{ut} (10)

�u} = b0{�u} + b1{ut} + b2{ut} (11)

here a0 = (6/�t2); a1 = (−6/�t); a2 = −3; b0 = (3/�t); b1 = −3;nd b2 = −(�t/2) and the superscript denotes the time.

Substituting Eqs. (10) and (11) in Eq. (9),

K]{�u} = {�P} − [D]{�F } (12)

here

K] = a0[M] + b0[C] + [K] (13)

Tesr

ig. 3. Comparison of the analytical and experimental hysteresis loops of the frictionotions (μmax = 0.15 and �μ = 0).

0.605 N00E 0.8430.608 N00E 0.570.629 N00E 0.834

�P] = −[M][r]{�ug} − [M](a1{ut}+ a2{ut}) − [C]({b1u

t} + b2{ut}) (14)

fter solving for incremental displacement vector from Eq. (12),he incremental acceleration and velocity vectors are obtainedrom Eqs. (10) and (11), respectively. Finally, the displacementnd velocity vectors are obtained using Eqs. (12) and (13),espectively, as given below.

ut+�t} = {ut} + {�u} (15)

ut+�t} = {ut} + {�u} (16)

o obtain the acceleration vector, the direct equilibrium equationxpressed by Eq. (6) is adopted. The incremental Eq. (9) can beolved by iterative technique along with Runge-Kutta method,equired to solve the coupled Eqs. (2) and (3), to obtain the

support in X-direction of the piping system under narrow-band random ground

129

F forceg

r

�

wrvf

7

ie

Fe

ig. 4. Comparison of the FFT spectra of analytical and experimental frictionround motions (μmax = 0.15 and �μ = 0).

esponse of the system at time, t + �t, where �t is expressed by

t = dt

N(17)

here dt is the time interval at which the ground motions areecorded and N are the number of divisions adopted for con-ergence of the structural responses due to the non-linearity ofrictional forces.

dtai

ig. 5. Comparison of the hysteresis loops for uni- and bi-directional idealizationarthquake (μmax = 0.15 and �μ = 0).

, Fx of the friction support for the piping system under narrow-band random

. Numerical study

The seismic response of piping system with friction supports investigated under four narrow-band random and four realarthquake ground motions. Details of the narrow-band ran-

om ground motions are mentioned in earlier sections. However,he specific components of the real earthquake ground motionspplied in the X- and Y-directions of the piping system are shownn Table 1. The response quantities of interest for the piping sys-

of the friction support in X-direction of the piping system under 1995 Kobe

1

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7

palhtottaomioAatefrons

pFttftmp

√tNp

Fig. 6. Numerical stability of the proposed analytical procedure for uni- andb

omfsvlqtb

30

em under consideration are the relative sliding displacementsxs or ys) at the friction support, absolute accelerations (xp or yp)t node B of the piping system and the support reactions (Rx ory) as indicated in Fig. 1(c). The absolute accelerations and reac-

ions at the support are directly proportional to the forces exertedn the piping system. On the other hand, the relative sliding dis-lacements are crucial from the design point of view of frictionupport. The x and y in the response quantities refer to the corre-ponding responses in the X- and Y-directions of the piping sys-em, respectively. The responses are obtained for (i) uncontrolledystem (i.e. piping system without friction support), (ii) con-rolled system (i.e. piping system with friction support but ignor-ng the bi-directional interaction of the frictional forces) andiii) controlled system (i.e. piping system with friction supportnd considering the bi-directional interaction of the frictionalorces).

.1. Validation of the proposed analysis procedure

To validate the proposed seismic analysis procedure for theiping system, analytical force-displacement hysteresis loopsre compared by considering and ignoring the variation of theimiting friction force with time along with the experimentalysteresis loops are shown in Fig. 3. The hysteresis loops ofhe friction support are shown in the X-direction under vari-us narrow-band random ground motions and duly consideringhe effects of the bi-directional interaction. It is observed fromhe Fig. 3 that there is a better comparison of the analyticalnd experimental hysteresis loop when the effect of variationf limiting friction force with time due to change in the nor-al reaction is considered. A comparison of the correspond-

ng FFT spectra of the frictional force of the friction supportbtained by analytically and experimentally is made in Fig. 4.s observed earlier, there is a better comparison of the analytical

nd experimental FFT spectra of the frictional forces when theime-dependent normal reaction at the friction support is consid-red. Thus, there is a close agreement of the proposed procedureor the analytical seismic response of the piping with the cor-esponding experimental results. The limiting frictional forcef the friction support must be modified due to change in theormal reaction during the earthquake excitation of the pipingystem.

The comparison of the hysteresis loops of the friction sup-ort in X-direction under 1995 Kobe earthquake is shown inig. 5 by ignoring and considering the interaction effects and

he time dependency of the limiting friction force. As expected,here are significant effects of the bi-directional interaction of therictional forces and time dependent limiting frictional force ofhe support under real earthquake excitation. Thus, these effects

ust be considered for seismic analysis of piping system sup-orted with friction device.

The peak resultant responses of the piping system (i.e. us =√ √

x2

s + y2s , up = x2

p + y2p and RR = R2

x + R2y) are plot-

ed against N (expressed by Eq. (17)) in Fig. 6 under 1994orthridge earthquake to study the robustness of the pro-osed analytical procedure and its sensitivity to time interval

ttqP

idirectional idealization of the friction support (μmax = 0.01 and �μ = 0).

f integration. It is observed from the figure that the mini-um value of N for numerically stable results is 20 and 40

or no-interaction and interaction cases, respectively. For rea-onable accurate response of the piping system the minimumalue of N is found to be 40 for both cases. Thus, the non-inear response of the piping system with friction support isuite sensitive to the time interval used for the integrations ofhe equations of the motion. The accuracy of the results shalle assured by taking the minimum time interval. In addition,he minimum time interval required for integration of equa-ions of motions is also found to be sensitive to typical earth-

uake ground motions having different frequency contents andGA.

131

F X- an(

7

Yg

m

Fa

ig. 7. Time variation of displacement, acceleration and support reaction inμmax = 0.15 and �μ = 0).

.2. Performance of the friction support

The time variation of various response quantities in X- and-directions of the piping system under a narrow-band randomround motion of PGA = 0.22 g and 1994 Northridge earthquake

aitc

ig. 8. Time variation of displacement, acceleration and support reaction in X- andnd �μ = 0).

d Y-directions of piping system under narrow-band random ground motion

otion are shown in Figs. 7 and 8, respectively. The responses

re plotted for the controlled piping system by considering andgnoring the effects of the bi-directional interaction along withhe responses of the uncontrolled piping system. There is signifi-ant reduction in the response quantities of the controlled piping

Y-directions of piping system under 1994 Northridge earthquake (μmax = 0.15

132

ultant

sfoctefteirqi

tugmeaot

TP

P

0

0

0

0

Fig. 9. Effects of bi-directional interaction of frictional forces on peak res

ystem in comparison to uncontrolled system implying that theriction support is very effective in reducing the seismic responsef piping systems. A comparison of the controlled response byonsidering and ignoring the bi-directional interaction indicateshat the response quantities are increased when the interactionffects are considered into account. This happens because of theact that the system with interaction effects starts sliding whenhe resultant value of the frictional force (i.e. F2

x + F2y = F2

s )xceeds the limiting friction force, which results in smaller limit-

ng friction force value compared to system without interaction,esulting in higher displacements. Thus, the friction support isuite effective for seismic response control of piping system andnteraction of frictional forces should be duly considered.

fftm

able 2eak response quantities of piping system on friction support under narrow-band ran

GA Direction Uncontrolled Controlled (no inter

xp or yp

(mm)xp or yp

(g)Rx or Ry

(kN)xp or yp

(mm)xp or(g)

.22 gX 6.714 0.682 1.1 2.474 0.522Y 6.11 1.971 2.982 1.372 0.445

.29 gX 9.031 0.932 1.46 3.795 0.743Y 7.989 2.513 3.866 3.234 0.995

.35 gX 11.58 1.174 1.882 5.331 0.911Y 10.033 3.166 4.825 5556 1.662

.56 gX 17.85 1.787 2.89 9.404 1.465Y 15.558 4.922 7.511 10.092 3.326

responses of piping system under narrow-band random ground motions.

Figs. 9 and 10 show the plot of peak resultant response quan-ities of the piping system against the friction coefficient (μmax)nder narrow-band random ground motions and real earthquakeround motions, respectively. It is observed that under all groundotions the sliding displacements are relatively more by consid-

ring the interaction effects in comparison to that without inter-ction effects implying that there is significant under estimationf the sliding displacements. Moreover, it is also observed thathe effects of the bi-directional interaction are dependent on the

riction coefficient and are relatively less for low values of theriction coefficient of the friction support. This is due to the facthat for lower values of friction coefficient, the system remains

ost of the time in the sliding phase for both cases of the excita-

dom waves (μmax = 0.15 and �μ = 0)

action) Controlled (interaction) Controlled(experimental)

yp Rx or Ry

(kN)xp or yp

(mm)xp or yp

(g)Rx or Ry

(kN)x (mm)

0.318 3.371 0.536 0.517 3.2740.644 3.504 0.974 1.595 2.7250.490 4.852 0.716 0.833 5.5921.447 5.207 1.506 2.366 4.0440.788 6.094 0.921 1.131 7.232.468 7.314 2.257 3.254 6.0111.711 10.862 1.386 1.897 12.1854.565 13.297 4.155 5.804 12.867

133

Fig. 10. Effects of bi-directional interaction of frictional forces on peak resultant responses of piping system under real earthquake ground motions.

Fig. 11. Effects of the velocity dependence of friction coefficient on peak resultant responses of piping system under narrow-band random ground motions.

134

k resu

tiTsm

seqioen

iesvapcsr

TP

E

I

N

L

K

Fig. 12. Effects of the velocity dependence of friction coefficient on pea

ion (i.e. with and without interaction). As a result, the differencen the sliding displacements for the two cases is relatively less.hus, for effective design of the friction supports for the pipingystems the bi-directional interaction effects of frictional forcesust be considered.In Tables 2 and 3, the peak response quantities of the piping

ystem are compared by considering and ignoring the interactionffect, respectively, under narrow-band random and real earth-uake ground motions. In Table 2, the responses in the two hor-

zontal directions are compared with corresponding responsesf the piping system without friction support along with thexperimental results. Displacements of the piping system underarrow-band random ground motion of PGA = 0.56 g, by ignor-

isss

able 3eak response quantities of piping system on friction support under real earthquakes

arthquake Direction Without friction support With f

xp or yp

(mm)xp or yp

(g)Rx or Ry

(kN)xp or y(mm)

mperial Valley, 1940X 9.366 0.686 1.334 4.23Y 3.636 0.84 1.677 1.07

orthridge, 1994X 12.982 0.874 2.091 8.67Y 4.821 1.348 2.676 4.13

oma Prieta, 1989X 15.103 1.283 2.121 12.40Y 4.462 1.213 2.094 3.17

obe, 1995X 22 1.391 3.132 12.90Y 4.51 1.069 2.426 2.83

ltant responses of piping system under real earthquake ground motions.

ng and considering the interaction effects, are observed to bequal to 10.092 and 13.297 mm, respectively, and the corre-ponding experimental value is 12.867 mm. Thus, the responsealues obtained by considering the interaction effects are in closegreement with the experimental results implying that the pro-osed analytical procedure yields analytical results those are inlose agreement with the experimental results. From Table 3,imilar observations are also noted for the piping system undereal-earthquake ground motions. The displacements of the pip-

ng system are required for designing the piping system andupplemental devices, the nodal accelerations are direct repre-entation of the body forces developed in the elements and theupport reactions are required to design the supporting system.

(μmax = 0.15 and �μ = 0)

riction support (no interaction) With friction support (interaction)

p xp or yp

(g)Rx or Ry

(kN)xp or yp

(mm)xp or yp

(g)Rx or Ry

(kN)

1 0.343 0.685 5.021 0.427 0.7577 0.462 0.742 2.001 0.625 0.7334 0.638 1.577 10.428 0.744 1.7284 0.986 2.433 4.352 1.092 2.4145 1.167 1.911 12.367 1.233 1.7892 0.905 1.106 3.529 1.307 1.3572 0.656 2.097 15.024 0.845 2.1664 0.913 1.701 3.372 0.891 1.929

TPrl

7

dvErTiqtdpicasBior

8

tgtiosposFc

(

(

(

(

(

(

A

DNSTes

R

A

B

B

C

C

F

H

J

K

K

M

M

hese are the key variables in the design of piping systems.iping systems, designed by considering the underestimatedesponse quantities can be unsafe in case of industrial instal-ations and nuclear power plants.

.3. Effects of velocity dependent friction coefficient

The friction coefficient of various sliding devices is typicallyependent on the relative velocity at the sliding interface. Theelocity dependence of the friction coefficient is modeled byq. (5). It will be interesting to study this effect on the peak

esponses of the piping system with sliding friction support.he peak resultant responses of the piping system are plotted

n Figs. 11 and 12 under narrow-band random and real earth-uake ground motions, respectively. The response is shown forhree values of �μ, (i.e. �μ = 0, 0.15 and 0.3). Note that �μ = 0enotes that friction coefficient of the friction support is inde-endent of the velocity (i.e. Coulomb-friction idealization). Its observed from Figs. 11 and 12 that dependence of frictionoefficient on the relative sliding velocity does not have notice-ble effects on the peak response of piping systems with frictionupport which were confirmed earlier by Fan et al. (1990) andhasker Rao and Jangid (2001) for buildings resting on sliding

solators. Thus, the effects of dependence of friction coefficientn sliding velocity may be ignored for finding out the peakesponses of piping systems with friction supports.

. Conclusions

The response of a piping system with friction support underwo horizontal components of narrow-band and real earthquakeround motions is investigated. The interaction between the fric-ional forces at the friction support in two orthogonal directionss duly considered. The dependence of the limiting friction forcen the time dependent normal reaction and relative velocity atliding interface are also considered. The proposed analyticalrocedure for seismic response of the piping system supportedn a friction support is validated by comparing the analyticaleismic responses with the corresponding experimental results.rom the trends of the results of the present study, the followingonclusions may be drawn

1) The friction support is very effective in reducing thedynamic response of the piping system under groundmotions with different frequency contents.

2) The proposed analytical procedure for seismic response ofthe piping system supported on a friction support is foundto be sufficiently robust and the results obtained are in closeagreement with the experimental results.

3) The analytical responses of the piping system obtained byconsidering bi-directional interaction are in close agreementwith the experimental results than those obtained by ignor-ing the interaction.

4) The sliding displacements of friction support isolating thepiping system are underestimated if the interaction of fric-tional forces in two orthogonal directions is ignored. Thiscan be crucial from design point of view. Similar effects of

OP

135

bi-directional interaction are also observed for nodal accel-erations and support reactions of the piping system.

5) The variation of the limiting frictional force of the frictionsupport due to change in the normal reaction had significanteffects on the response of the piping system supported onthe friction support.

6) The dependence of friction coefficient on the relative slid-ing velocity at the friction support does not have noticeableeffects on the peak responses of the piping system. There-fore, these effects may be ignored for finding out the peakresponses of the piping system.

cknowledgements

This study is a part of R & D project sponsored by BRNS,epartment of Atomic Energy, Government of India (Sanctiono. 2002/36/7-BRNS). The authors would like to thank Prof. K.uzuki and his team, Department of Mechanical Engineering,okyo Metropolitan University, Tokyo, Japan, for providing thexperimental results for comparison of the results of the presenttudy.

eferences

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