Support Stiffness Effect on Piping Dynamic Response

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    K2/4Support Stiffness Effects on Piping Dynamic Response

    H. UyeiEDS Nuclear, 220 Montgomery Street, San Francisco, California 94104, U.S.A.

    G.V. MillerEDS Nuclear, 350 Lennon Lane, Walnut Creek, California 94598, US.A.

    ABSTRACTA typical assumption employed in production piping analysis is to assume

    that struts, rod hangers, and other so-called rigid restraints may be modeledwith either infinite or very large stiffness values. Recent researchindicates that this may not be appropriate. We review conclusions derivedboth from the literature and from studies by the authors about the effect ofneglecting these stiffness values. We then suggest a method to check theassumption's validity in the form of a simple equation.

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    INTRODUCTIONA traditional assumption used in the analysis of piping systems is that

    the pipe supports do not dynamically interact with the pipe. As such they canbe modeled with either infinite or very large stiffnesses. This assumption islikely valid for sparsely supported systems, where the flexibility of the pip-ing system decouples it dynamically from the stiffer supports: however, it canlead to inaccurate and unconservative results for rigidly supported systems.

    Several "rule of thumb" techniques are commonly used to insure decouplingbetween the dynamic response of the pipe and its supports. Typical among theseare: 1) specifying some lower bound on a support's fundamental frequency toinsure that its lower frequencies are much higher than those of the modescontributing to the piping response; 2) imposing some lower bound on thesupport's stiffness; or 3) checking that the deflection of the support underits calculated reaction does not exceed a prescribed value. Severalresearchers have studied support stiffness effects by performing parametricnumerical studies on existing piping systems. Others (see for example Lee [lJand Stevenson and Bergman [2J) used theoretical models. The principal conclu-sions are the same for both approaches. The inclusion of support stiffnessaffects the piping response in two ways. The first is that all frequenciesshift to lower values, and often the modes change shape as well. The secondeffect depends on the input response spectrum. If the spectra have sharp peaksat one or more frequency points, drastic changes in the acceleration ordinateread may occur with even small frequency shifts. The term "frequency shift" isdefined as the difference between any given mode's frequency calculated withrigid supports and with the actual supports' stiffnesses included. Thus,including support stiffnesses may lower some modes out of the peak range of aninput spectrum while at the same time lowering some higher modes into the peakrange. The net effect is therefore uncertain. changing in an intractable waywith frequency and input spectra shape changes.

    In this paper, we first present detailed results of a numerical studyconducted by the authors. These show that the changes in the piping's dynamicresponse are most influenced by the realitive magnitude of the supports'stiffnesses as they are distributed along the pipe. We then present a deriva-tion of a simple equation that may be used to monitor the accuracy of therigid-supports assumption, and aid the engineer in identifying criticallyflexible supports.

    NUMERICAL STUDIESWe analyzed five piping systems which are presently installed in a

    nuclear power plant. For space considerations, results are presented onlyfor two of the lines: these results are typical.

    The study used the response spectrum method to calculate maximum supportreactions and piping stresses. Each of the five lines was analyzed with threesupport conditions: 1) with all support stiffnesses equal to lxE+lO lbs/in,a typical value used in production piping: 2) all stiffnesses equal to lxE+06;and, finally, 3) stiffnesses calculated from the "as built" supports. These

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    supports were originally proportioned based on reactions from a rigid-supportsanalysis and which satisfied the deflection criteria discussed above. A flatinput response spectrum as well as the actual response spectra was used toisolate the input load changes due to the frequency shifts.

    Figures I through 6 summarize the effect of the support stiffnesses. Ineach case the y-axis gives the percent change in the particular value whencalculated under one assumption (say, with rigid supports) to the valuescalculated under some other assumption (say, with actual "as built" supportstiffnesses). Figures I through 3 correspond to a 16" low pressure coolantinjection line. Figures 4 through 6 give results for a 4" diesel generatorservice water pipe line. Note the small effect of substituting a uniformvalue of lxE+06 for the much higher value of lxE+lO. This suggests thatstiffness magnitude alone plays only a small role.

    CONCLUSIONS FROM NUMERICAL STUDIESThe major conclusions support those arrived at by others. These are: 1)The imposition of a deflection criteria alone does not insure the accuracy ofthe analysis and sometimes gives unconservative results. 2) Pipe responses,especially support reactions, are sensitive to the distribution of supportstiffnesses along the pipe. Thus, any reasonable uniform value for supportstiffness does not significantly improve the accuracy. 3) The percent changein frequency correlates well with the change in all other reponses.

    These conclusions identify the reason for the sensitivity of responses tostiffness values. Changes in response quantities, if the input spectrum isflat, are due only to a change in the 'shape' of'the modes. Changing thesupport stiffness values in any uniform way, i.e. by a scale factor, does notsig~ificantly change the shape of the modes calculated from a rigid supportanalysis. so that using a lower uniform stiffness does not significantlyimprove the results. The distribution along the pipe of support stiffnessesdetermines the overall shape of the modes, and thus of support reactionmagnitudes.

    We can infer from this that a very few number of relatively flexible sup-ports in a large system can invalidate the results of a rigid supports analy-sis. One reliable method to insure the reasonableness of the rigid supportassumption would be to monitor the difference between the "actual" and, say,"rigid-supports model" mode shapes. Unfortunately. we cannot measure thischange without first running the analysis with the actual support stiffnessesincluded. But because of conclusion number 3. we can measure the mode shapechange indirectly by measuring the shift in any mode's frequency. That is,limiting the frequency shift will indirectly limit the change in all otherresponse quantities of interest.

    An equation with this attribute is derived below. It allows the computa-tion of any modal frequency shift for any support suspected of being tooflexible.

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    FREQUENCY SHIFT EQUATIONLet the combined pipe-support system have a frequency vector (ooJ, and modeshape matrix []. Figure 7 illustrates one support in the system.

    We impose a prescribed displacement U (t) to the support s. Thesequation of motion for the system is then given by[M]{i} + [KJ{X} = {6} Us(t) (1)

    where [M] and [KJ are the mass and stiffness matrices of the piping systemwith rigid sUPforts. The vector {o J has a form

    {a}ku,s-lk e,s-lke,s-lku,s-k e , s

    - ke,s

    us-les-lB sus+19s+1

    Corresponding DOF (2 )

    wherek .U,1

    l2EI.1~ 1

    (3)

    6El.1T1 (4)Let

    {x} = ( < I > J {q}Substituting (5) into (1), and premultiplying [~JT.[M"]{q}+ [K"":!H}= ku,S-ltPu,s-l(l) + k6,5-l$e,5-l(1) + (ks.s-l-Ke.5)1>e,s(llt ku,s $u,S+l(l) - ke,stPe,5+l(l}

    (5 )

    whereMi* the generalized mass of the piping system with rigid supports.Ki* Mi the generalized stiffness.

    We now choose and Us in the form{x} = {yp} cos(oo t +eUs = YS cos ( 0 0 t +e )

    where {~:} isa mode shape of the combined piping-support system andcorresponding natural frequency. Then, from (5) and (7)

    {q} = {q} cos (0 0 t +e )

    (7 )(8 )

    is the

    (9)where

    {yp} = [ < I> J {q}Substitution of (8) and (9) into (6) and solving for qi results in

    (10)

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    q.1k A. (i) +u,s-l't'u,s-l (k -k)$ (i)8S,-1 8,s 8,S

    Ys~MiWe now write the equation of motion for the mass ms of the combined piping-support system.

    + k A. (i) -u,s't'u,s+l

    2 2W pi -W

    k (i)e,s'Pe,s+l(ii)

    (12)- ke , s8t ku,sUs_l - ka ,s8s+l

    From (8), (10), (11), and (12), we haveku, s-1 { E $ 1(i)< .} + 'k, -1 0: 4 l 8 -1(i l ! $ }1 U,s- ' l : : J ,s 1 ,S -

    where K = the left-hand side of (11).Equation (13) is the exact frequency equation for the piping-support

    system. If we assume that the support is almost rigid, the first term of theleft-hand side of (13) can be neglected compared to the second. We also con-servatively neglect the terms ku,s-l and ku,s of the second term of theleft-hand side of (13).

    Since the resulting equation is for support s, the influence of allsupports requires a summation over s. Then the equation for frequency shiftbecomes

    2 2 1 s 2R 'sW j =w pj - - - - - ; r c E k (l4)M. s=l sJwhere

    kU,s-1'PU,S-1(j)+ke,s-1'Pe,s-1(j)+(ke,S-1-k8,s)$e,s(j)+ku,s

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    identified to be the change in the mode shapes. This discrepancy can be mini-mized by controlling the frequenc

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