displacement method for determining acceptable piping vibration

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PVP-Vol. 313-2, International Pressure Vessels and PipingCodes and Standards: Volume 2 - Current PerspectivesASME 1995

DISPLACEMENT METHOD FOR DETERMININGACCEPTABLE PIPING VIBRATION AMPLITUDES

J. C. WachelEngineering Dynamics Incorporated

San Antonio, Texas

ABSTRACTMethods are presented in the ANSI! ASME Operation

& Maintenance Standards/Guides Part-J 1991"Preoperational and Initial Startup Testing of NuclearPower Plant Piping Systems" (OM-3) for determining ifthe vibration-induced stresses in a vibrating pipe span areacceptable. The displacement and velocity methods arepresented in OM-3. These methods are applicable forpiping spans vibrating at resonance in lateral bendingmodes. The background and theory necessary forunderstanding the lateral bending vibration modes oftypical piping systems and the procedures for evaluatingstresses are discussed.

The stress factors for characteristic spans given inOM-3 do not cover the entire range of configurationsencountered in actual piping systems. In addition, data forcalculating the lateral bending natural frequencies ofpiping spans and bends are not included in the OM-3guidelines. Frequency factors, deflection stress factorsand velocity stress factors for calculating the mechanicalnatural frequencies and the vibration-induced stresses ofmost common piping spans have been published (Wachelet al., 1990). This information was used to develop anomogram which can be used to calculate the naturalfrequency, the vibration-induced stress, and the allowablevibration amplitude of most piping spans.

The information presented herein on piping naturalfrequencies and vibration-induced stresses should allowthe vibration analyst to evaluate piping systems under theOM-3 Vibration Monitoring Group 2 (VMG-2) category inless time and with increased confidence in the results.

INTRODUCTIONMost piping will experience some amount of steady-state vibrations as a result of the excitation sources in thesystem. Piping vibration excitation mechanisms includepressure pulsations in the fluid being transported by thepiping, and vibrations mechanically transmitted byattached or adjacent equipment, as discussed in Part 3,Appendix E of OM-3. Figure I summarizes the mostcommon excitation mechanisms for piping vibration.The vibratory response of the piping system is a

function of (1) the piping mechanical natural frequencieswhich are dependent on the pipe diameter, span length,effectiveness of the pipe supports, restraints, guides,anchors, and snubbers, (2) the acoustical resonantfrequencies of the piping system which are influenced bythe acoustical properties of the fluid, (3) the piping systemoperational conditions, such as pressure, temperature,flow, multiple unit operation and the kind of operationaltransient (startup, shutdown, etc.), (4) the flow-inducedpulsations which are dependent upon the flow velocity,Strouhal number, main and branch pipe diameters, and thepresence of obstructions such as orifices, valves,condenser tubes, etc, and (5) pulsations generated fromreciprocating machinery.

In the design stage, it is difficult to calculate thepulsations that will exist in the piping due to flow-excitedmechanisms. The acoustical natural frequencies of pipingsystems and the potential for the flow excitation of theacoustical natural frequencies can be calculated; however,the amplitudes of the flow induced pulsation and theseverity of the problem are difficult to accurately predict.

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Pulsation characteristics from reciprocating machinery,such as positive displacement charging pumps, can becalculated by using digital computer programs or analogsimulators which model the acoustical properties of thepump and the piping system. Many piping vibration-induced failures have occurred in charging pump systems(Olson, 1985). These could have been avoided by takingadvantage of the improvements in the state-of-the-art inthe design of proper pulsation control in reciprocatingpump systems (Wachel et al., 1993). For example, thelatest revision (scheduled to be published in 1995) of theAmerican Petroleum Institute Code for ReciprocatingPumps API 674 will require that an acoustical analysis ofthe pump and piping system be made to develop adequatepulsation and vibration control for critical systems.

Steady state piping vibration in nuclear power plantsis considered in the design stage by using the experiencewith previous systems and good engineering practices inthe layout and supporting of the piping systems tominimize vibration problems. The piping vibrations mustbe qualified to ensure that fatigue failure will not resultThe predominant way of accounting for these vibrations isto monitor the piping vibration levels during startup andactual operation. Because of the immense amount ofpiping typically found in a power plant, simplifiedqualification techniques are needed to keep the task ofaddressing the vibrations within manageable limits.

Most dynamic fatigue failures in piping systems arecaused by individual piping spans vibrating at resonance.The stress in a piping span which is vibrating at resonanceis directly proportional to the maximum vibrationamplitude (displacement, velocity, or acceleration) in thespan. In order to determine if measured vibrationamplitudes of piping systems are acceptable, the dynamicstresses caused by the vibrations should be compared tothe applicable endurance limit for the piping material.SIMPLIFIED METHOD FOR QUALIFYING PIPINGSYSTEMS

In OM-3, the maximum allowable vibrationamplitudes for a piping system are evaluated in VibrationMonitoring Group 2 (VMG-2) by dividing the pipingsystem into characteristic or analogous beam segments. Itis assumed that piping vibration measurements are madeon the characteristic piping span while the span isvibrating at its mechanical natural frequency and that themeasured mode shape exactly matches the theoreticalmode shape of the selected characteristic beamconfiguration. The simplified methods in OM-3 forcalculating the vibration-induced stresses and theacceptable vibration amplitudes are based on the modeshape of the analogous piping characteristic span. Theacceptance criteria are further based on limiting the peak

vibratory stresses to a value less than or equal to the stressendurance limit for the piping material.OM-3 (1991) uses the stress amplitude at 10 6 cyclesfrom Figures I-9.1, I-9.2.1 and I-9.2.2 of the Boiler andPressure Vessel Code to develop endurance limit stresses.The endurance limit for carbon steels at 107 cycles is

obtained by reducing the value at 106 cycles by 0.8. Toobtain an endurance limit at 1011 cycles, the value at 10 7cycles is divided by 1.3. This essentially reduces thestress amplitude at 106 cycles by a factor of 1.625.The measured vibrations on a piping span during a test

reflect the conditions at the time of the test; however,experience has shown that an additional factor may beneeded to compensate for the uncertainty in the testingconditions which may not have caused vibrations whichare the highest that will be experienced.The development of the vibration-induced stress insimple piping characteristic spans is shown below.

STRESS AS A FUNCTION OF THE VIBRATIONAMPLITUDE

The dynamic stress in a vibrating pipe is related to thedynamic bending moment.MDs=-21 (1)

where: S = stress. psiD = pipe diameter, inI = pipe moment of inertia. in "M = moment. in-Ib

The moment is related to the piping vibration mode shape.

(2)

where:E = elastic modulus. psi

d2 ~ = second derivative of vibration mode shapedx-If equation (2) is substituted in equation (I), we obtain:

s = -ED d2y2 d x "

Thus, it can be seen that the dynamic stress is afunction of the material, the diameter, and the secondderivative of the vibration mode shape. The vibration

(3)

198



mode shape depends upon the piping configuration and theend conditions.

To illustrate, let us calculate the vibration-inducedstress in a simply supported beam. It ca n be shown thatthe vibrating mode shape for a simply supported beam forthe first mode is a sine wave. Thus,

. 1 C %y= yosm-ld2y ; r 2 . Jr X--=-Yo-sm-dx2 [2 l

Yo= maximum vibration amplitude, inl = span length. in

The maximum occurs at midspan ( . t " = f )

,; r -=-Yo- [2Substituting into the stress equation:

ED ( ; r 2 )s=-- -Yo-2 l 2Thus, for any pipe span configuration, it is feasible to

calculate the stress which results from a given vibrationamplitude providing the end conditions and vibratorymode shape are known.

The general equation relating maximum unintensifiedstress in a pipe to the maximum vibration amplitude alongthe span ca n be obtained by dividing equation (3) by themaximum measured vibration amplitude. Note that themaximum stress and maximum deflection are generallynot at the same point

where:S = stress at maximum stress point, psi

1 > ' ' ' 1 = d2;, evaluated at maximum stress pointdxN =deflection, inches, at maximum deflection point

(4)

(5)

(6)

( 7 )

(8)

(9)

(10)

Based on the characteristic mode shapes of uniformstraight beams as presented by Blevins (1979) it can beshown that:

(11)

where:

y" = second derivative of characteristic mode shapefor mode n

-; n = characteristic mode shape for mode nl = span length, inA . = frequency factor

Tables of -; , ," and -;" are given for the first five modesfor straight beams. Ifwe define a deflection stress factor:

(12)

where:K = Deflection Stress Factor for Mode nIn OM-3 analyses, an acceptable vibration amplitude

of the piping is determined by breaking the piping systeminto a series of characteristic lengths and determining thestresses of the individual resonant spans. The allowablevibration can be expressed in terms of the vibrationamplitude required to cause 10,000 psi stress as givenbelow:

(13)where:

5allow = Allowable zero to peak vibration deflection limit,inches.

Sd = SA, where SA is the alternating stress at 106 cyclesfrom Fig. 1-9.1, or at 1011 cycles from Fig. 1-9.2.2of the BPV Code. The user shall consider theinfluence of temperature on the modulus ofelasticity.

O n = value of deflection from OM-3 Fig. 1, P 29.Cz = Secondary stress index as defined by the BPY

Code.K 2 = Local stress index as defined by the BPY Code.

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a = Allowable stress reduction factor: 1.625 formaterials covered by Fig 1-9.1; or 1.0 formaterials covered by Fig. 1-9.2.1- or 1-9.2.2 ofthe BPV Code.

Figure 1 in OM-3 is a nomogram which gives thenominal vibrational deflection values. The K value isgiven for the Characteristic Span Models in Figures 2 - 9ofOM-3. The equation for O n is given below:

L2s =K-n DThe relationship between the K factor given in theOM-3 Code and the deflection stress (Ka) factors byWachel et al., (1990) is given below:

The vibration-induced stresses can be calculated bvthe following equation: -

where:S =Dynamic stress (psi),

K .! = Deflection stress factory =Vibration (mils), maximum amplitude measured

between nodes (normally supports),D = Outside diameter (inches),L = Span length (ft),

The equations outlined above show the basic theorythat was used to develop the methods given in OM-3 forVMG-2 analyses to determine if the measured vibrationamplitudes are acceptable on the basis of vibration-induced stresses. The OM-3 guidelines allow the analystto use the displacement or velocity method to make thecomparison of vibration-induced stresses to theconservative endurance stress limits defined in theguidelines. The author believes that the deflection(displacement) method is easier to use and is less prone toinaccuracies. The stress correction factors are discussed inOM-3; however, additional stress correction factors mustbe considered when using the velocity method asdiscussed by Wachel et al., (1990). The additionalconsiderations include concentrated weights, otT-resonantexcitation frequencies. and contents and insulation. Thedisplacement method can be used to develop conservative

(14)

(15)

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allowable vibration amplitudes even though the naturalfrequency and mode shape do not exactly match theresonant frequency condition. However, it is very criticalthat the piping span be exactly on resonance if the velocitymethod is used. There are many difficulties in properlyapplying the velocity method and this can lead to non-conservative errors. Therefore. the author recommendsthat the displacement method be used.

The author and his associates have zeneratedfrequency factors, deflection stress factors andovelocitystress factors which can be used to calculate the acceptablepiping vibration amplitudes for most pipingconfigurations, The piping spans were analyzed using afinite element code which included piping elbows. Thevarious factors were calculated as a function of the aspectratios of the leg lengths of the L-bend, Z-bend, U-bend,and the 3-



The basis of the nomogram is listed below:1 . The piping span is vibrating on resonance at its first

natural frequency. The first natural frequency ofpiping spans with bends is assumed to correspond tothe out-of-plane mode.

2. The vibration mode shape is that associated with thefirst natural frequency.

3. The maximum vibratory stress occurs at the locationof the theoretical maximum moment.

4. The piping span is constant diameter and the radiusof gyration is approximately 0.34 times the outsidediameter.

5. There are no concentrated weights within the spanand the weight of the fluid and insulation do notaffect the vibration mode shape.

The following equations are solved by the nomogram.Each of these will be discussed below:

f = 75.8,{~) (17)

e K d ( E _ ) ( 18)Y L23000 (19)Yo/I = -S-Y

USE OF NOMOGRAMDetermine the Piping Span Natural Frequency

To determine the first natural frequency of the pipingspan; first, connect the piping total span length (L) to theoutside diameter (0) of the pipe. Note that the outsidediameter of the nominal pipe sizes has been noted on theoutside diameter scale. The length L is the sum of thedistances from the boundaries to the intersection of the leglengths (A + B + C). (When the piping spans wereanalyzed, the piping elbows were modeled; however, theresults have been plotted in terms of the sums of thelengths to the intersections as opposed to the developedlengths.) The line between the total span length and theoutside diameter locates a point R on the reference line.

The second step to determine the mechanical naturalfrequency of the span is to connect the point on thereference line to the frequency factor A . for thecharacteristic piping span or piping bend. Several of thetypical piping spans have been noted on the A . scale;

however, ifthe span is one of the piping spans which hasbends in it, the frequency factor can be determined fromthe curves on the four graphs included on the nomogram.The graphs give the frequency factors and the deflectionstress factors for the L-bend, Z-bend, U-bend, and the 3-dimensional bend. The factors are plotted as a function ofthe leg length aspect ratios (B/A and CIA). Two curvesare shown for the CIA ratios, the 0.5 and the 1.0 ratios. Ifthe aspect ratio is within this range, the intermediate ratiosca n be interpolated.

Locate the frequency factor on the .t scale andconnect it with the point on the reference line. The pipingspan natural frequency is located on the f scale which islocated adjacent to the right of the reference line.Calculate the Stress per Mil of Vibration

To determine the stress per mil that is caused by thepiping span vibrating at its first natural frequency, thepoint on the reference line is connected to the det1ectionstress factor for the characteristic piping straight span orpiping bend. If the appropriate det1ection stress factor ismarked on the ~ scale, it can be connected with the pointon the reference line. Ifthe piping span has piping bends,the appropriate deflection stress factor can be determinedfrom the four graphs provided. Again, the aspect ratiosare used to determine the deflection stress factors.

To determine the stress per mil of vibration for thepiping characteristic span, connect the deflection stressfactor to the point on the reference line. The stress per milscale ( S l Y ) is adjacent to the right of the frequency scale.The stress per mil determined from the ( S l Y ) scale is thenominal stress and does not include the C2K2intensification factors. If it is desired to calculate themeasured vibration-induced stresses, the value of stressper mil from the ( S l Y ) scale must be multiplied by theC 2 K Z and the measured maximum vibration at the antinodeof the piping span.

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The safety factor can then be calculated by comparisonof the calculated vibration-induced stress with theappropriate endurance limit given in OM-3 guidelines.Allowable Vibration Amplitudes

The allowable vibration amplitudes can be calculatedusing the allowable endurance limits given in the OM-3guidelines. For carbon steel piping with an ultimatetensile strength less than 80,000 psi, the allowable is 7690psi. For the higher strength low alloy steel of 115,000 to130,000 psi ultimate tensile strength, the allowable is

201



equal to 12,300 psi. For stainless steels, the lowestallowable endurance limit is equal to 13,600 psi.The value of stress per mil (SlY) read from the scale

can be multiplied by the e2K] value and the measuredvibration in mils ( Y _ o s ) to determine the vibration-induced stress. If the calculated stress is less than theallowable given in the OM-3 guidelines, the measuredvibrations are acceptable and the piping span meets theacceptance criteria given in OM-3.

The BPV criteria indicates that the maximum stressintensifier that normally needs to be used in evaluatingintensified stress is a factor of approximately 5, except fornotch-like defects. When this information is consideredwith the documentation of the acceptable dynamicstrainlstress given by Wachel (1982), it is possible todetermine an acceptable vibration limit from the stress permil value for the piping span. An acceptable level ofdynamic strain for most piping systems is 50 x 10"; inlin,zero to peak. Based on this dynamic strain and a modulusof elasticity of 30 x 106 psi, it can be seen that if thecombined stress intensity factors e]Kl are 5.12, theallowable dynamic strain would be 50 x 10";inlin which isequal to 1500 psi, zero to peak stress amplitude or 3000psi, peak to peak stress range. The normal measurementfor vibration displacement (deflection) is in mils, peak topeak, therefore, the allowable vibration displacement inmils peak to peak can be determined directly from thestress per mil scale, ifthe value of 5.12 for elK] is used.

Therefore, once the stress per mil is determined, anormally conservative value of the acceptable vibrationdisplacement can be determined at the same location onthe stress per mil scale by reading on the opposite side ofthe scale where the allowable vibration amplitude islabeled. This scale assumes that the e2K2 value is 5.12,and is the maximum that has to be used. If it is knownthat the value ofe2K] is less than 5.12, then the allowablevibration level read from the nomogram (Yall-n) can beratioed by 5.12 over the actual e2K] values. Since mostpiping spans will have fairly high safety factors, itnormally is faster to determine a conservative allowablevibration amplitude and compare the measured value.Most piping spans will have maximum vibration levelslower than the conservative allowable vibration amplitudegiven on the scale, therefore, the acceptability of the spanvibration and stress amplitudes are documented.

( 5.12 )Y ol/=Y ol/-n C 2K 2Deviations in the Span Natural Frequency

Since the basis of the characteristic spans forVibration Monitoring Group 2 assumes that the piping

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span IS vibrating at its first lateral natural frequency, thenomogram also makes this assumption. Therefore, theresults of the stress calculations can not be verified unlessthe measured natural frequency matches the calculatednatural frequency. The derivation of the stress deflectionfactors discussed above indicates that the deflection stressfactors can be adjusted if the measured natural frequencydoes not match the calculated natural frequency. It isknown that differences in the boundary conditions fromthe ideal end conditions can affect the natural frequencyand dynamic stress. Ifthe basic mode shape of the pipingspan vibration matches the expected mode shape, but, thenatural frequency deviates from the calculated naturalfrequency by 25 percent, the stress deflection factor canbe adjusted by the ratio of the measured natural frequencyto the calculated natural frequency.

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Using the new value of the deflection factor [ ( K d ) _ o s ] 'connect this value to the point on the reference line andobtain the stress per mil value and the conservativeallowable vibration amplitude.Pipinq Spans With Weights

The nomogram can be used to calculate the stressper mil and the allowable vibration amplitude for pipingspans with concentrated weights by using the stressdeflection factors and the point on the reference line. Thecorrection factors that can be used to correct the naturalfrequency calculations for the concentrated weights aregiven in Figure 4. When using the displacement methodfor calculating stresses, the correction factor for the use ofconcentrated weights changes linearly from 1 to 2 as theratio of the concentrated weight to the span weightincreases from 0 to 20. For most practical cases, theweight ratio will result in a stress weight correction factorofless than 1.3.

Note that the span natural frequency for spans withconcentrated weights can be calculated using the weightcorrection factors given in Figure 4.Allowable Velocity Amplitude

The allowable vibration velocity can be calculatedby calculating the allowable vibration displacement andconverting this value to velocity by the relationshipbetween the displacement and velocity. The allowablevelocity in in/sec ( V o il ) is equal to the allowable vibrationdisplacement times the natural frequency divided by318.3. Using this value, the vibration measurement couldbe made in velocity units.

202



Vall = Yall(3~3)

CONCLUSIONS

Based on the theory used in ASME Operations andMaintenance Standards/Guides Part 3 for evaluating thepiping span vibration under steady state operation,. anomogram has been developed which provides help inimplementing the procedures. The nomogram allows theanalyst to check to see if the measured natural frequencymatches the calculated natural frequency of thecharacteristic span. This is needed to verify that theassumptions used to calculate the vibration-inducedstresses are valid. Information is provided on thenomogram for additional characteristic spans to coverpiping configurations which are not now included in OM-3. The stress per mil for practically any type of prpingconfiguration and boundary conditions ca n be calculated.The maximum dynamic stress can be calculated using themeasured maximum vibration displacement and theintensifiers C2 K 2 . Based on a normally conservative valueof C2K 2 of 5.12, the maximum allowable vibrationamplitude ca n be read directly from the scale directlyacross from the stress per mil value. This nomogramshould allow the analyst to quickly evaluate complexpiping systems to determine which of the spans have anadequate margin of safety.

(23) ACKNOWLEDGMENTThe author acknowledges the contributions of hiscolleagues, especially Troy Feese for writing the computerprogram to develop the nomogram and to Mark Broomwho drew the nomogram.

REFERENCES

"Preoperational and Initial Startup Vibration Testingof Nuclear Power Plant Piping Systems: 1991,ANSI! ASME Operations & MaintenanceStandards/Guides Part-3, New York,NY.

Blevins, R. D., 1979, "Formulas for Natural Frequencyand Mode Shape," Van Nostrand Reinhold Company, NewYork,NY.Olson, David E., 1985, "Piping Vibration Experience

In Power Plants," Pressure Vessel and Piping Technology1985.A Decade of Progress. Book No. H00330, (ASME).

"Positive Displacement Pumps, Reciprocating, APIStandard 674," 1987, American Petroleum Institute,Washington, D.C.

Wachel, J. C., 1992, "Field Investigation of PipingSystem for Vibration - Induced Stress and Failures,"Pressure Vessel and Piping Conference. ASME BoundVolumeNo. H00219.Wachel, J.C., Morton, S. J., and Atkins, K. E., 1990,

"Piping Vibration Analysis," Proceedings o[ 19thTurbomachinery Symposium. Texas A&M University,College Station, TX.Wachel, J. C., Szenasi, F. R, et, al., 1993, "Vibrations

in Reciprocating Machinery and Piping," EngineeringDynamics Incorporated Report 41450.

203



GENERATION MECHANISM DESCRIPTION OF EXCITATION EXCITATION FREQUENCIESFORCES

1. MECHANICAL INDUCED INHigh Level, Low Frequency

1=60A. Machinery Unbalanced Forces & 2NMoments 12=6 0

1 N=-B. Structure - Bourne Low Level 602. PULSATION INDUCED

High Pressure Pulsations, 1 =..!!I::!.. n= 1,2,3 ... (modes)A. Reciprocating Compressors 60Low Frequency N =Speed. rpm1 =nNP P = Number of Pump60

High Pressure Pulsations. PlungersB. Reciprocating Pumps Low Frequency 1 =..!!I::!..60

C. Centrifugal Low Pressure Pulsations. l=nBN B = Number of BladesCompressors & Pumps High Frequency 60

l=nvN v=Number of Volutes60or Diffuser Vanes

3. GASEOUS FLOW EXCITEDA. Flow Through Pressure Letdown

Valves or Restrictions/Obstructions

B. Flow Past Stubs

4. LIQUID (OR MIXED PHASE)FLOW EXCITED

A. Flow Turbulence Due to Quasi SteadyFlow (e.g. Fluid Solids Lines)

B. Cavitation and Flashing

5. PRESSURE SURGEJ HYDRAULICHAM :MER

1= S!.. S = Strouhal Number 0.2~.5Digh Acoustic Energy,Mid to High Broad BandFrequencies v =Flow Velocity ftlsec

D = Restriction Diameter, ft.vI=S-DModerate Acoustic EnergyMid to High Frequencies S = 0.2 - 0.5D = Stub Diameter. ft.

Random Vibrations, LowFrequency 10 - 30 H z (Typically)

High Acoustic Energy,Mid to High FrequenciesTransient Shock Loading Broad Band

Discrete Events

FIGURE 1PIPING VIBRATION EXCITATION

SOURCES

204



Piping Frequency Deflection stress Velocity StressConfiguration Factor Factor Factor1st 2nd 1st 2nd 1st 2nd

J I Fixed-Free 3.52 22.4 366 2295 219 219LSimply 9.87 39.5 1028 4112 219 219F Supported

~Fixed- 15.4 50.0 2128 6884 290 290Supported

1 ~ Fixed-Fixed 22.4 61.7 2935 8534 275 290L

, L-Bend Out 16.5 67.6 1889 9690 241 301A+8 ,. L L-Bend In 59.4 75.5 7798 9575 276 266A-=88n-Bend Out 18.7 35.6 2794 4628 314 273U-Bend In 23.7 95.8 3751 8722 332 191A+8+C -= LA=8=C~

Z-Bend Out 23.4 34.2 3522 4133 317 254

Z-Bend In 22.4 96.8 3524 8933 331 194A+8+C -= LA=S"CA

83-D Bend 20.6 27.8 3987 4752 407 359C ).-

A+S+C :a LA=S..CFormula f D S.. D S .. K vY C %K%. 75.8.\17 KdY ?C%K%

.Steel Piping (E,..30 x 106 psi, p,. 0.28.3 Ib/in3)

FIGURE 2 FREQUENCY FACTORS AND STRESS FACTORS FORUNIFORM STEEL PIPE CONFIGURATIONS

205



f 10 Piping Weight 'I V eigh t Correction'P.~1+a : Configuration Location Factora

) Cantilever L 3.93L/4 1.7L~

Simply L/2 2.0Supported L/4 1.1

~Fixed- L/2 2.3Supported 3L/4 1.6

~ ~ Fixed-End L/2 2.7L/4 0.9L

, L-Bend First ModeOut-of-PlaneBIA B/A0.5 1.0A/2 1.24 0.63A+B = L A 2.69 3.25A=B B/2 0.39 0.63B U-Bend First Moden B/A ... 1.0Out In

A/2 0.26 0.33A+B+C = L A 2.24 1.79A=B=C B/2 2.31 2.00

~

Z-Bend First ModeB/A = 1.0Out In

A/2 0.40 0.29A+B+C .. A 2.76 1.77L B/2 2.21 2.09A=B=C

A 3-D Bend First ModeB/A ==1.0

B 1st 2ndC V A/2 0.31 0.58A 2.35 1.58B/2 2.06 2.77A+B+C ==LA==B=C

FIGURE.3 WEIGHT CORRECTION FACTORS AND UNIFORM PIPING CONFIGURATIONS

206



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