C-27 PIPING FLEXIBILITY ANALYSIS

TRAINING MANUAL- PIPING

PIPING FLEXIBILITY ANALYSIS

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DOC No. : 29040-PI-UFR-0027

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CONTENTS

Page

0.0 Cover Sheet 1

1.0 Scope 2

2.0 Piping Codes 2

3.0 Introduction 2-3

4.0 Definitions 3-4

5.0 Sustained and Displacement Stresses 4-6

6.0 Allowable Stresses 6-9

7.0 Stress Intensification 9-11

8.0 Easily Analyzed Piping Systems 11-18

9.0 Piping Flexibility Analysis 18-23

10.0 Analysis of Complex Systems 23

11.0 Support Selection 24-25

12.0 Reducing Stresses 25-26

13.0 Designing with Expansion Joints 26-28

14.0 Sample data for Expansion rate and Allowable stress 29-31

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1.0 SCOPE

This design guide presents concepts and principles for calculating the strains and resultant stresses inpiping system to determine whether the system has sufficient flexibility to safely accommodatechanges in length resulting from temperature variations while simultaneously providing adequatesupport for all loadings present.

1.1 Limitations :This guide is not applicable to the design of non-metallic piping systems or to systems whichincorporate brittle materials, e.g. glass lined steel.

1.2 Application :This guide is in conformance with the requirements of ASME B31. process piping, hereinafterreferred to as the code.

2.0 PIPING CODESIn addition to ASME B31.3, which is applicable to all piping within the property limits of a chemicalplant, petroleum refinery, gas processing plant or tank farm which is not covered by other codes, thestress analysis methods set forth in this design guide can be used with the codes for other classes ofpiping. The scope of each of these codes is briefly stated in the following.

ASME B31.1. Power Piping :Covers piping directly associated with power boilers, i.e. vessels which conform to Section I of theASME Boiler and Pressure Vessel Code.

ASME B31.4 Pipeline Transportation System for Liquid Hydrocarbons and other liquids :Applies to piping carrying liquid petroleum products between refineries, plants, tank farms, etc. outsideplant boundaries.

ASME B31.5 Refrigeration Piping :Covers requirements for refrigeration piping for services as low as -320°F, both field erected andfactory assembled.

ASME B31.8 Gas Transmission & Distribution Piping Systems :Applies to fuel gas piping systems not federally regulated, from the source to the user's meter butexcludes piping on process plant property.

ASME B31.9 Building Services Piping :This code, when issued, will cover piping normally associated with industrial, commercial and multi-unit residential buildings.

ASME Boiler & Pressure Vessel Code, Section III :Covers piping systems of nuclear power plants.

3. INTRODUCTION :Piping systems are stress analysed for three reasons :1) to prevent over-stressing the material of construction,2) to prevent joint leakage caused by excessive forces and moments,3) to prevent failure or malfunction of attached equipment caused by excessive piping reactions. This

design method permits a designer to accomplish these objective by :4) To calculate forces and displacements at support points for support design.

3.1 Assuring adequate support to prevent excessive sag and stresses in the piping system.

3.2 Incorporating sufficient flexibility to accommodate stresses resulting from changes in pipe length dueto thermal effects and movement of the connections at the ends of the pipe.



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3.3 Designing the piping system to prevent its exerting excessive forces and moments on equipment such aspumps and tanks or on other connection and support points.

Figure 1 schematically illustrates the flexibility analysis system.It is discussed in detail in Para. 9

Para.4 through 7 provide detailed explanation of the basic considerations involved in piping stressanalysis Para.8 explains how to determine which piping systems justify computer modelling Para.9deals with Piping Flexibility Analysis. Paras.10 through 13 discuss other aspects of evaluating andcontrolling the stress in piping system.

4. DEFINITIONS :The following symbols and definitions are used in this document.Aw Cross-sectional area of corroded pipe wall, inch2

C Cold spring factor.D Outside diameter of pipe, inch.dp Pitch diameter of bellows expansion joint, inch.E Casting quality or weld joint factor.Ea Modulus of elasticity at installation temperature, psi.Ej Weld joint factor for welded pipe.Em Modulus of elasticity at design maximum temperature, psi.FA Axial force, lbf.f Stress range reduction factor based on the total number of full temperature cycles (f =1 for

7000 or fewer cycles).h Flexibility characteristic.ii Inplane stress intensification factor.io Outplane stress intensification factor.K Spring constant, lbf / inch.LC Cold load exerted by spring hanger, lbf.LH Hot load exerted by spring hanger, lbf.M Bending moment, ft-lbf.Mi Inplane bending moment, in-lbf.



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Page : 4Mo Outplane bending moment, in-lbf.Mt Torsional moment, in-lbf.P Concentrated force, lbf.Pr Internal pressure, psi.PT Test pressure, psi.R Total reaction forces & moments, lbf or ft-lbf.RT +P Variable reaction caused by thermal & pressure effects, lbf, ft-lbf.R W Weight reaction, lbf or ft-lbf.S Basic allowable stress from Appendix A of B31.3, psi.S A Allowable stress range, psi.S b Resultant bending stress, psi.

0.5

(i i M i )2 + (i o M o ) 2

= _______________ zS c Basic allowable stress at minimum metal temperature during the displacement cycle, psiS E Computed displacement stress range, psi.S h Basic allowable stress at maximum metal temperature during the displacement cycle, psi.S L Sum of longitudinal stresses caused by pressure, weight and other sustained loadings, psi.S t Torsional stress, psi = M t / 2ZTp Pressure thrust of bellows expansion joint, lbf.t Pipewall thickness, inch.y Deflection in y-direction, inch.Z Section modulus of pipe, inch3.

5. SUSTAINED AND DISPLACEMENT STRESSES :Piping flexibility analysis in accordance with the basic assumptions and requirements of B31.3 isconcerned with two types of stress called sustained stress and displacement stress. Each type of stressmust be considered separately; these are the two criteria by which the adequacy of a piping system isevaluated. They are considered separately because sustained stresses are associated with sustainedforces while displacement stresses are associated with fixed displacements.

5.1 SUSTAINED STRESSES :Sustained Stresses are defined as stresses caused by loads that are not relieved as the piping systemdeflects. An example is the stress induced by the weight of the valve at the end of the cantilevered pipesegment shown below.

Regardless of the magnitude of the displacement ∆, the magnitude of the load (the weight of the valve)which causes the stress is unchanged. Therefore, to avoid catastrophic failure, the magnitude of anysustained stress must not exceed the yield strength of the material. Another example of a sustained stress isthe hoop and longitudinal stresses induced by pressure. The loadings, which induce sustained stresses, aretermed sustained loadings.

The sustained stress principle is expressed as a Code requirement . The sum of the longitudinal stresses dueto pressure, weight and other sustained loadings SL must not exceed the hot allowable stresses Sh.



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This requirement is written as :

SL ≤ Sh (1)

SL is computed by the following equation : FA

SL = Sb + ------ (2) Aw

5.2 DISPLACEMENT STRESSES :Displacement stresses are defined as those stresses caused by fixed displacements, i.e., caused by loads thatare relieved as the piping system deforms. Consider the cantilevered beam shown below.

Assume that the end of the beam is displaced, its elastic limit and its elastic range is δ. As long as anydisplacement cycle is within the elastic range of the beam, no yielding will take place.

Consider the same cantilever beam displaced from its original position to position (1).

Its elastic limit is exceeded and the beam will yield. However, as long as D does not exceed δ, no furtheryielding of the beam will take place provided all successive displacement cycles are within the displacementrange D. If the beam is made of a relatively ductile material, yielding only in the first half cycle will notcause failure of the beam. Therefore, fixed displacements can be allowed to cause displacement stressesthat exceed the yield strength of the material as long as the elastic range of the material is never exceeded.

Consider the stresses induced in a piping system caused by thermal expansion, as illustrated by the followingexample.



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In Figure 5A, if the pipe end at position (1) is assumed free, then when the piping is heated it would move tothe unrestrained hot shape with the free end at position (2). Figure 5B illustrates how an increase intemperature of the restrained piping system is equivalent to displacing the unrestrained hot pipe fromposition (2) to position (1). Therefore, stresses caused by thermal expansion are displacement stresses.

Any yielding or permanent strain, with attendant relaxation or reduction of stress in the hot condition, leadsto the creation of a stress reversal when the piping system returns to the cold position. This reversal ofstress, referred to as self-springing, is similar to cold springing in its effect. See Para.9.3.

The code requires that the calculated stress range SE (sometimes known as the expansion stress range) mustnot exceed the allowable stress range SA.

SE ≤ SA (3)

When piping system statisfies Eq.3, it is judged to be adequately flexible against thermal expansion andrestraint displacement because the elastic range of the system will never be exceeded even though the systemmay yield.

Since the inherent flexibility is most piping systems is provided by changes in direction, the code considersonly bending and torsional stresses significant in the calculation of SE and gives the following equation forits computation.

SE = (S b2 + 4S t

2)½ (4)

In summary, two types of stress are of concern in a piping flexibility analysis. Sustained stresses are limitedby Eq.1 and the displacement stress range is limited by Eq.3. All piping systems must satisfy Eqs.1 and 3and a separate analysis must be performed for each equation.

6. ALLOWABLE STRESSES :The stresses computed by the program must be compared to Code allowable stresses to determine theadequacy of piping systems. The Code differentiates between stresses caused by pressure and othersustained loads and stresses caused by displacement strains. These allowable values are a function of thebasic allowable stresses.



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6.1 BASIC ALLOWABLE STRESSES :The code sets forth the rules for deriving basic allowable stresses for metals in the clause, 302.3.2 - Bases fordesign stresses. The significant aspect of these rules is that B31.3 permits the use of one third the tensilestrength, rather than one fourth the tensile strength as used by Section VIII, Division I, of the ASME Code,using section VIII allowable stresses for piping is unduly conservative.

For piping flexibility analysis, longitudinal stresses, i.e. stresses which act in the direction parallel to thecenterline of the pipe, are the primary concern.

Appendix A Table 1 of the Code tabulates basic allowable stresses.

6.2 ALLOWABLE SUSTAINED STRESS :The allowable stress for sustained loads is the basic allowable stress at the maximum metal temperature, Sh.See Para.5.1

6.3 ALLOWABLE DISPLACEMENT STRESS RANGE :The computed displacement stress range SE in a piping system shall not exceed the allowable displacementstress range. SA calculated by equation No.5.

S A = f ( 1.25 S c + 0.25 S h ) (5 )

When S h is greater than S L, the difference between them may be added to the term 0.25 S h in equation 5.In that case, the allowable stress range is calculated by equation No.6.

S A = f [ 1.25 ( S c + S h ) - S L ] (6 )

6.3.1 EXPLANATION FOR STRESS RANGE :Note that an allowable stress range rather than an allowable stress are considered for displacement loads. Inaddition, the allowable stress range can exceed the yield strength but not the elastic range of the material.(S A can be as large as 1/3-2/3 times the yield strength of the material. The code allows the piping materialto yield during its initial cycle provided it afterwards stays within its elastic range. The stress-strain diagramillustrates this concept.

During the initial cycle, the material is strained along the path represented by the line OA and strainedbeyond yield by as much as 0.1%. When the displacement load is released, the strain relaxes along the lineAB. Repeated cycles cause the material to behave elastically, i.e., follow the elastic line represented by theline AB.

If the stress range for a piping system is greater than the elastic range of the material (about twice the yieldstrength), the material would yield repeatedly at each end of the displacement load cycle (line CDEF),causing premature failure.



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6.3.2 TYPICAL EXAMPLE OF STRESS RANGE

Let it be assumed that the 90º turn shown above is to absorb 6" of expansion between anchors and that thecalculated maximum stress is 24,000 psi.

Supposing the material at the particular operating temp. has 18,000 psi yield stress, further thermalexpansion will be associated with yielding equal to ∆Y. After this when system is cooled down to ambientconditions it induces stress in opposite direction equivalent to yielding i.e. from 18,000 to 24,000 equal to -6,000 psi.

This -ve cold stress is again limited to cold yield stress Syc (otherwise there will be repetitive hot and coldyielding and pipe will fail).

Therefore actual stress range available to us, SR=Syc + Syh



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Page : 9The allowable stress values given in the code are, S = Sy/1.6Sy = 1.6 S, therefore SR = 1.6 Sc + 1.6 Sh.

However to have safety factor, code (ASME B31.1 / 31.3) allows SR = 1.25 Sc + 1.25 Sh which includes allstresses that is expansion, pressure and weight stress and other sustained loads. Therefore the stress rangedue to Thermal Expansion SE shall not exceed the value given below :

SE = f [1.25 Sc + 0.25 Sh + ( Sh - SL )] = SA + f (Sh - SL)

where f = Stress range reduction factor for cyclic conditions for total number of full temperature cycle overthe design period.

Sc = Allowable stress at minimum (cold) temperature.

Sh = Allowable stress at maximum (hot) temperature.

SL = The sum of longitudinal stresses due to pressure, weight and other sustained loads. This valueshall not exceed the allowable stress Sh. i.e SL < Sh.

SA = f (1.25 Sc + 0.25 Sh)

6.4 DUCTILE Vs BRITTLE MATERIALS :Most piping failures of ductile materials are caused by repeated yielding at relatively low strains, i.e., thepipe cracks after a successful period of operation; a small leak results. The failure is not catastrophic unlessthe escaping fluid causes a hazardous situation. When large deformations in piping systems are present, thedeformations are generally noticed and the situation corrected.

Brittle piping materials (cast iron, glass) behave differently. Failures in these systems often occur shortlyafter start-up and are catastrophic. The pipe breaks through its entire cross section. Large deformationscannot occur without failure of the pipe. Consequently, there is no warning, as there often is, for ductilepiping materials. On this account, allowable stress ranges must be very cautiously applied to brittlematerials.

7. STRESS INTENSIFICATION :Local stresses in fittings such as tees and elbows are generally higher than stresses in the adjoining pipesegments. The code allows these stresses to be calculated by multiplying the stresses in the adjoining pipesegments by a Stress Intensification Factor (SIF).

The stress analysis of elbows has been the subject of many studies since 1910 when a German named Bantlindemonstrated that curved pipe segments behaved differently than predicted by simple curved beam theory.The curved beam theory assumes the elbow cross section remains circular when subjected to bendingmoments. In fact, the cross section becomes oval when subjected to load, increasing the flexibility and thestress magnitude in the curved portion of the elbow.



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Page : 10The figure below illustrates the cross section distortion of an elbow when subjected to bending moments.

The circumferential bending stresses caused by the cross sectional distortion of the elbow are generally higherthan the bending stresses in the adjoining pipe segments.

The magnitude of the stresses and the degree of flexibility of an elbow have been determined by a number ofresearchers. These analytical solutions have been verified by full scale tests of pressurized and nonpressurized elbows.

The equations for SIFs for the different types of elbows and tees are given in Appendix D of the Code. Theseformulas determine the flexibility characteristics h and then, using that value, calculate the inplane andoutplane SIFs.

The Code also includes equations for computing SIFs for branch connections.

There can be stress intensification at flange and other seemingly innocent connections because of the changein geometry. The Code suggests the following :

Description SIF

Butt-welded joint, reducer or weld neck flange 1.0Double welded slip-on flange 1.2Socket weld flange or fitting, fillet welded joint 1.3 - 2.1Lap joint flange 1.6Threaded pipe joint or threaded flange 2.3



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Page : 11Omission of appropriate SIFs from the input for any piping stress analysis can result in a gross under-estimation of the actual stresses. It is important to recognise that SIFs should be applied at most locations inthe piping net work where there is a change in geometry or a hole in the pipe.

8. EASILY ANALYZED PIPING SYSTEMS :A formal piping flexibility analysis using the computer is not required for every piping system. Analysis isnot required for systems that duplicate or replace existing systems which have a satisfactory service record orfor systems which may be readily compared to previously analysed systems. Also, approximate or simplifiedmethods may be used for configurations for which adequacy has been demonstrated thus obviating a formalanalysis.

For an approximate analysis, the two types of stresses, sustained and displacement, are calculated by differentmethods and compared to different allowable stresses.

Stresses due to sustained loads are calculated by the weighted cantilever method which employs basic beamequations.

Stresses resulting from displacement loads are calculated by the guided cantilever method for simple systems.

Caution must be exercised in applying approximate methods. Assumptions or simplifications for modellingthe piping system must be conservative. The stresses values generated by an approximate analysis are notnecessarily actual. They should be used only to indicate whether a formal analysis is necessary.

8.1 SUSTAINED LOAD STRESSES :The stresses resulting from sustained loads are calculated by the weighted cantilever method. Maximumstress is determined by dividing the maximum moment by the section modulus of the pipe and adding thelongitudinal stress due to pressure. The weight of the pipe and its fluid contents is considered but the weightof insulation is usually ignored for approximate analysis.

Figure 8 illustrates the application of two of the maximum moment equations to model a simple pipingsystem.

For which M = PL + WL (L/2)From this maximum moment, the stress can be determined.

Frequently a piping system is too complicated to analyse as a whole. In these instances, it can be brokendown into sections for analysis.

In analysing the system shown in Figure 9, a conservative approach is to break it into three sections andconsider each separately as shown in Figure 9. For section 3, to assure conservatism, the moment for thelonger leg at its anchor, rather than that for the shorter one, is computed.



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Figure 11, illustrates the conservative consideration of another situation frequently encountered in pipingsystem analysis.

Because both cantilever legs deflect the same due to the weight of the vertical section, the moment is largerat the anchor for the shorter cantilever. Therefore, to be conservative, only the moment on the leg isconsidered for calculating the stress in the piping system. To conservatively calculate that moment, it ismodelled as a uniform load on the shorter cantilever plus a concentrate load acting at its end. Theconcentrated load P is composed of two components, viz, the actual weight of the vertical leg and theconservatively estimated effective weight of the longer cantilever leg. Experience has shown that includinghalf the weight of the longer leg is sufficient to assure a conservative analysis.

wL2s + PLs

M = -------- 2

Where : P = w ( cLL + Lv ) c = 0.5



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8.1.1 MATHEMATICAL EXAMPLE :Figure 12 shows the application of the foregoing concepts to an actual piping system numerical values.

20' 16' Section A 15'

0Data Pipe NPS 12 Sch 10 S 30' Section Bw = 25 lb / ftZ=22.0 inch3 24'P = 200 psi

FIGURE 12 - EXAMPLE PROBLEM

To simplify the problem, the analyst separated the system into two sections at Point 0. The two ends atPoint 0 are both considered to be anchored. The maximum longitudinal stress in each section is calculated.

FOR SECTION A (Figure 12A) :In this case, the weight of the 16 feet section is ignored, and the maximum moment for the 20 feet length iscalculated on the basis of uniformly loaded cantilever beam.

20'

16' LL

FIGURE 12 A - SECTION A

wLL2

M = 2

25 ( 202 ) = 5000 FT-lb 2

M 5000 (12 in./ft.)Sw = =

Z 22

= 2727 psi



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FOR SECTION B (Figure 12 B) :

The weights of the 30 feet vertical leg Lv and the 24 feet horizontal leg LL are combined into a concentratedload P acting at the end of the shorter 15 feet cantilever leg LS. In this case the analyst considered theeffective weight of L L to be one half its actual weight.

P = w (Lv + L L/2 ) = 25 ( 30 + 24/2 ) = 1050 lb M = PLs + w ( Ls / 2) Ls = (conc. load ) (uniform load) = 1050 (15) + 25 ( 15/2 ) 15 = 18563 ft - lb 18,563 x 12 in / ft

Sw = M/Z = = 10,125 psi 22

The longitudinal stress resulting from internal pressure in the system is then calculated by the formula.

PrD 200 ( 12.75) Sp = = = 3542 psi

4 t 4( 0.18)

This value is added to the largest calculated weight stress to give the total longitudinal stress in the system.

SL = Sw (max.) + Sp = 10,125 + 3542 = 13,667 psi.SL is then compared to the allowable stress for the material used at operating temperature Sh to decidewhether a formal analysis is required because of sustained loads.



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8.2 DISPLACEMENT LOAD STRESSES :Stresses produced by displacement loads are calculated by the guided cantilever method. Consider the

where LS = length of shorter leg LL = length of longer leg

This system may be conservatively modelled as follows :

where M = bending moment at end of LS

P = force exerted by expansion of LL.= free expansion of LL.

This is the "guided cantilever" model - a beam with one end fixed and other free to deflect but locked againstrotation.



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8.2.1 EXAMPLES :

The following illustrate the application of the guided cantilever method for evaluating flexibility.

Example in Figure 15

Header : NPS 6, A53 Gr.BBranches : NPS 2, A53 Gr.BSA = 29.6 ksiTCOLD = 70 º FTHOT = 275º FExpansion = 1.61 in/100 ft.SIF at stub-ins (2), (3) & (4) = 3.36

3 E D ∆ f = ------------- where 144 L2

f = Stress. lb / in2

E = 30 x 106 lb / in2

D = Pipe OD, inches ∆ = Deflection, inches L = Span, ft.



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The point on the system which will undergo the largest deflection is node (4). The horizontal deflection willbe :

4 = (30 + 15 + 15) (0.0161) = 0.966 inch

Considering branch 4-7 to be a guided cantilever, the approximate stress is f4 = 6370 psi from aboveformulae including the SIF raises the stress to 21.4 ksi. This stress is less than SA, therefore the system isadequately flexible and need not be formally analysed for displacement stresses.

Example in figure 16 :Pipe : NPS 4, Sch 40, A53 Gr.B.SA = 29.0 ksiTCOLD = 70 º FTHOT = 550º FExpansion = 4.11 in/100 ft.SIF at LR ells = 1.95

Since the system is of uniform pipe size, an imaginary anchor may be assumed to exist at node (3). Thisbreaks the system into two parts and each part can be further simplified into a guided cantilever. Thehorizontal movement at node (2) can be approximated as ∆2 = (13) (0.0.411) = 0.53" and at node (4) it canbe approximated as ∆4 = (12)(0.0411) = 0.49". From above formulae, the guided cantilever stresses are SE2= 80,740 psi and SE4 = 45,420 psi. Because SE2 and SE4 is higher than the allowable stress SA, a formalanalysis is required.

9.0 PIPING FLEXIBILITY ANALYSIS :Design of safe, functionally acceptable piping systems requires that they be adequately supported whileretaining sufficient flexibility to accommodate thermal and external displacements without imposing forcesand moments that will overstress the pipe, piping components or attached equipment. The design involvesthree basic steps as described in Para3 and illustrated by Figure 1.

It is important to note that when one step in the design process exceeds its limit, the analysis must berepeated beginning with Step 1 regardless of which step failed. Each step must be completed for everyanalysis, but for some problems one step may be more significant than the others. For example, for theanalysis of an NPS 24 steam header on a pipe bridge, the thermal analysis would be the most important step.When a piping system is connected to sensitive equipment, such as a pump, turbine etc., the reaction analysisis the most important; but each step in the process must be successfully completed.



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9.1 WEIGHT-PLUS-PRESSURE ANALYSIS :

PurposeThe purpose of a weight-plus-pressure analysis is to determine whether the piping system satisfies EQ.1,SL ≤ Sh. If it does, the piping system is sufficiently stiff against weight and pressure loads.

Data requiredThe process and mechanical data required for the weight-plus-pressure analysis of a piping system include :

1. Appropriate piping drawings or sketches to give the geometry of the system (Isometric).2. Piping specification to provide wall thickness, materials of construction, types of branch connections, etc.3. Specific gravity of the fluid in the pipe.4. Maximum and minimum design temperature and pressure.5. Insulation specification to obtain insulation density.6. Ambient temperature.7. Young's modulus of elasticity 'Ec' at ambient temperature.8. Young's modulus of elasticity 'Eh' at flex. temperature.9. Bend radius.10. Weight of valves, control valves, flanges & other items.11. Support locations and type.12. Allowable stress ranges SA, Sh, Sc.13. End point movements, and type of restraints.14. Weight of valves and special items.15. Type of fittings.16. Operating requirements

9.2 THERMAL ANALYSIS :

PurposeA thermal analysis determines whether the piping system satisfies Eq.3, SE ≤ SA . If it does, the system issufficiently flexible against thermal expansion and fixed anchor displacements.

Data requiredThe process and mechanical data required for the thermal analysis of a piping system consist of :

1. All data as listed for weight-plus-pressure analysis.2. Other process information pertaining to expected number of cycles and possible extremes or upset

conditions.

After a thermal analysis has been performed, the system is checked for adequate flexibility by comparing thedisplacement stress SE to the allowable stress range SA.



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9.2.1 RESTRAINST :The restraints are the most difficult components to model adequately in the thermal analysis of a pipingsystem. Restraints are hangers, guides, anchors, attached equipment, or other devices which can constrain apiping system. A restraint can impose a displacement on a piping system and cause displacement stresses;and it can respond to loads imposed by the system if the restraint has inherent flexibility. Both displacementand inherent flexibility should be incorporated in a thermal analysis. However, because the flexibility of arestraint is usually very small when compared to that of the piping system, it may be ignored withoutintroducing too much conservatism into the analysis. This is not the case with displacements.

Spring hangers are restraints which usually are very flexible relative to the piping system. Therefore theirstiffness may be ignored without the analysis becoming too unconservative. See Para11.2 Spring hangers.

Restraints usually have the effect of raising the levels of thermal stresses and lowering the levels of weightstresses. Therefore it is important to include all hangers, guides, and anchors in the thermal analysis.

Figure 17, below illustrates how restraint displacements affect the thermal analysis of a piping system.

The pipe is attached to heat exchanger "A" at nozzle (1) and to vessel "B" at nozzle (2). These nozzles arerestraints which constrain the piping system. When a cycle begins, the heat exchanger will thermallyexpand from its point of fixed support in the +X direction and the vessel will grow upward from its point ofsupport attachment. Vessel "B" will grow in a direction that will tend to reduce the displacement stresses inthe pipe, while heat exchanger "A" grows in the direction that increases these stresses. Sometimes the hotfluid in the pipe can cause the pipe to heat up much more quickly than either the heat exchanger or thevessel because they are more massive than the pipe. If that is the case, the extremes of the thermal cycle canbe conservatively approximated by including the movement of the heat exchanger but not the movement ofthe vessel in the thermal analysis.



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9.3 REACTION ANALYSIS :A piping system is adequate stress-wise if it satisfies Eqs. 1 & 3. However, unless the restraints are at leastas strong as the pipe, they may be overstressed or overstrained even though the piping is not. Oftenallowable loads on certain pieces of equipment are very low relative to the strength of the pipe attached toit.

Therefore, for piping systems connected to load-sensitive equipment such as pumps, a complete analysisincludes the calculation of maximum restraint reaction loads.

Weight, temperature, and pressure superimpose reaction loads on the restrains. Weight is always present,but thermal and pressure loads may or may not be present. It may be necessary to analyse several cases tofind the loading combination that produces the maximum restraint reactions. For instance, the forces in thesystem caused by weight can partially or completely cancel the forces caused by temperature. Therefore, themagnitude of the sum may be less than the magnitude of one or more of the individual forces. This meansthat to determine the maximum value of reaction loads, each load combination that the system may encountermust be separately examined.

The constant portion of the reaction analysis consists of a weight only analysis. This generates the weightreaction Rw.

A thermal reaction analysis of a piping system differs from a thermal stress analysis in that the installationtemperature is used as the cold temperature instead of the design minimum temperature. Either the designmaximum or design minimum temperature, depending upon which produces the greatest temperaturedifference, is used as the other temperature.

The forces and moments on the restraints determined by a thermal-plus-pressure reaction analysis comprisethe reaction range R. A thermal-plus-pressure analysis can always be used to find the reaction range unlessthe change in pressure and the change in temperature have opposite signs. In this case the effect of pressureand of temperature must be evaluated separately.

To accommodate variation in the modulus of elasticity caused by temperature change, the maximum reactionforces and moments at design temperature can be estimated by multiplying RT+P by the ratio of the modulusof elasticity at design temperature Em to the as-installed modulus Ea, that is

EmRT+P

Ea

This total reaction R is obtained by adding the constant and variable reactions.

EmR = RW + RT + P ( 7 )

Ea

This equation assumes elastic behaviour of the entire piping system. This assumption is sufficiently accuratefor systems where no plastic deformation occurs or where it takes place at many points in the piping system.It does not reflect the actual strain distribution in an unbalanced system where only a small portion of thepiping system undergoes plastic strain, or where creep is uneven. Under these conditions, the weaker ormore highly stressed portions of the sytems will suffer strain concentrations due to an elastic follow-up ofthe stronger or lower stressed portions. This local overstrain can be produced by :

1) The use of small branches with larger headers with the branch lines relatively highly stressed.

2) Local reduction of pipe size or cross-sectional area or the transition to a weaker material.



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Page : 22Local overstrain should be avoided where possible, especially when using materials of relatively lowductility. Where unavoidable, its effects can be lessened by cold spring or controlled by expansion joints.Following is an illustration of local overstrain and how it can affect the estimation of reaction loads.

Assume, a thermal-plus-pressure analysis has been performed and the reaction range at anchors A, B and Cdetermined, i.e. RA, RB and RC. By inspection, it can be seen that branch intersection point (1) willthermally displace to the right. The displacement is caused by the expansion of branch A and is resisted bybranches B and C. Branch B will plastically deform at smaller displacements than branch C. For suchinelastic displacement, the actual reaction R at point B will be less than the calculated reaction RB, evenwhen modified by Eq.7. That is, the actual reaction R at anchor C will be greater than the calculatedreaction RC when modified by Eq.7. R at anchor A will be about equal to the value calculated by Eq.7.

Cold spring is defined by the code as the intentional deformation of piping during assembly to produce adesired initial displacement and stress. Some of the benefits of intentional cold spring are :

1) It can serve to limit the amount of initial overstrain in the system. This is especially good for pipingmaterials of limited ductility.

2) It helps to assure minimum departure from as-installed hanger settings.

3) Credit for cold spring can be taken when calculating the total reaction load R. See Eq.8.

On the negative side :

1) Credit for cold spring is not permitted in stress range calculations. This restriction applies because the life ofa system under cyclic operation depends on the stress range rather than on the stress level at any given time.

When the effect of cold spring is included in Eq.7 it becomes : Em

R = RW + RT + P (1 - 2/3 C) (8) Ea

The value of C, the cold spring factor, ranges from zero for no cold springing to 1.0 for 100% coldspringing. The additional 2/3 factor is included because a specified cold spring cannot be assured even withelaborate precautions.

The effect of misalignment during erection is similar to that of intentional cold spring except it may bedetrimental instead of beneficial because misalignment may be in a direction opposite to that produced bythermal expansion. (Note that piping design may have to include special inspection requirements or thepiping designer must arrange to do the necessary inspection where alignment can be critical.

The possibility of non-simultaneous or uneven heating of a piping system must always be kept in mind whenperforming an analysis because this condition can significantly affect the results. The piping stress analystshould be sufficiently familiar with the process under normal and upset conditions to determine the worstpossible combination of uneven equipment and pipe branch heating. If the worst case is not obvious, suchcombination should be analysed individually.



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Page : 23In summary, a reaction analysis is performed to find the maximum reaction loads on the restraints. Themaximum reaction load on the system results from the highest combination of loads to which the piping isexposed and, except for weight, can be modified by cold spring and the ratio of the module, Em/Ea, inaccordance with Eq.8.

10.0 ANALYSIS OF COMPLEX SYSTEMS :Frequently piping systems are so geometrically complicated that a stress analysis of the system as a wholewould be impossible, even with the help of the computer program. The solution is to make simplifyingassumptions which permit considering a part of the piping system at a time.

One common technique is to separately consider branches in which small size branch lines are attached tolarger size main lines. It is called "the tail does not wag the dog" assumption. After the main line isanalysed, the displacement (linear and angular) at the intersection points found in the thermal analysis of themain line are input as extraneous anchor displacements in the analysis of the branches. The SIFs at thebranch connections must be included in the analysis of both the main line and the branch lines.

This can be done because the stiffness of a piece of pipe is proportional to its moment of inertia, whichincreases geometrically with increasing pipe size. For example, an NPS 3 Sch 40 pipe is over 4½ times asstiff as an NPS 2 Sch 40 pipe and an NSP 12 Sch. 40 pipe is almost 100 times stiffer than an NPS 3 Sch 40pipe. This means that smaller branch lines usually have such a small effect on the larger main lines that theycan be ignored. This will unestimate the displacement stresses in the main line and overestimate thedisplacement stresses in a branch. This technique is not appropriate for sustained stresses.

Another technique to simplify the analysis is to use anchors to break the system into smaller parts. Find asuitable place to restrain the system, anchor it, and then perform the analysis separately on each part of thesystem. This is easier than trying to analyse the entire system at once and is appropriate for both sustainedand displacement stresses. The anchor must be installed when the system is built.



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11.0 SUPPORT SELECTION :Sustained loadings on a piping system often necessitate the use of supports between anchors. To providethis support, three types of hangers are generally employed :

a) Rigid Supports :Rigid supports are those which prevent movement in one or more directions. Typical rigid supports are rodhangers, pipe shoes, longitudinal guides, transverse guide and intermediate anchors.

b) Variable supports (spring hangers )Variable supports or spring hangers provide a supporting force that changes with thermal deflection. Thistype of hanger is used at points where both support and flexibility are required to some degree.

b) Constant support spring hangers :Constant spring hangers provide a constant supporting force throughout the thermal cycle. This type hangerdoes not resist thermal deflection and therefore will not increase either displacement stresses or restraintreaction loads. This hanger is used where larger thermal displacements are encountered.

11.1 RIGID SUPPORTS :

When modelling rigid pipe supports keep in mind that :

1) Rigid supports usually are not really rigid and cannot be designed to prevent a movement of less than 1/16".They can have significant inherent flexibility. This can be an important consideration when a support toprevent movement is needed. It is not practical to reduce the restraint reaction loads on a nozzle bypreventing displacement at the nozzle with a rigid support.

2) Thermal forces may tend to make a pipe lift off its supports. If the thermal forces pushing up on a pipe aregreater than the weight forces holding it down, the pipe will lift off when hot thereby rendering the supportineffective during that part of the cycle.

3) Sliding type supports can cause friction loads that significantly affect the piping system. Because thermalloads are cyclical, friction loads will be in one direction when the pipe is warming up and in the otherdirection when the pipe is cooling down. The coefficient of friction for steel on steel ranges from 0.25 to0.50. Friction forces are particularly important in bridge piping as the system can have a tendency to snakebecause of friction. It is usually good practice to guide and anchor these systems whenever possible toeliminate the tendency to snake.

11.2 SPRING HANGERS :Spring hangers are used to reduce stresses and reactions in piping systems. Different methods of analysis forsizing the hangers are employed depending upon whether stress reduction or reaction reductions is theprimary concern.

11.2.1 STRESS REDUCTION :The following method is used to size spring hangers to reduce the stress in a piping system.

1) Run a weight analysis with the points at which the spring hangers are located modelled as rigid in thevertical or Y-direction. From this analysis the hot loads LH or the loads which the spring hangers are tosupport in the hot condition are determined.

2) Run a thermal-plus-pressure analysis with the points at which spring hangers are located, free to deflect -unrestrained - in the vertical direction. From this analysis the Y deflection (∆Y), the deflection rangethrough which the spring hangers will operate, are determined.

3) Using the hot loads and Y deflections determined in Step 1 & 2, size and select the appropriate hanger usingthe method set forth in vendor catalogues.



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Page : 25In the sizing process, the spring constant K is developed and the cold load calculated by the followingequation.

LC = LH + K ∆Y (9)

Note :The cold load is the load to which the spring hanger should be preloaded when delivered and installed. Thecold load must always be included in spring hanger specifications and requisitions.

To check the effect of the selected spring hangers on the stresses in the piping, analyse the system forsustained and displacement stresses by the following procedure.

1) Run a weight-plus-pressure analysis. Specify the spring constants and cold loads for each point at which aspring hanger is located. Compare the computed stresses with the hot allowable stress Sh.

2) Run a thermal analysis specifying the spring constant for each spring hanger. Compare these stresses withthe allowable stress range SA.

12. REDUCING STRESSES :To simplify the process for determining the best way to reduce stresses in piping systems, the problem isbroken into two parts. These are : 1) reducing stresses resulting from sustained loads; and 2) reducingstresses due to displacement loads. The two considerations require significantly different approaches andfrequently counteract each other. The general considerations for each are :

TO REDUCE STRESSES DUE TO SUSTAINED LOADS :

- Add supports to relieve stresses caused by weight.- Use thicker wall pipe to reduce stresses caused by pressure.

TO REDUCE STRESSES DUE TO DISPLACEMENT LOADS :

- Replace rigid supports with spring hangers.- Revise the geometry of the system to increase flexibility.- Add expansion joints.

12.1 REDUCING STRESSES DUE TO SUSTAINED LOADS :

The obvious first step to reduce the stresses is to ascertain its cause. Many times the cause can be identifiedby visual inspection of the system. Long spans of pipe, for example, are suspect. In other instances, adetailed interpretation of the computer analysis is required to identify the best solution - location and type ofrestraints for example.

If the stresses are caused by the weight of the system, adding supports is the solution. If pressure is thecause, the pipe wall thickness must be increased. Occasionally thicker wall pipe is also the best solution to aweight problem.

12.2 REDUCING STRESSES DUE TO DISPLACEMENT LOADS :

As with sustained loads, the initial step in the correction process is to identify the source of the stress. Thisusually requires a detailed interpretation of the computer analysis which should include :

- Checking the input data to assure that the system is modelled exactly as intended.- Inspecting the computed stresses to identify the types of forces and moments causing those stresses.- Evaluating the nature of the movement of the pipe. The translations and rotations, given for each mode,

often tell the story. In some cases, the pipe moves too much.- Analysing the reactions on the system restraints. Both the moments and the forces should be checked to

assure that the restraints are not overloaded. These reactions can provide information to help determine thecause of stresses.



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Page : 26After the cause of the stresses is understood, the stress reduction process can begin. To reduce displacementload stresses, flexibility is added to the system. Knowing the cause of the stresses permits adding flexibilitywhere it will be most effective. The three ways most frequently employed to add flexibility are to :

- Replace rigid supports with spring hangers.- Add expansion joints and- Revise the system geometry.

The usual approach to changing the system geometry to increase flexibility is to add expansion loops. Anexpansion loop inserted in a straight run adds four elbows to the system. Elbows are much more flexiblethen straight pipes. Whenever increased flexibility is required the addition of elbows should be considered.A couple of elbows inserted at strategic places is frequently the economical solution to a flexibilityproblem.

When flexibility is added to a system, the displacement load stresses are decreased but the sustained loadstresses are increased. Therefore these must be checked to see whether they remain within allowable limits.The analyst must be aware that

- When sustained load stresses are reduced (system stiffness increased) displacement load stresses areincreased.

- When displacement load stresses are reduced (system flexibility increased) sustained load stresses areincreased, and

- Whenever one type of stress is reduced, the other must be checked to see if it is within its allowable value.

13. DESIGNING WITH EXPANSION JOINTS :Expansion joints can solve many problems encountered by the pipe stress analyst. Properly applied,expansion joints can simplify layouts and are more economical than other solutions to expansion problems.On the other hand, improper application of expansion joints can result in expensive repairs andmodifications, as well as costly shut-downs. The piping network must be carefully designed when usingexpansion joints. There are three types of expansion joints ball type, slip type and bellows or corrugatedtype. The ball- type and slip-type joints are seldom used because the packing is difficult to maintain orbecause leaks through the packing are intolerable.

13.1 BELLOWS EXPANSION JOINTS :Bellows expansion joints (also known as corrugated expansion joints) are commonly used in piping systemsand are quite versatile. They are best suited for absorbing direct axial movement but can be used to absorblateral or angular movement. See Figure 19.



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Page : 27Bellow expansion joints are essentially rigid in torsion.

By using appropriate hardware and two or more bellows to form an expansion joint, one can constructuniversal, hinged, gimbals or pressure balanced joints as illustrated in Figure 20.

Many designers try to avoid the use of expansion joints because of previous un-fortunate experiences. Thereare inherent problems which can be minimised with proper selection and application. Characteristics ofbellows expansion joints are :

1. Bellows are made of relatively thin metal. Corrosion can be a real problem because there is little metalavailable. The bellows must be fabricated from a material resistant to corrosion attack by the process fluid.An internal sleeve may be required to protect the bellows from erosion.

The relatively thin metal must be protected from abuse during shipment and construction, as well as after it isinstalled. External blows on a bellows can render it useless. External protective covers should beconsidered.

2. Bellows are designed to deform plastically. Because the bellows repeatedly yield as they move throughtheir rated cycle, they can be subject to premature fatigue failures. The normal high state of stress in thebellows makes them prime candidates for stress corrosion attack. Austenitic stainless steel bellows shouldnot be specified for steam or water service, for example.

3. Bellows expansion joints can be easily misapplied. Without a thorough understanding of the limitations ofa bellows expansion joint, a designer may make a number of mistakes which lead to failures.

An example is over looking the pressure thrust from an expansion joint. This can result in specifying a jointwhose rated displacement is not sufficient to accommodate the movement of the system. Significanttorsional loads can also cause failure.

13.2. SELECTION AND SPECIFICATION :

The first step in designing with expansions joints is to make sure that the expansion joint is the practical andeconomical means for providing flexibility. This is most often the case with large diameter pipe (greaterthan NPS 12) in compact layouts, and when there are low allowable reactions, such as at pump and turbinenozzles.



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To protect the expansion joint from failure, it must be protected from excessive movement unless the pipe isproperly anchored and guided. The anchors and guides must take the test pressure thrust. The pressurethrust is defined by :

dp2

TP = π PT (10) 4

The pressure thrust can be considered as two equal and opposite forces acting at each end of the expansionjoint, as illustrated in Figure 21.

Without the guide marked (1) the pipe might be overstressed at anchor A and the expansion joint could beoverextended. Guide (2) & (3) are necessary to prevent excessive lateral displacement and rotation of theexpansion joint. In general, the expansion joint must be free to move in the direction intended and must besecured against deformation in other directions.

Pressure thrust can be eliminated from the piping system by using gimbals or hinged expansion joints.

If the bellow is near a machine which vibrates, the vibration amplitude and frequency should also bespecified. Occasionally, the limit rods (or tie rods) are subjected to thermal loads in addition to the loadscaused by pressure thrust. These loads must be described in detail to the expansion joint manufacturer.Additional information is required to specify hinged, gimbals, or pressure-balanced expansion joints.



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14.0 SAMPLE DATA FOR EXPANSION RATE & ALLOWABLE STRESSES

14.1 MATERIAL : ASTM-A53GR.B, A106 GR.B, API 5L GR.B TEMPERATURE ∆ E ANSI B31.3 - 1987 Ed

˚ F ˚ C ( mm / M ) Kg / Cm2x106 σ, Kg / Cm2 σA, Kg / Cm2

-425 -254 - - - --375 -226 - - - --325 -198 -2.0 2.110 - --300 -184 -1.9 2.102 - --250 -157 -1.6 2.088 - --200 -129 -1.4 2.074 - --150 -101 -1.2 2.057 - --100 -73 -1.0 2.039 - --50 -46 -0.7 2.017 - --20 -29 -0.5 2.005 1406 21100 -18 -0.4 1.994 1406 2110

32 0 -0.2 1.980 1406 211070 21 0.0 1.962 1406 2110

100 38 0.2 1.959 1406 2110150 66 0.5 1.954 1406 2110200 93 0.8 1.948 1406 2110250 121 1.2 1.937 1406 2110300 149 1.5 1.927 1406 2110350 177 1.9 1.913 1406 2110400 204 2.2 1.899 1406 2110450 232 2.6 1.878 1368 2100500 260 3.0 1.856 1329 2090550 288 3.4 1.831 1273 2076600 316 3.8 1.807 1217 2062650 343 4.2 1.775 1195 2057700 371 4.7 1.744 1160 2048750 399 5.1 1.694 914 1986800 427 5.6 1.645 759 1948850 454 6.0 1.473 612 1910900 482 6.5 1.301 457 1872950 510 6.9 1.192 316 1837

1000 538 7.4 1.083 176 18021025 552 7.6 1.041 144 17941050 566 7.9 0.999 113 17861075 579 8.1 0.957 91 17811100 593 8.4 0.914 70 17761125 607 8.6 - - -1150 621 8.8 - - -1175 635 9.0 - - -1200 649 9.2 - - -1250 677 9.7 - - -1300 704 10.2 - - -1350 732 10.6 - - -1400 760 11.1 - - -1450 788 - - - -1500 816 - - - -

σA = ( 1.25 σ c + 0.25 σ h ) i. e. Allowable stress range ∆ = Total thermal expansion (from 70º F / 21º C to indicated temp.) E = Young's modules ; σ = Allowable stress.



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14.2 MATERIAL : ASTM-A335 GR.P11 (SEAMLESS) COMPOSITION : 1.25 Cr. -0.5 MoTEMPERATURE ∆ E ANSI B31.3 - 1987 Ed


-425 -254 - - - --375 -226 - - - --325 -198 -2.0 2.180 - --300 -184 -1.9 2.174 - --250 -157 -1.6 2.163 - --200 -129 -1.4 2.151 - --150 -101 -1.2 2.145 - --100 -73 -1.0 2.138 - --50 -46 -0.7 2.128 - --20 -29 -0.5 2.121 1406 21100 -18 -0.4 2.117 1406 2110

32 0 -0.2 2.111 1406 211070 21 0.0 2.102 1406 2110

100 38 0.2 2.096 1406 2110150 66 0.5 2.085 1361 2098200 93 0.8 2.074 1315 2087250 121 1.2 2.056 1290 2080300 149 1.5 2.039 1266 2075350 177 1.9 2.025 1248 2070400 204 2.2 2.011 1231 2066450 232 2.6 1.990 1220 2063500 260 3.0 1.969 1210 2060550 288 3.4 1.948 1192 2056600 316 3.8 1.927 1174 2051650 343 4.2 1.899 1139 2043700 371 4.7 1.870 1097 2032750 399 5.1 1.839 1069 2025800 427 5.6 1.807 1055 2022850 454 6.0 1.765 1020 2013900 482 6.5 1.723 900 1983950 510 6.9 1.670 774 1951

1000 538 7.4 1.617 549 18951025 552 7.6 1.571 468 18751050 566 7.9 1.525 387 18551075 579 8.1 1.479 334 18411100 593 8.4 1.434 281 18281125 607 8.6 1.350 229 18151150 621 8.8 1.265 176 18021175 635 9.0 1.181 130 17901200 649 9.2 1.097 84 17781250 677 9.7 - - -1300 704 10.2 - - -1350 732 10.6 - - -1400 760 11.1 - - -1450 788 - - - -1500 816 - - - -

σA = ( 1.25σc + 0.25σ h ) i. e. Allowable stress range

∆ = Total thermal expansion (from 70º F / 21º C to indicated temp.) E = Young's modules ; σ = Allowable stress.



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14.3 MATERIAL : ASTM-A312 TP 304 SEAMLESS) COMPOSITION : 18 Cr. - 8 NiTEMPERATURE ∆ E ANSI B31.3 - 1987 Ed


-425 -254 - - 1406 2110-375 -226 - - 1406 2110-325 -198 -3.2 2.138 1406 2110-300 -184 -3.0 2.131 1406 2110-250 -157 -2.6 2.117 1406 2110-200 -129 -2.3 2.103 1406 2110-150 -101 -1.9 2.085 1406 2110-100 -73 -1.4 2.067 1406 2110-50 -46 -1.0 2.044 1406 2110-20 -29 -0.8 2.031 1406 21100 -18 -0.6 2.022 1406 2110

32 0 -0.4 2.006 1406 211070 21 0.0 1.989 1406 2110

100 38 0.3 1.980 1406 2110150 66 0.7 1.964 1406 2110200 93 1.2 1.948 1406 2110250 121 1.7 1.927 1406 2110300 149 2.2 1.905 1406 2110350 177 2.7 1.888 1360 2098400 204 3.2 1.871 1315 2086450 232 3.7 1.853 1272 2076500 260 4.2 1.835 1230 2065550 288 4.7 1.810 1191 2055600 316 5.2 1.786 1153 2046650 343 5.7 1.765 1139 2042700 371 6.3 1.744 1125 2039750 399 6.8 1.720 1097 2032800 427 7.4 1.695 1069 2025850 454 7.9 1.670 1048 2020900 482 8.4 1.645 1026 2014950 510 9.0 1.620 998 2007

1000 538 9.6 1.596 970 20001025 552 9.8 1.584 914 19861050 566 10.1 1.572 858 19721075 579 10.4 1.560 770 19501100 593 10.7 1.547 682 19281125 607 10.9 1.535 612 19111150 621 11.2 1.523 541 18931175 635 11.5 1.511 482 18781200 649 11.8 1.498 422 18631250 677 12.4 1.477 330 18401300 704 13.0 1.456 260 18231350 732 13.5 1.406 204 18091400 760 14.1 1.357 162 17981450 788 14.7 - 127 17891500 816 15.4 - 98 1782

σA = ( 1.25σ c + 0.25σ n ) i. e. Allowable stress range ∆ = Total thermal expansion (from 70º F / 21º C to indicated temp.) E = Young's modules ; σ = Allowable stress.

C-27 PIPING FLEXIBILITY ANALYSIS

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