table 7-11 computing stressespvmanage.com/wp...from-pages-from-pressure-vessel...table 7-11...

Table 7-11Computing stresses

Figure b Value from Figure Forces and Moments Stress

Radial Load

Membrane 7-21A NfRm

Pr¼ ð Þ Nf ¼ ð ÞPr

Rm

sf ¼ KnNf

t

7-21BNxRm

Pr¼ ð Þ Nx ¼ ð ÞPr

Rmsx ¼ KnNx

t

Bending 7-22AMf

Pr¼ ð Þ Mf ¼ ð ÞPr sf ¼ 6KbMf

t2

7-22BMx

Pr¼ ð Þ Mx ¼ ð ÞPr sx ¼ 6KbMx

t2

Longitudinal Moment

Membrane 7-23A NfR2mb

ML¼ ð Þ Nf ¼ ð ÞCLML

R2mb

sf ¼ KnNf

t

7-23BNxR

2mb

ML¼ ð Þ Nx ¼ ð ÞCLML

R2mb

sx ¼ KnNx

t

Bending 7-24AMfRmb

ML¼ ð Þ Mf ¼ ð ÞML

Rmbsf ¼ 6KbMf

t2

7-24BMxRmb

ML¼ ð Þ Mx ¼ ð ÞML

Rmbsx ¼ 6KbMx

t2

Circumferential Moment

Membrane 7-25A NfR2mb

Mc¼ ð Þ Nf ¼ ð ÞCcMc

R2mb

sf ¼ KnNf

t

7-25BNXR

2mb

MC¼ ð Þ Nx ¼ ð ÞCcMc

R2mb

sx ¼ KnNx

t

Bending 7-26AMfRmb

Mc¼ ð Þ Mf ¼ ð ÞMc

Rmbsf ¼ 6KbMf

t2

7-26BMxRmb

Mc¼ ð Þ Mx ¼ ð ÞMc

Rmbsx ¼ 6KbMx

t2

Shear Stresses

• Stress due to shear loads, VL or Vc .Round attachments:

ss ¼ VL

prot

ss ¼ VC

prot

Square attachments:

ss ¼ VL

4Ct

ss ¼ VC

4Ct

Rectangular attachments:

ss ¼ VL

4C1t

ss ¼ VC

4C2t

• Stress due to torsional moment, MT.Round attachments only!

sT ¼ MT

2pr2ot

Notes

1. Figure 7-15 should be used if the vessel is in brittle(low temperature) or fatigue service. For brittlefracture the maximum tensile stress is governing. Thestress concentration factor is applied to the stresseswhich are perpendicular to the change in section.

2. Subscripts q and C indicate circumferential direc-tion, X and L indicate longitudinal direction.

3. Only rectangular shapes where C1/C2 is between 1/4and 4 can be computed by this procedure. The chartsand graphs are not valid for lesser or greater ratios.

4. Methods of reducing stresses from local loads:a. Add reinforcing pad.b. Increase shell thickness.c. Add partial ring stiffener.d. Add circumferential ring stiffener(s).e. Kneebrace to reduce moment loads.f. Increase attachment size.

5. See Procedure 7-3 to convert irregular attachmentshapes into suitable shapes for design procedure.

6. For radial loads the stress on the circumferential axiswill always govern.

7. The maximum stress due to a circumferentialmoment is 2–5 times larger than the stress due toa longitudinal moment of the same magnitude.

8. The maximum stress from a longitudinal moment isnot located on the longitudinal axis of the vesseland may be 60�–70� off the longitudinal axis. Thereason for the high stresses on or adjacent tothe circumferential axis is that, on thin shells, thelongitudinal axis is relatively flexible and free todeform and that the loads are thereby transferredtoward the circumferential axis which is less free todeform. Figures 7-23 and 7-24 do not showmaximum stresses since their location is unknown.Instead the stress on the longitudinal axis is given.

9. For attachments with reinforcing pads: This appliesonly to attachments that are welded to a reinforcingplate that is subsequently welded to the vessel shell.Attachments that are welded through the pad (likenozzles) can be considered as integral with the shell.

Moment loadings for nonintegral attachments must beconverted into radial loads. This will more closelyapproximate the manner in which the loads are distributedin shell and plate. Stresses should be checked at the edge ofattachment and edge of reinforcing plate. The maximumheight of reinforcing pad to be considered is given by:For radial load:

2d2 max ¼ 2C2d1C1

For longitudinal moment:

2d21 max ¼ 4C2d13C1

For circumferential moment:

2d11 max ¼ 4C1d23C2

Moments can be converted as follows:

Pr ¼ 3ML

4C2or

Pr ¼ 3Mc

4C1

10. This procedure is based on the principle of “flex-ible load surfaces.” Attachments larger than one-half the vessel diameter (b>0.5) cannot be deter-mined by this procedure. For attachments whichexceed these parameters see Procedure 7-1.

456 Pressure Vessel Design Manual

70

50

30

2015

108

6

4

3

2

1.00.80.60.5

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

NφR

mP r

NxR

mP r

γ = 300γ = 100γ = 50

γ = 15

γ = 5

β

β

70

5040

30

2015

10

8

643

2

1

0.050.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

γ = 300

γ = 100

γ = 50

γ = 15

γ = 5

(a)

(b)

Figure 7-21. Membrane force in a cylinder due to radial load on an external attachment. (Reprinted by permission fromthe Welding Research Council.)

Local Loads 457

0.6

0.4

0.2

0.10.080.06

0.04

0.02

0.010.0080.006

0.004

0.0020.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

β

β

γ = 5

γ = 15

γ = 50γ = 100

γ = 300Mφ

P rM

xP r

γ = 5

γ = 15

γ = 50

γ = 100

γ = 300

0.3

0.20.15

0.10.08

0.06

0.04

0.02

0.010.0080.006

0.004

0.0020.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

(a)

(b)

Figure 7-22. Bending moment in a cylinder due to radial load on an external attachment. (Reprinted by permission fromthe Welding Research Council.)


0.05 0.150.1 0.2 0.25 0.3 0.35 0.4 0.50.45

108

6

4

3

2

10.8

0.6

0.4

0.3

0.2

0.15

0.1

0.080.06

0.150.2

0.3

0.4

0.50.7

1.0

1.5

2

3

456789

10

15

20

30

40

γ = 5

γ = 300

γ = 100

γ = 50

γ = 15

γ = 5

γ = 15

γ = 50

γ = 100

γ = 300

β

0.05 0.150.1 0.2 0.25 0.3 0.35 0.4 0.50.45β

ΝφR

m2β

ML

ΝxR

m2β

ML

(a)

(b)

Figure 7-23. Membrane force in a cylinder due to longitudinal moment on an external attachment. (Reprinted bypermission from the Welding Research Council.)

Local Loads 459

0.05 0.1 0.15 0.3 0.35 0.4 0.50.2 0.25

0.10.08

0.06

0.04

0.02

0.015

0.010.0080.006

0.004

0.002

0.0010.0008

γ = 50

γ = 100

γ = 300

γ = 15

γ = 5

β

0.05 0.1 0.15 0.3 0.35 0.4 0.50.2 0.25β

MφR

mβ

ML

MxR

mβ

ML

0.45

0.45

0.1

0.08

0.06

0.04

0.02

0.010.008

0.006

0.004

0.002

0.001

γ = 5

γ = 15

γ = 50

γ = 300

γ = 100

(a)

(b)

Figure 7-24. Bending moment in a cylinder due to longitudinal moment on an external attachment. (Reprinted bypermission from the Welding Research Council.)


0.05 0.1 0.15 0.2 0.25 0.3 0.4 0.45 0.5

141210

8

6

4

2

1.5

10.80.60.4

0.2

0.15

0.10.080.060.05

252015

1086

4

2

10.80.6

0.4

0.2

0.15

0.1

γ = 100

γ = 50

γ = 15

γ = 5

γ = 300

γ = 5

γ = 15

γ = 50

γ = 300

β

β0.05 0.1 0.15 0.2 0.25 0.3 0.4 0.45 0.5

0.35

0.35

NφR

mβ

2

Mc

NxR

mβ

2

Mc

(a)

(b)

Figure 7-25. Membrane force in a cylinder due to circumferential moment on an external attachment. (Reprinted bypermission from the Welding Research Council.)

Local Loads 461

Maximum Allowable Nozzle Loads

This procedure is an alternative work process fordeveloping and analyzing nozzle loads on pressurevessels. It establishes the minimum criteria for design byproviding the maximum allowable nozzle load by size andclass of flange rating. This is an alternative work processto determining each individual nozzle load after the pipingsystem is designed. This procedure eliminates much of thelate design changes that occur when late data on nozzleloads impact either the design of the piping system or thestresses in the vessel shell.

The allowable loads and moments listed in Table 7-14do not represent any actual loading or a real maximumallowable load or moment. Rather they are “arbitrary”maximum allowable nozzle loads. This procedure doesnot take into account any of the vessel parameters such asdiameter, thickness, material, temperature, allowablestress, internal pressure, etc. Since the basis of Table 7-14is the “flange rating”, the associated nozzle loads aregeneric only.

There are two reasons for the implementation of thisprocedure as follows;

1. To provide the vessel fabricator with nozzle loadswith which to design the vessel shell or head to whichthe nozzle is attached, prior to design of the pipingsystems.

2. To provide the piping designer with guidelines fordesign of piping that terminates at a vessel nozzle.Therefore, as long as the piping does not exceed theloads in Table 7-14, they automatically know thatthe vessel shell or head is not overstressed for thiscondition.

If the piping department cannot design the piping insuch a way as to not exceed the values in Table 7-14, thenthe work process must revert back to the original workprocess of analyzing each specific, individual nozzle forthe actual loads and resultant shell stresses. This wouldbecome an iterative work process between the vesseldesigner and the piping designer where actual loads willdictate the ultimate design.

In the event that the nozzle loads still result in excessiveshell stresses after the iterative work process is conducted,the loadings may be reduced by recalculating the pipingloads utilizing the “vessel spring rate”. Often times theloads determined using the vessel spring rate will be much

0.20.15

0.10.08

0.06

0.040.03

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.08

0.06

0.04

0.03

0.02

0.015

0.01

β

β

γ = 50

γ = 5γ = 15

γ = 300

γ = 5

γ = 15

γ = 300

γ = 50γ = 100

γ = 100MφR

mβ

Mc

MxR

mβ

Mc

(a)

(b)

Figure 7-26. Bending moment in a cylinder due to circumferential moment on an external attachment. (Reprinted bypermission from the Welding Research Council.)


less than that determined by the normal process ofconsidering the vessel as a “rigid anchor”.

In general the following notes apply to this procedure;

1. Each nozzle, including those designated as “spare”,but with the exception of manways and instrumentnozzles, shall be designed to withstand the forcesand moments specified in Table 7-14. The indicatedloads are to be considered to act at the shell/head tonozzle intersection and to be true normal andtangential to the shell at that point. The effect on theshell/head shall be analyzed per an acceptable localload procedure such as WRC # 107.

2. With regard to radial load (Pr), calculations shall bemade first with the force acting radially outwards in

conjunction with the internal pressure and then withthe force acting inwards. In the second instance, theinternal pressure shall not be used to oppose thecompressive stresses due to the force acting radiallyinwards; for this load condition a null pressurecondition is to be considered to exist.

3. Values in Table 7-14 were computed by the coef-ficients and equations given in Table 7-13. In Table7-13, the variables shown are “D”, the nominaldiameter in inches and “b”, the value listed againstthe nozzle flange rating. These variables and equa-tions were used in the development of Table 7-14.

4. Whenever shell or head stresses exceed the allow-able stress for local loadings, the vendor shall apply

Table 7-13Coefficients used for determination of maximum allowable nozzle loads

Flange Rating Class 150 300 600 900 1500 2500

b Value 0.6 0.7 0.8 0.9 1 1.1

Equations:

Longitudinal Bending Moment ( Ft- Lbs) ML ¼ b X 110 X D2

Circumferential Bending moment ( Ft-Lbs) M4 ¼ b X 85 X D2

Resultant Bending Moment, (Ft-Lbs) MR ¼ [ML2 þ M4

2] .5 ¼ b X 140 X D2

Radial Load (tension or compression) (Lbs) Pr ¼ b X 500 X D

Table 7-14Maximum allowable nozzle loads

NPS

(Inches)

FLANGE RATING

CLASS 150 FLANGES CLASS 300 FLANGES

Force, Lbs Bending Moment, Ft-Lbs Force, Lbs Bending Moment, Ft-Lbs

Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

2 600 264 204 336 700 308 238 392

3 900 594 459 756 1050 693 536 882

4 1200 1056 816 1344 1400 1232 952 1568

6 1800 2376 1836 3024 2100 2772 2142 3528

8 2400 4224 3264 5376 2800 4928 3808 6272

10 3000 6600 5100 8400 3500 7700 5950 9800

12 3600 9504 7344 12096 4200 11088 8568 14112

14 4200 12936 9996 16464 4900 15092 11662 19208

16 4800 16896 13056 21504 5600 19712 15232 25088

18 5400 21384 16524 27216 6300 24948 19278 31752

20 6000 26400 20400 33600 7000 30800 23800 39200

24 7200 38016 29376 48384 8400 44352 34272 56448

Local Loads 463

adequate reinforcement and or increase thickness ofshell and /or nozzle locally.

5. For nozzles on a formed head or sphere, the resultantbending moment is to be compared with MR.

6. Shear and torsion effects are omitted from consid-eration because they have negligible effects on finalstress resultants.

7. The loadings computed from these equations shallbe considered as caused by 67% thermal and 33%dead weight load.

8. Under vacuum conditions, the deflection. 2, adjacentto the nozzle should be limited to the following;

2 < .0025 R where R is the radius of the shell orhead.

NPS

(Inches)

FLANGE RATING



Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

2 800 352 272 448 900 396 306 504

3 1200 792 612 1008 1350 891 689 1134

4 1600 1408 1088 1792 1800 1584 1224 2016

6 2400 3168 2448 4032 2700 3564 2754 4536

8 3200 5632 4352 7168 3600 6336 4896 8064

10 4000 8800 6800 11200 4500 9900 7650 12600

12 4800 12672 9792 16128 5400 14256 11016 18144

14 5600 17248 13328 21952 6300 19404 14994 24696

16 6400 22528 17408 28672 7200 25344 19584 32256

18 7200 28512 22032 36288 8100 32076 24786 40824

20 8000 35200 27200 44800 9000 39600 30600 50400

24 9600 50688 39168 64512 10800 57024 44064 72576

NPS

(Inches)

FLANGE RATING



Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

Radial Load,

Pr

Longitudinal,

ML

Circumferential,

M4

Resultant,

MR

2 1000 440 340 560 1100 484 374 616

3 15000 990 765 1260 1650 1089 842 1386

4 2000 1760 1360 2240 2200 1936 1496 2464

6 3000 3960 3060 5040 3300 4356 3366 5544

8 4000 7040 5440 8960 4400 7744 5984 9856

10 5000 11000 8500 14000 5500 12100 9350 15400

12 6000 15840 12240 20160 6600 17424 13464 22176

14 7000 21560 16660 27440 7700 23716 18326 30184

16 8000 28160 21760 35840 8800 30976 23936 39424

18 9000 35640 27540 45360 9900 39204 30294 49896

20 10000 44000 34000 56000 11000 48400 37400 61600

24 12000 63360 48960 80640 13200 69696 53856 88704


Procedure 7-5: Stresses in Spherical Shells from External Local Loads [11–13]

Notation

Pr ¼ external radial load, lbM ¼ external moment, in.-lbRm ¼ mean radius of sphere, crown radius of F & D,

dished or ellipsoidal head, in.ro ¼ outside radius of cylindrical attachment, in.C ¼ half side of square attachment, in.Nx ¼ membrane force in shell, meridional, lb/in.Nf ¼ membrane force in shell, latitudinal, lb/in.Mx ¼ internal bending moment, meridional, in.-lb/in.Mf ¼ internal bending moment, latitudinal, in.-lb/in.

Kn,Kb ¼ stress concentration factors (See Note 3)U,S ¼ coefficientssx ¼ meridional stress, psisf ¼ latitudinal stress, psiTe ¼ thickness of reinforcing pad, in.s ¼ shear stress, psi

MT ¼ torsional moment, in.-lbV ¼ shear load, lb

Procedure

To calculate stress due to radial load (Pr), and/ormoment (M), on a spherical shell or head:

1. Calculate value “S” to find stresses at distance xfrom centerline or value “U” at edge of attachment.Note:At edge of attachment, S¼U. Normally stressthere will govern.

2. From Figures 7-29 to 7-32 determine coefficients formembrane and bending forces and enter values inTable 7-15.

3. Compute stresses in Table 7-15. These stressesare entered into Table 7-16 based on the type ofstress (membrane or bending) and the typeof load that produced that stress (radial load ormoment).

4. Stresses in Table 7-16 are added vertically to total atbottom.

ro

Nx

Nx

x

Nx

Mx

M

Latitudinal

Mer

idio

nal

Pr

Mx

Nφ

Nφ

φσ

σx

Figure 7-27. Loadings and forces at local attachments inspherical shells.

Stress indices180°

90°270°

0°

Mφσφσ

φσ φσ

σx

σx

σx

σx

r o

ro

Rm

M

T

Pr

x

Figure 7-28. Dimensions and stress indices of localattachments.

Local Loads 465

Table 7-15Computing stresses

Figure Value from Figure Stresses

Radial Load

Membrane 7-29A NxT

Pr¼ ð Þ sx ¼ ð ÞKnPr

T2

7-29B NfT

Pr¼ ð Þ sf ¼ ð ÞKnPr

T2

Bending 7-30A Mx

Pr¼ ð Þ sx ¼ ð Þ6KbPr

T2

7-30B Mf

Pr¼ ð Þ sf ¼ ð Þ6KbPr

T2

Moment

Membrane 7-31A NxTffiffiffiffiffiffiffiffiffiffiRmT

pM

¼ ð Þ sx ¼ ð Þ KnM

T2 ffiffiffiffiffiffiffiffiffiffiRmT

p

7-31B NfTffiffiffiffiffiffiffiffiffiffiRmT

pM

¼ ð Þ sf ¼ ð Þ KnM


p

Bending 7-32A Mx

ffiffiffiffiffiffiffiffiffiffiRmT

pM

¼ ð Þ sx ¼ ð Þ 6KbM


p

7-32B Mf

ffiffiffiffiffiffiffiffiffiffiRmT

pM

¼ ð Þ sf ¼ ð Þ 6KbM


p


Figure 7-29. Membrane force due to Pr. (Extracts from BS 5500:1985 are reproduced by permission of British Stan-dards Institution, 2 Park Street, London, W1A 2BS, England. Complete copies can be obtained from national standardsbodies.)

Local Loads 467

Figure 7-30. Bending moment due to Pr. (Extracts from BS 5500:1985 are reproduced by permission of the BritishStandards Institution, 2 Park Street, London, W1A 2BS, England. Complete copies can be obtained from nationalstandards bodies.)


Figure 7-31. Membrane force due to M. (Extracts from BS 5500:1985 are reproduced by permission of the BritishStandards Institution, 2 Park Street, London, W1A 2BS, England. Complete copies can be obtained from nationalstandards bodies.)

Local Loads 469

Figure 7-32. Bending moment due to M. (Extracts from BS 5500:1985 are reproduced by permission of the BritishStandards Institution, 2 Park Street, London, W1A 2BS, England. Complete copies can be obtained from nationalstandards bodies.)


Formulas

• For square attachments.

ro ¼ C

• For rectangular attachments.

ro ¼ ffiffiffiffiffiffiffiffiffiffiffiCxCf

p• For multiple moments.

M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

1 þM22

q

• For multiple shear forces.

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV21 þ V2

2

q

• General stress equation.

s ¼ Ni

T� 6Mi

T2

• For attachments with reinforcing pads.

T at edge of attachment ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 þ T2

e

qT at edge of pad ¼ T

• Shear stresses.Due to shear load

s ¼ VproT

Due to torsional moment, MT

s ¼ MT

2pr2oT

Stress Indices, Loads, and Geometric Parameters

ro ¼Rm ¼T ¼

Kn ¼Kb ¼Pr ¼M ¼

S ¼ 1:82xffiffiffiffiffiffiffiffiffiffiRmT

p

U ¼ 1:82roffiffiffiffiffiffiffiffiffiffiRmT

p

Notes

1. This procedure is based on the “Theory of ShallowSpherical Shells”.

2. Because stresses are local and die out rapidly withincreasing distance from point of application,this procedure can be applied to the sphericalportion of the vessel heads as well as to completespheres.

3. For “Stress Concentration Factors” see “Stresses inCylindrical Shells from External Local Loads”,Procedure 7-4.

4. For convenience, the loads are considered asacting on a rigid cylindrical attachment of radiusro. This will yield approximate results for hollowattachments. For more accurate results for hollowattachments, consult WRC Bulletin 107 [11].

5. The stresses found from these charts will be reducedby the effect of internal pressure, but this reductionis small and can usually be neglected in practice.Bijlaard found that for a spherical shell with Rm/T¼100, and internal pressure causing membrane stressof 13,000 psi, the maximum deflection wasdecreased by only 4%–5% and bending moment by2%. In a cylinder with the same Rm/T ratio, thesereductions were about 10 times greater. This smallreduction for spherical shells is caused by thesmaller and more localized curvatures caused bylocal loading of spherical shells.

Local Loads 471

References

[1] Roark RJ. In: Formulas for Stress and Strain. 5thEdition. New York: McGraw-Hill Book Co; 1975.

[2] Isakower RI. Ring Redundants. Machine DesignMarch 1965.

[3] Blake A. Stresses in Flanges and Support Rings.Machine Design September 1974.

[4] Samoiloff A. Investigation of Stress in CircularRings. Petroleum Refiner July 1947.

[5] Blake A. Rings and Arcuate Beams. Product Engi-neering January 1963.

[6] Blodgett OW. Design of Welded Structures. TheJames F. Lincoln Arc Welding Foundation 1966.Section 66–4.

[7] Harvey JF. Theory and Design of Modern PressureVessels. 2nd Edition. Van Nostrand Reinhold Co;1974.

[8] Roark RJ. Stresses and Deflections in Thin Shellsand Curved Plates Due to Concentrated and Vari-ously Distributed Loading. Technical Note 806,National Advisory Committee on Aeronautics; 1941.

[9] Tentative Structural Design Basis for Reactor Pres-sure Vessels and Directly Associated Components.PB 151987, United States Dept. of Commerce;December 1958. pp. 62–81.

[10] Dodge WG. Secondary Stress Indices for IntegralStructural Attachments to Straight Pipe. WeldingResearch Council Bulletin No. 198; September1974.

[11] Wichman KR, Hopper AG, Mershon JL. LocalStresses in Spherical and Cylindrical Shells Due toExternal Loadings. Welding Research CouncilBulletin No. 107; April 1972.

[12] Bijlaard PP. Computation of the Stresses from LocalLoads in Spherical Pressure Vessels or PressureVessel Heads. Welding Research Council BulletinNo. 34; 1957.

[13] BS 5500: Specification for Unfired Fusion WeldedPressure Vessels. London: British Standards Insti-tute; 1985.


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