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chapter

5

133

DISCONTINUITY STRESSES

Pressure vessels usually contain regions where abrupt changes in geometry, material, or loading oc-

cur. These regions are known as discontinuity areas, and the stresses associated with them are called

discontinuity stresses by the code [1, 2]. These codes have outlined a general procedure or analyzing

the discontinuity stresses and discussed examples o some common occurrences. This chapter will

discuss some o the applications relevant to the design o piping systems.

Because o dissimilar characteristics, each o the adjacent parts joining at a discontinuity area be-haves dierently to an applied load, such as internal pressure or temperature. The deormations o the

disconnected ree bodies are dierent rom each other. Because these parts are joined together, they

share a common displacement that is dierent rom their ree displacements. The dierence between

the ree displacement and the actual joint displacement is a orced displacement, which produces

orces and stresses. These additional stresses are reerred to as discontinuity stresses.

Calculation o discontinuity stresses is generally based on the behavior o the longitudinal strip o

the cylindrical shell. Because a longitudinal stripe o a vessel behaves like a railroad sitting on an elastic

oundation, the discontinuity stresses at the vessel are generally calculated based on the theory o Beams

on Elastic Foundation [3, 4]. A beam on an elastic oundation receives a lateral reaction orce that is

proportional to the displacement. The rail track is a typical example o a beam on elastic oundation.

5.1 DIFFERENTIAL EQUATION OF THE BEAM DEFLECTION CURVE

Some basic principles o strength o materials have been discussed in Chapter 2. However, beam

deection equations were skipped or simplicity. As these equations are essential or discussing beams

on elastic oundation, they are summarized in this section.

The sign conventions are as given in Fig. 5.1. The positive y-axis is assumed to be pointing down-

ward. This is dierent rom the convention used by piping stress analyses, but ollows the convention

used in traditional treatment o beams on elastic oundation. All positive values o orces and mo-

ments are as shown.

Figure 5.1 shows the relationship between the changes in orces and moments at an infnitely small

beam element defned by two adjacent transverse parallel planes a-b and c-d. The positive shear orces

and moments are as shown. The uniorm orce, q, per unit length, acts upward representing the reac-

tion to the downward movement o the beam. Because the beam is sitting on an elastic oundation,

this reaction orce is proportional to the downward displacement. That is, q= ky, where kis the spring

constant o the oundation per unit length o the beam. The changes in orces and moments across

these two planes can be ound by the equilibrium o them acting on the entire element. First, by takingthe equilibrium o the yorces, we have

-V- qdx+ V+ dV= 0

or

dV

dx q (5.1)

Downloaded 08 Feb 2012 to 130.159.158.129. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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134 Chapter 5

The relation o the moments and orces are determined by the equilibrium o the moments. Taking

the moment around the base plane a-b and setting the sum to zero, we have the equilibrium equation

as ollows

M (M dM) (V dV)dx qdxdx

2 0

By ignoring the higher-order terms o very small quantities, the above becomes

dM

dx V (5.2)

Combining Eqs. (5.1) and (5.2), we have

d2M

dx2= q = ky (5.3)

The deormation o a slender beam can be entirely attributed to the bending moment. The shear

orce eect on the displacement is generally small in slender beams and can be ignored. The basic

elastic curve o a beam in pure bending, as can be ound in a textbook on strength o materials, is given

as ollows:

EId2y

dx2= -M (5.4)

By dierentiating the above equation with respect to xtwice and substituting Eq. (5.3), we have the

dierential equation or the beam on elastic oundation

EId4y

dx4= -ky or

d4y

dx4+

k

EIy = 0 (5.5)

This is a linear dierential equation o ourth order with constant coefcients and right member

zero. It can be solved by letting y= ex, where is an exponent constant yet to be determined. Di-erentiating and substituting y= ex to Eq. (5.5), and dividing both sides with y, we have

a

4

+

k

EI = 0 or a =

4

-

k

EI =

4 k

EI

4

-1

The value, involving the ourth roots o negative one, has the ollowing our values:

4-1 =

1

2(1 i) ; i = -1

The square root o 2 actor can be combined with k/EI to become the characteristic actor

Fig. 5.1

Beam on elastic Foundation


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Discontinuity Stresses 135

4 k

4EI (5.6)

The our solutions o Eq. (5.5) become

y1 exix

; y2 exix

; y3 exix

; y4 e xix

The general solution o the equation is

y Ay1 By2 Cy3 Dy4

whereA, B, C, and Dare integration constants. Ater some mathematical manipulations, we have the

working solution as

y e

x(C1cosx C2 sinx) e

x(C3cosx C4 sinx) (5.7)

The integration constants C1, C2, C3, and C4 are to be determined rom the known boundary and

physical conditions o the beam. This chapter will use Eq. (5.7) to solve some o the discontinuity

stress problems related to piping design.

5.2 INFINITE BEAM ON ELASTIC FOUNDATION WITH

CONCENTRATED LOAD

The infnite beam with a concentrated load is the frst situation to be investigated. Equation (5.7)

applies to the entire beam except at the loading point, which is considered the origin with x = 0.

Because o this discontinuity point, the constants C1, C2, C3, and C4 will be dierent at the let-hand

and right-hand sides o the beam. However, due to the condition o symmetry, it is only necessary to

investigate one side o the beam. By investigating the right-hand (+x) side, the integration constants

are determined by the ollowing conditions.

Because either C1 or C2 would have made the deection infnite at an infnite distance away (x ),

both C1 and C2 have to be zero. That is, C1 = C2 = 0. The deection curve or the right-hand portion

o the beam becomes

y e

x(C3cosx C4 sinx) (5.8)

The frst two constants have been determined by the boundary condition at a point infnitely away

rom the loading point. The other two constants can be determined by the boundary condition at the

loading point. Due to the condition o symmetry, at the loading point the slope must be zero and the

shear orce equals one-hal o the applied orce. To satisy these conditions, we have

y dy

dx e

x[(C3 C4)cosx (C3 C4)sinx]

withx=0,y=(-C3+C4)=0; hence,C3=C4

and

V =dM

dx

= - EId3y

dx3

= - EIb3e-bx[2(C3 + C4)cosbx + 2(-C3 + C4)sinbx]

withx = 0, V= - EIb3[2(C3 + C4)] = -

P

2

From the above two relations we have

C3 C4 P

8EI3

P

2k


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136 Chapter 5

From Eq. (5.8), the deection curve or the right-hand portion o the beam becomes

y =Pb

2k

e-bx(cos bx + sin bx) (5.9)

From which we have the ollowing relations

q =dy

dx= -

Pb2

ke

-bxsin bx (5.10)

M = -EId2y

dx2= -

P

4be

-bx(sin bxcos bx) =P

4be

-bx(cos bx - sin bx) (5.11)

V = - EId3y

dx3= -

P

2e

-bxcos bx (5.12)

The above equations can be written with American Society o Mechanical Engineers (ASME) nota-

tions [1, 2] as ollows:

y P

2kf3(x) (5.13)

P2

kf4(x) (5.14)

Fig. 5.2

inFinite Beam with concentrated Force, P curves reFlect only the

general shaPes


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M P

4f2(x)

(5.15)

V P

2f1(x) (5.16)

where

f1(bx) = e

-bxcos bx (5.17a)

f2(bx) = e

-bx(cos bx - sin bx) (5.17b)

f3(bx) = e

-bx(cos bx + sin bx) (5.17c)

f4(bx) = e

-bxsin bx (5.17d)

Figure 5.2 shows the deections, slopes, bending moments, and shear orces along the beam in

terms o the dimensionless parameter x. The bending moment curve represents the attenuation othe stress rom the loading point, and the shear orce curve represents the load carrying capacity o

the beam element at the points away rom the loading point. The attenuation behaviors o these two

curves are enlarged in Fig. 5.3. This fgure serves as the guideline or setting the eective zones or

reinorcement, nozzle separation, load carrying area, and other actors. The eect o the concentrated

load diminishes to less than 20% o that at the loading point when x is greater than 1.0. This x=1.0 appears to be a good cuto point and has been used in the pressure design o the miter bends

discussed in Chapter 4.

Fig. 5.3

attenuation oF shear Force and Bending moment


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138 Chapter 5

5.3 SEMI-INFINITE BEAM ON ELASTIC FOUNDATION

A very long uniorm beam with the loads applied at the edge is called a semi-infnite beam. Most o

the discontinuity problems involve two pieces o dissimilar parts joining together. It can be the joiningo two pieces o dissimilar semi-infnite pipe segments, or a semi-infnite pipe segment with a non-pipe

dissimilar body. Thereore, the semi-infnite beam model is the main work orce or calculating the

discontinuity stresses in the piping systems.

As shown in Fig. 5.4, the semi-infnite beam generally involves a shear orce and a bending moment.

The general solution or the beam on elastic oundation as given in Eq. (5.7) is applicable to the entire

beam. Again, because either C1 or C2 would have made the displacement infnite as the x value ap-

proaches infnity, both C1 and C2 have to be zero. Hence, the general solution is the same as the one

previously given or the infnite beam (Eq. 5.8). For convenience, the general solution, as given by Eq.

(5.8), is duplicated as ollows:

y = e-bx(C3cos bx +C4 sin bx)

The constants C3 and C4 can be determined by the boundary conditions at the loading point where

x= 0. From Eqs. (5.4) and (5.2), we have

EId2y

dx2 x0 M0

EId3y

dx3 x0 V P

Dierentiating the general solution (5.8) and substituting the dierentials into the above relations,

the integration constants are ound as

C3 =1

2b3EI(P- bM0); and C4 =

M0

2b2EI

Substituting C3 and C4, the solution becomes

y =e-bx

2b3EI[Pcos bx - bM0 (cos bxsin bx)]

(5.18)

or, using ASME notations given in Eqs. (5.17a) to (5.17d) and substituting 44EI = k, the abovebecomes

y 2

kP f1(x) M0f2(x)]

2P

kf1(x)

2M02

kf2(x) (5.19)

Similarly, by successive dierentiation o (5.18), we have

dy

dx

2P2

kf3(x)

4M03

kf1(x) (5.20)

Fig. 5.4

semi-inFinite Beam


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M EId2y

dx2

P

f4(x) M0 f3(x)

(5.21)

V EId3y

dx3 P f2(x) 2M0 f4(x) (5.22)

Two working ormulas or the condition at the edge o the beam can be ound by setting x= 0 or

Eqs. (5.19) and (5.20). Noting that f1 = f2 = f3 = 1.0 at x= 0, we have

y0 2P

k

2M02

k (5.23)

0 2P2

k

4M03

k (5.24)

By using Eqs. (5.23) and (5.24) together with the principle o superposition, many discontinuity

problems regarding pressure vessels and piping segments can be solved.

5.4 APPLICATION OF BEAM ON ELASTIC FOUNDATION TO

CYLINDRICAL SHELLS

One o the most important applications o the theory o beams on elastic oundation is the calcula-

tion o stresses produced at discontinuity junctions o thin-wall cylindrical shells. Beore any shell

application can be perormed, we have to frst establish that the shell behaves the same way as beam

on elastic oundation and the oundation spring constant can be readily calculated.

Figure 5.5 shows a cylindrical shell subjected to a radial load uniormly distributed along any circle

perpendicular to the axis o the shell. Because o the symmetrical loading, the section normal to the

axis remains circular. The load causes the shell to move y-distance in the radial direction toward the

center. Thisydisplacement changes along the axis, and thus creates bending stresses on the shell. Due

to the symmetry o the cross-section and loading, the situation can be investigated with a longitudinal

strip o unit width, b = 1, as shown in the fgure.

Fig. 5.5

cylindrical shell with axially symmetrical loading


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140 Chapter 5

Radial displacement ymust be accompanied with a compression displacement in the circumeren-

tial direction. The situation is similar to a shell subjected to an external pressure. In conjunction with

the reduction in radius, the circumerence is reduced. A reduction in circumerence means compres-

sion in the circumerential direction. The amount o compression strain is the same as the rate ochange in radius, y/r. This gives a circumerential compressive orce per unit strip length as ollows

N Et

ry

The strip having a width b = r is considered a beam supported by N urnished by the rest o theshell, (2-). The resultant o these Norces has a radial direction component as

P 2Nsin

2 2N

b 2

r

N

rb

1

r

Et

ry b

Et

r2y for b 1

Because the support orce, P, is proportional to the displacement, y, the strip is considered a beam

on elastic oundation. With the unit width o the beam, the spring constant o the oundation per unit

length o beam is determined by

k Py

Etr2

(5.25)

When the strip o the beam is subjected to a bending moment, it produces linearly varying longitu-

dinal stresses as shown in Figure 5.6. The maximum stress equals the moment divided by the section

modulus. These bending stresses produce linearly varying strains in the longitudinal direction. I the

side suraces were ree to move, these stresses would have also produced circumerential varying

strains as shown. However, under the symmetrical condition, the side suraces have to remain in the

radial direction, preventing the varying deormation rom taking place. In the actual condition, the

circumerential strain due to longitudinal bending moment is zero. Putting ez = 0 in Eq. (2.8) and

ignoring Sy, we have

ex Sx

E

Sz

E; ez

Sz

E

Sx

E 0 Sz Sxor

Combining the above two equations, the stress strain relation or the bending stress becomes

ex 1

ESx

2Sx =

1 2

ESx

This means that in a laterally constricted beam, the strain or displacement produced by a bending

moment is smaller than that predicted by Hooks law by a actor o (1 -2). The beam becomes stier

Fig. 5.6

sidewise constriction


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than usual. In practical applications, either the modulus o elasticity or the moment o inertia can be

increased by a actor o 1/(1 -2) to match the reduction o the strain and displacement. That is, wecan choose either

E E

1 2or I

I

1 2, but not both

Generally, it is less conusing to modiy the modulus o elasticity, because a modifcation o the

moment o inertia may propagate to the section modulus, which remains the same. However, to avoid

conusion between the modulus o elasticity used in the oundation spring constant and the one used

or the bending moment, the traditional treatment o beam on elastic oundation chooses to modiy

the moment o inertia as

I I

1 2

t3b

12(1 2)

t3

12(1 2)

or a rectangular cross-section with width b = unity. Substituting the above I or I and k rom Eq.

(5.25) in Eq. (5.6), we have

4 k

4EI

4 3(1 2)

r2t2

1285

rtfor 0.3

, (5.26)

Equation (5.26) shows the characteristic actor o pipe considered as beam on elastic oundation.

This beam on elastic oundation analogy is strictly applicable only to axially symmetric and uniormly

distributed circumerential loading and deormation. However, it has been extended to some applica-

tions with localized loading, presumably as a conservative approach.

5.5 EFFECTIVE WIDTHS

In dealing with discontinuity, it is important to know the extent o its eect. This includes the es-

timate o the width o the reinorcement required to spread out the load, the eective orce-carrying

zone to share the load, and the eective reinorcement zone. The eect o the discontinuity attenu-

ates inverse exponentially with distance. It reduces very quickly extending outward, but never exactly

reaches the zero point. Because the defnition o the eective zone is not very clear-cut, some engi-neering judgment is exercised in defning it. Depending on the purpose it serves, the eective zone is

defned somewhat dierently or each type o application. However, they all ollow the same pattern

based on beams on elastic oundation. The responses o beams on elastic oundation are generally

expressed in terms o the combined dimensionless location parameter x. Considering the discontinu-ity as an infnite beam subjected to a concentrated load, the response curves given in Figs. 5.2 and 5.3

can be used as guidelines or determining the eective width. Figure 5.3 shows that at x= 1, the shearorce reduces to about 20% o the orce at the discontinuity point. Because shear orce represents the

load carrying capacity, it appears reasonable to choose x= 1 or the load carrying zone and eectivereinorcement zone. Assuming = 0.3, we have

x 1285

rtx 10 i.e., effective width x 0778 rt (5.27)

This number has been adopted in the pressure design o miter bends discussed previously. As orthe eective reinorcement zone, some codes use a smaller constant o 0.5 instead o 0.778.

The eective width or providing enough stress attenuation is defned based on the bending moment

curve. Figure 5.3 shows that the bending moment reaches zero at x= /4, and reverses its sign aterthat. It reaches negative maximum at x= /2 with the peak magnitude reducing to 21% o the origi-nal moment. The moment reduces to negative 10% at x= 2.6. With this 10% residual stress criterion,the separation or reinorcement width required is


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142 Chapter 5

bx =1.285

rtx = 2.6, i.e., required width x= 2.0 rt

(5.28)

In calculating the reinorcement width, the thickness also includes the pad thickness.

5.6 CHOKING MODEL

One basic application o shell as beam on elastic oundation is the choking model. In this model,

a uniorm radial line load is applied around the circumerence o the shell as shown in Fig. 5.7. This

is, in essence, the same model given in Section 5.4 or deriving the characteristic parameter . In thissection, we will use the model to fnd the stress produced. From this model, the load per unit length

around the circumerence is f, and the corresponding displacement due to this load is Dy. This is

equivalent to an infnite beam on elastic oundation subjected to concentrated orce as discussed in

Section 5.2. By taking a unit circumerential width as the beam and assuming = 0.3, then rom Eqs.(5.15) and (5.26) we have the maximum bending moment occurring at orcing pointx= 0 as

M f

4 f

4 1285 rt 01946frt

Dividing the bending moment with the section modulus Z= t2/6, we have the maximum bending

stress at orcing point as

Sb M

Z

1167frt

t2 1167

rt15

f (5.29)

This ormula has been extensively used or calculating local attachment stresses in the design o

piping systems [5].

The bending stress shown in Eq. (5.29) is in the longitudinal direction. It is also reerred to as the

longitudinal shell bending stress. Because its average across the shell thickness is zero, it is not in-

cluded in the membrane stress category. As discussed previously, due to the nature o symmetry, the

circumerential bending strain due to Poissons eect is suppressed thus producing a circumerential

shell bending stress equal to Sb = 0.3Sb or = 0.3. Because this Poissons stress is always in the same

sign as the longitudinal shell bending stress, it is subtractive to the longitudinal stress when combining

Fig. 5.7

choking model on cylindrical shell


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with other circumerential stresses. Thereore, it is conservatively ignored in most practical applica-

tions. See also Fig. 5.9.

In addition, there is also a circumerential membrane stress due to choking displacement, Dy. From

Eqs. (5.13) and (5.25), we have the displacement at orcing point equal to

y f

2k

f

2

1285

rt

r2

Et 0643

r

t

15 f

E

This displacement produces a circumerential strain oDy/r, thus producing a circumerential mem-

brane stress o

Smc Ey

r 0643

r

t15f (5.30)

This circumerential membrane stress may be additive to the pressure hoop stress depending on the

direction o the orce.

5.7 STRESSES AT JUNCTIONS BETWEEN DISSIMILAR MATERIALS

A piping system may involve more than one material, or may connect to an equipment o dier-

ent elastic or thermal properties. The most common junctions between dissimilar materials are those

involving an austenitic stainless steel section connected to a erritic steel section. Austenitic steel

is needed at high temperature zones directly in contact with the radiant ame inside a urnace or a

boiler, although erritic steel is more economical or outside piping. The same thing applies in a pip-

ing system with a combination o internally insulated and externally insulated sections. The internally

insulated portion uses carbon steel, whereas the externally insulated portion requires the use o stain-

less or other high alloy steel pipe.

When two pipes o dierent materials are joined together, the joint produces additional disconti-

nuity stresses due to dierences in expansion rate, thickness, and modulus o elasticity. Figure 5.8

shows a simple bimetallic welded joint o a pipe with a uniorm thickness. When the pipe temperature

changes rom the construction temperature, reerred to as ambient, the pipe expands. Due to dier-

ences in expansion rate, the radius o Mat-a would have expanded at an amount oDa, whereas Mat-b

would have expanded byDb

, i the two sides were not joined together. Because the two pieces arejoined together, the actual displacement at the junction is somewhere in between these two values.

Fig. 5.8

Junction Between dissimilar materials


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144 Chapter 5

This requires an internal squeezing orce at the piece with the higher expansion rate to reduce its

displacement, and an internal aring orce at the piece with the lower expansion rate to increase its

displacement. The squeezing and aring internal orces, acting in opposite directions, have the same

magnitude and are denoted as P. In general, an internal shell bending moment is also produced alongthe pipe. The shell bending moments or both sides o the piping are the same M0, but in opposite

directions, at the junction.

Each piece o the pipe segment can be considered a semi-infnite beam on elastic oundation, sub-

jected to the concentrated end orce Pand end bending moment M0. By comparing Fig. 5.8 with Fig.

5.4, it is clear that Mat-a can use the standard ormulas or a semi-infnite beam as presented, whereas

Mat-b needs an adjustment in the ormulas. Mat-b can be viewed as Fig. 5.4 turned upside down with

the y-axis pointing upward, but with the sign o the bending moment reversed that is, when using

the standard semi-infnite beam ormulas, the sign oM0 is reversed or Mat-b. The working ormulas

or the dissimilar joints are then derived using two boundary conditions: (1) the slopes are the same

or both joining pieces at junction; and (2) the sum o the displacements rom both pieces is equal to

the dierential expansion D = Da-Db. From Eq. (5.24) with a = b = , we have

2P2a

ka

4M03a

ka

2P2b

kb

4M03

b

kb

or

2P2a

ka

2bkb

4M03a

ka

3b

kb (5.31)

The sum o the displacements equals the dierential thermal expansion that is, D = Da-Db =

r(DT)(a-b). DT is the temperature dierence between construction and operation, and a and bare thermal expansion coefcients or Mat-a and Mat-b, respectively. From Eq. (5.23), we have

r(T) (a b) 2Pa

ka

2M02a

ka

2Pbkb

2M0

2b

kb

or

r(T) (a b) 2Pa

ka

b

kb

2M02b

kb

2a

ka (5.32)

Equations (5.31) and (5.32) can be easily applied to junctions with two segments o pipes having

identical cross-sections. They can also be applied to a pipe connecting to a relatively rigid section.

Two special cases will be discussed in the ollowing to investigate the general behaviors o the junc-

tions between dissimilar materials. In Sections 5.7.1 and 5.7.2, we use = 0.3 to obtain some com-parative stress values.

5.7.1 Uniform Pipes With Similar Modulus of Elasticity

The modulus o elasticity maintains relative uniormity or all errous and some non-errous materi-

als at temperatures within the allowable working range. Thereore, a simplifed model assuming both

Mat-a and Mat-b having the same EI, and hence the same kand values will be used to quick checkthe general behaviors o the junction. By assuming ka = kb = kand a = b = , Eq. (5.31) gives M0 =0 that is, the bending moment at the junction is zero. With M0 = 0, Eq. (5.32) becomes

r(T)(a b 4P

kor P

1

4

k

r(T)(a b)

Substituting the above M0 and P to Eq. (5.21), we have the bending moment at a point x-distance

away rom the junction as


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M P

f4(x)

k

42r(T)(a b)f4(x)

The maximum bending moment Mmax occurs at x= /4 (i.e., x= /(4) = 0.611(rt)1/2), and f4(x)= 0.3224. Using this x, Eq. (5.25) or k, and Eq. (5.26) or , the maximum shell bending stress canbe calculated as

Sbmax Mmax6

t2

03224 6

4t2Et

r2

rt

12852r(T)(a b)

or

Sbmax 029E(T)(a b) (5.33)

In addition to the shell bending stress, there is the hoop stress caused by the radial displacement at

the junction. With the same kand , intuitively we can conclude that the displacement contributed byeach side o the pipe is one-hal o the dierential expansionD. That is,y0,a =y0,b = 0.5D= 0.5r(DT)(a-b). This can also be obtained by substituting the above Pand M0 = 0 to Eq. (5.23). The choking or

aring o the radius generates a circumerential strain o 0.5D/r, and thus a circumerential membranestress o

Sc E05

r 05E(T)(a b) (5.34)

This thermal circumerential stress is tensile in the piece with the lower expansion rate, and is com-

pressive in the piece with the higher expansion rate that is, both tension and compression natures

have to be considered, the same as in the case o bending. Because o this dual tensile and compressive

nature, it is additive to the pressure hoop stress.

Because both shell bending stress and circumerential hoop stress have tensile and compressive

characteristics, they will have to be added together directly to calculate the stress intensity. How-

ever, as discussed previously in this chapter, the longitudinal bending stress comes together with a

circumerential bending stress due to Poissons eect. Because Poissons stress is always in the same

sign as the longitudinal stress, it is subtractive to the longitudinal stress when we are dealing with the

combined eect.

Figure 5.9 shows the stress situation at the pipe shell. Based on maximum shear ailure theory, stress

intensity is taken as the dierence o the two perpendicular principal stresses. For the two directions

o stresses to be additive, they have to be in the opposite directions that is, one in tensile and the

other in compressive. Assuming that at a given point o the pipe wall the longitudinal bending stress

Fig. 5.9

suBtractive nature oF Poissons stress


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146 Chapter 5

is tensile, then it will be additive to the circumerential stress only when the circumerential stress is

compressive. However, a tensile longitudinal bending stress generates a tensile circumerential Pois-

sons bending that reduces the net compressive circumerential stress. Thereore, the combined stress

intensity becomes SE = (1 -)Sb + Sc.For the dissimilar material connection, the maximum circumerential stress occurs at the junction,

whereas the maximum longitudinal bending stress occurs at 0.25/away. As a quick estimate, wewill assume that both maximum circumerential stress and maximum longitudinal bending stress oc-

cur at the same point, in which case the potential maximum combined stress intensity is SE = [0.29

(1 - 0.3) +0.5]E(DT)(a-b) = 0.703E(DT)(a-b). Because their maximum values do not oc-cur at the same place, the actual maximum stress intensity is somewhat smaller, but never less than

0.5E(DT)(a-b), which occurs at the junction with zero longitudinal bending stress. We can use anaverage value o 0.6E(DT)(a-b) as the actual maximum combined stress intensity [6]. Thereore,the thermal discontinuity stress to be added to the general thermal expansion stress is

SE,t 060E(T)(a b) 060E(T2 T1)(a b) (5.35)

Because expansion rate changes considerably with temperature, the average value between T1 andT2 should be used. The piping codes generally provide the average expansion rates data, rom ambient

temperature to operating temperature, or most piping materials.

As an example, or austenitic stainless steel joined with carbon steel at 800F (427C). a = 10.110-6 in./in./F (18.18 10-6 mm/mm/C); b = 7.8 10

-6 in./in./F (14.04 10-6 mm/mm/C), E=

106(26.0 + 24.1)/2 = 25.05 106 psi (172.72 103 MPa), (T2-T1) = 800 - 70 = 740F (411.11C),

then SE,t = 0.6 25.05 740 (10.1 - 7.8) = 25,581 psi (176.376 MPa). It should be noted that or

uniorm thin pipe connections, the stress is independent o size and thickness.

Although the B31 codes adopt a base o using one-hal o the theoretical secondary stresses, the

above stress occurs at the weld joint, which is the basis o the adjusted stress. Since the above stress

is superimposed on a weld joint, its ull value shall be used in evaluating the sel-limiting stress. The

thermal discontinuity stress should be either added to the general thermal expansion stress range or

subtracted rom the allowable value.

From the above example, it is clear that at 800F (427C) the thermal discontinuity stress at the

austenitic-carbon steel junction is very close to the nominal allowable displacement stress range o30,000 psi (206.8 MPa). This leaves very little margin or the expansion stress range rom thermal ex-

pansion o the piping system. To reduce the thermal discontinuity stress, some critical piping may use

a two-step connection by inserting a short piece o nickel-chrome-iron steel pipe (e.g., ASTM B-407)

in between the austenitic-carbon steel junction. Because the expansion rate o nickel-chrome-iron is

roughly the average o the expansion rates o austenitic steel and carbon steel, it can eectively reduce

the junction thermal discontinuity stress by 50%. The length o insert shall at least be 5/or 4 in.(100 mm), whichever is greater. This is to ensure the separation o the discontinuity stress felds and

the weld aected zones.

5.7.2 A Pipe Connected to a Rigid Section

In comparison to the uniorm junction discussed above, a pipe connected to a rigid section is an-

other extreme o application. In this discussion, the pipe segment Mat-b is considered as rigid thatis, kb = . Substituting this kb into Eqs. (5.31) and (5.32), we have

P 2M0; and r(T)(a b) 2Pa

ka 2M0

2a

ka

Solving the above two equations and dropping subscript a on and k, we have


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M0 1

2

k

2r(T)(a b) and P

k

r(T)(a b)

The maximum shell bending stress, which occurs at the junction, is

Sb M06

t2

3

t2

Et

r2

rt

12852r(T)(a b 1817E(T)(a b) (5.36)

The pipe segment Mat-a in this case has to deect in the radial direction the ull dierential expan-

sion D. Thereore, the circumerential membrane stress is

Sh E

r 10E(T)(a b) (5.37)

Because shell bending stress and circumerential hoop stress occur at the same point, they need to

be added directly together to become part o the sel-limiting stress. Ater subtracting circumerential

Poissons bending, we have the stress due to discontinuity as

SE,t 1817(1 03) 1]E(T)(a b) 2272E(T2 T1)(a b) (5.38)

Equation (5.38) is applicable to nozzle inserts and socket welds. Since the stress rom (5.38) is 3.7

times as large as the one given by (5.35), a dissimilar material junction at a socket weld or other insert

should be avoided.

5.8 VESSEL SHELL ROTATION

Movements at boundary points are very important when calculating the stress o a piping system.

Normally, it is easy to visualize and to apply the translation movements at a vessel connection. Trans-

lation movements are calculated based on direct thermal expansion o the vessel. However, due to the

choking eect at the discontinuity area, a pressurized vessel shell may also have signifcant rotation

at the vicinity o the discontinuity. The rotation can create a much more severe stress in the piping

system compared to the translation movement. Under normal piping and vessel confgurations, thediscontinuity zone is too small to accommodate any nozzle or branch connection. However, the dis-

continuity zone at the bottom o a large storage tank can reach as much as 5 t (1.524 m) in the verti-

cal direction. In act, most tank connections are located at the bottom discontinuity zone. Thereore,

signifcant shell rotations may occur at the pipe connections located near the bottom portion o the

tank.

Figure 5.10 shows the bottom portion o the shell o a large storage tank, which has a diameter

o 180 t (54.86 m) and a liquid height o 70 t (21.34 m). The stored liquid produces a hydrostatic

pressure that varies linearly along the height. The maximum pressure occurs at the bottom course o

the shell. I the tank shell were not attached to the bottom plate, the pressure would have produced

proportionally a radial displacement with the maximum occurring at the bottom. However, because

the shell is attached to the bottom plate, the shell radial displacement is choked to zero at the bottom.

As the choking only aects a small portion o the shell, the shell can be considered having a uniorm

pressure and thickness in the area being investigated. The ree displacement o the shell constitutes

pressure and thermal two parts. That is,

Sh

Er (T2 T1)r

p

Etr2

(T2 T1)r (5.39)

wherep is the hydrostatic pressure. For water at 70 t deep,p = 70 62.4/144 = 30.33 psi (0.209 MPa).

The amount o displacement choked by the bottom plate depends on the mobility o the bottom plate.


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148 Chapter 5

For the pressure part, the radial displacement o the bottom plate is always considered zero. For the

thermal expansion part, the bottom plate has the potential to move as much as the shell does. How-

ever, because the uid is cooler at the bottom and the riction orce rom the oundation can greatly

restrict the expansion, no thermal expansion rom the bottom plate is generally assumed. Thereore,

the bottom plate will exert a ull choking displacement oD. This is a conservative assumption or the

piping stress analysis, because more choking creates more rotation, and thus more stress.

Deormation o the shell depends on the exibility o the bottom connection. Two extreme cases

(Fig. 5.10(a) and (b)) were investigated. The actual condition shall be somewhere in between these

two conditions.

Case (a) shows that the bottom plate is fxed allowing no rotation at the shell. This will exert a bend-

ing moment M0 as well as the radial orce Pa at the shell junction in order to choke the expanding shell

to the fxed bottom plate. This is the same as one side o the infnite beam subjected to a concentrated

orce. The deection, slope (rotation), and moment curves given in Fig. 5.2 are all applicable.

Case (b) shows that the bottom plate does not oer any rotational stiness. This is the case o thesemi-infnite beam with zero bending moment at the junction. Substituting M0 = 0 and y0 = D to Eqs.

(5.23) and (5.24), we have

P k

2and 0

2P2

k (5.40)

Substituting Pto Eqs. (5.19) and (5.20), we have the choked displacement and the slope at the point

locatedx-distance away rom the junction as

y 2P

kf1(x) f1(x) and

2P2

kf3(x) f3(x) (5.41)

Because the tank bottom is generally not anchored, the actual case is somewhat closer to case (b).

Due to the existence o the inection in the deection curve, case (a) is more sensitive to the shell

parameters that can only be estimated approximately. In addition, a pipe connection constitutes arigid area on the shell, thus making a sharp curvature as in case (a) not very likely. Thereore, unless

a more accurate assessment is available, a smoother curve as given by the case (b) is generally used in

the design analysis.

Because o the choking at the bottom, the circumerential stress at the shell changes along the height

o the shell. Figure 5.11 shows the relation between deection and stress. At point a, the un-aected

area, the shell deects D, which constitutes the hydrostatic pressure portion DP and thermal expan-

Fig. 5.10

deFormation oF tank shell


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sion portion DT. Since ree thermal expansion does not produce any stress, the circumerential hoop

stress is entirely due to the hydrostatic pressure. It is in tension and is proportional to DP. As the shell

is choked gradually, the deection o the shell reduces gradually, thus also reducing the tensile hoop

stress. Stress is reduced to zero at point b when the deection equals the thermal expansion DT. Fur-

ther reduction o the shell deection produces a compressive hoop stress. At point cthe compressive

hoop stress reaches maximum level. The maximum compressive hoop stress corresponds to squeezing

the thermal expansion DT to zero. Compressive hoop stress is oten ignored in the design analysis. A

design curve oten assumes that the tensile hydrostatic hoop stress reduces gradually to zero at the

bottom junction. This circumerential membrane stress at the choked area has to be taken into ac-

count when evaluating the interace with the piping. Further discussion on tank connections is givenin Chapter 8.

REFERENCES

[1] ASME Boiler and Pressure Vessel Code, Section VIII, Pressure Vessels, Division 2, Alterna-

tive Rules, Article 4-7, Discontinuity Stresses, American Society o Mechanical Engineers,

New York.

[2] ASME Boiler and Pressure Vessel Code, Section III, Rules or Construction o Nuclear Facil-

ity Components, Division 1, Article A-6000, Discontinuity Stresses, American Society o

Mechanical Engineers, New York.

[3] Hetenyi, M., 1946, BeamsonElasticFoundation, Chapter II, University o Michigan Press, Ann

Arbor, MI.

[4] Timoshenko, S., 1956, StrengthofMaterials, Part II, 3rd edn., Chapter I, McGraw-Hill, NewYork.

[5] M. W. Kellogg Company, 1956, DesignofPipingSystems, revised 2nd ed., Chapter 3, John Wiley

& Sons, Inc., New York.

[6] ASME Boiler and Pressure Vessel Code, Section III, Rules or Construction o Nuclear Facil-

ity Components, Division 1, Sub-article NB-3600, Piping Design, American Society o Me-

chanical Engineers, New York.

Fig. 5.11

tank deFlection and stress


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