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Mechanical Design 2

School of Aerospace, Mechanical and Manufacturing Engineering 1MIET2072 2013 Ver.1

MIET2072 - Topic 7College of Science, Engineering and Health Learning Package

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Topic 7: DETERMINATION OF MINIMUM

THICKNESS FOR CIRCULAR

UNSTAYED FLAT ENDS AND COVERS

Learning Outcomes

Upon successful completion of this topic you will be able to:

• determine the thickness of circular flat cover plates on pressure vessels.

Introduction to the Topic

Circular flat plates provide a convenient means of sealing a flanged opening. Whenexposed to pressure, flat surfaces experience bending stresses similar to those inbeams you have previously studied.

In this topic you will learn how the equations in the standard for determining thethickness of such plates have been developed and applied.

Background Skills and Knowledge

Students will require the following:

• Familiarity with force and moment equilibrium.

• Familiarity with bending stresses

• Familiarity with Hooke’s law

Acti vi ty 7A - Reading

Read all of Chapter 7 below

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Chapter 7: Determination of Minimum Thickness for

Circular Unstayed Flat Ends and Cover

Section 7.1 Introduction

A common method of covering the inspection opening in pressure vessels is tobolt a thick flat circular cover plate to a flanged joint such as is illustrated inFigures 7.1a and b.

Fig. 7.1a Inspection opening with flat cover plate bolted to flange © RMIT University, 2013, (Dixon C.)

Fig. 7.1b U.S. Department of Agriculture, Arsenic removal absorber vessels, at the Freer WaterControl and Improvement District (FWCID) Arsenic Removal System Site, on Tuesday, June 18, 2013, in Freer,

Texas. http://www.flickr.com/photos/usdagov/9404352421/

http://www.flickr.com/photos/usdagov/9404352421


http://www.flickr.com/photos/usdagov/9404352421/





http://creativecommons.org/licenses/by/2.0/deed.en

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Section 7.2 General Comments on the thickness determination

The thickness of metal used in flat covers must, as with all parts of the pressurevessel, be sufficient to safely accommodate:

• the expected corrosion over the anticipated life

• the stresses caused by the fluid pressure (or vacuum if applicable) andapplied loads, this thickness is called the minimum calculated thickness

• handling and transport loads.

Allowance must also be made when ordering the metal for possible mill undertolerance, and reduction of thickness during fabrication such as facing off in alathe.

“AS1210-2010 Pressure Vessels”, published by Standards Australia isrelevant to these matters and, as previously mentioned, you should have thatto hand while reading these notes.

The Australian Standards can be accessed via the RMIT Library - SAI Globallink to the Australian Standards.1

NOTE: You will need to login with your student login and password.

Reference 1: Section 3.4 of AS1210

Refer to Section 3.4 of AS1210 (“Australian Standard AS1210 Pressure Vessels”),published by Standards Australia which gives some further generalinformation on the issue of required thickness in pressure vessels.

Reference 2: Section 3.15 of AS1210

Refer to Section 3.15 of AS1210 which deals with unstayed flat ends and coversin particular.

Reference 3: Figure 3.15.1 of AS1210Refer to Figure 3.15.1 of AS1210, which illustrates a variety of acceptable typesof flat ends and covers. Please look at Fig 3.15.1 rows (k), (l) and (p) to see someflat plate covers used with various types of flanged joints such as were studiedin the previous chapter.

1 https://login.ezproxy.lib.rmit.edu.au/login?url=http://www.saiglobal.com/online/autologin.asp

https://login.ezproxy.lib.rmit.edu.au/login?url=http://www.saiglobal.com/online/autologin.asp






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“AS2129-2000 Flanges for pipes, valves and fittings”, published by StandardsAustralia is also relevant to these matters as it has the dimensions of “blank”

flanges which are used as cover plates The Australian Standards can be accessed via the RMIT Library - SAI Globallink to the Australian Standards.2


Reference: TABLE E Fig (a) (i)

Refer to TABLE E Fig (a) (i) on page 26 of AS2129 (2000).

Section 7.3 Minimum Calculated thickness for Circular FlatUnstayed Ends

From studies of uniformly distributed loads of “q”(such as pressure) on rectangularbeams (Appendix 7A) or circular flat plates (Appendix 7B) it has been shown thatthe equation for maximum bending stress is of the following form:

2

2

hqak mzx =σ where: “a” is half the span, “ h” is the thickness, and “k” is a co-

efficient whose value depends on the edge support.








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Fig. 7.3b Rectangular beam with various edge supports exposed to a uniformly distributed load q

© RMIT University, 2013, (Marchiori G., Dixon C.)

For the case of a beam simply supported at its end k = 12/4

For the case of a beam with built in ends k = 8/4

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Fig. 7.3b Circular flat plate with various edge supports exposed to a uniformly distributed load q


For the case of a circular plate with a simply supported edge : k = 4.95/4(for metal with Poissons ratio of 0.3 )

For the case of a circular plate with a built in edge : k = 3/4

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Changing now the equation2

2

h

qak mzx =σ to the symbols used in clause 3.15 of

AS1210 : q becomes P; h becomes t; a becomes (D/2); and the co-efficient is put in

the denominator and given the symbol K . Retaining the symbol σmax gives:

maxσ =

2

21

t

PD

K

The magnitude of the co-efficient K of course depends on the type of edge supportand needs to be recalculated to allow for the effects of (i) D2 being used instead ofa2 and (ii) the co-efficient being put in the denominator. Recalculating gives:

for the case of a circular plate with a simply supported edge :K = 4 x 4/4.95 = 3.23(for metal with Poissons ratio of 0.3 );

for the case of a circular plate with a built in edge : K = 4 x 4/3 = 5.3

For safe operation the maximum stress calculated from:

maxσ =

2

21

t

PD

K

must be kept less than or equal to the allowed design tensile strength f of the platematerial if there are no welds in it, and less than or equal to f η if there are welds

(η being the weld efficiency).

i.e.max

σ ≤ f η

∴

1

2

2

t

PD

K ≤ f η

∴ the minimum calculated t = f K

P D

η ….. eqn. 3.15.3(1) from AS1210

If corrosion is present a corrosion allowance must be added to the valuedetermined from this equation.

Values of K for various types of flat ends and covers are given in the 2nd column ofof Figure 3.15.1 in AS1210. The diagrams in Figure 3.15.1 also indicate where tomeasure D. For example for a circular flat cover joined to a full face flange, K is 4.0,which can be seen to lie between the two pure theoretical cases above; and D istaken as the diameter of the bolt pitch circle, which lies between the inner and outerportions of the gasket.

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Some covers are of the type that have an edge moment that exacerbates further thebending stress, such as for a blind flange using a narrow face gasket system asshown in Figure 3.15.1 row(k) in AS1210 and in Figure 7.3c below. Here the edgemoment is W x hG generated by the bolt force W. Note that in the case of the fullface flange joint this edge moment is not present because the inner and outer gasketmoments are assumed to be equal as discussed in Topic 6. i.e. '

G G G Gh h′=H H

Return now to the case of the cover on a narrow face flanged joint as shownschematically in Figure 7.3(c).

Fig. 7.3c Forces acting on circular flat cover joined to a narrow face flange.

© RMIT University, 2013, (Kissane M., Dixon C.)

It can be shown ( page 99 “Strength of Materials, Parts II”, by Timoshenko, S., 3rd edition, 1956, published by Van Nostrand Reinhold) that for a simply supportedcircular plate of diameter D the maximum bending moment, which is at the centre,

is:

M 1 = M 2 =2

216

3

+ DP

µ

Superimposed on this is the edge bending moment per unit edge circumference

D

WhG

Π= if the disc is of outside diameter D [in fact for the situation shown above it is

larger] (Jacobs, W.S., “Fundamentals of Pressure Vessel Design”, in “PressureVessels – A Workbook for Engineers”, published by ASME, 1981.)

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∴ M max at middle = D

Wh DP G

Π+

+2

216

3 µ

&max

σ =

6

2

max

t

M

∴ max

σ =( )

22

2

62

8

33

Dt

Wh

t

DP

G

Π+

+ µ

and if µ = 0.3

maxσ =

22

291.1

23.3

1

Dt

Wh

t

PD G+

and ifmax

σ is to be f η ≤

⇒ t fD

Wh

f

DP G

91.1

23.3

2

η η +≥

Taking D outside the

⇒ t ≥ 3

91.1

23.3 D

Wh

f

P D G+

η

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The equation in AS1210 for the minimum calculated thickness for covers with suchan edge moment is:

t =

+

3

78.1

D f

Wh

Kf

P D G

η η eqn. 3.15.3.2 in AS1210

with K =3.3 as shown Figure 3.15.1 row(k) in AS1210 and a 1.78 term instead of 1.91(the author of these notes presumes this is because the actual plate diameter isgreater than D).

Since the bolt force required for seating the gasket during assembly is in some casesmore than the bolt force required during operation, it is necessary to use the aboveequation twice:

a) For the seating condition, using the appropriate bolts' force (see Clause 3.21and the chapter in these notes on Flanges), P=Zero and the value of "f " foratmospheric temperature.

b)

For the operating condition, using the appropriate bolts' force, the designpressure P, and the value of "f " for the operating temperature.

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Appendix 7A –

Bending stress in a beam with uniformly distributed load

Consider a simply supported beam with a uniformly distributed pressure load“q” on its upper surface, such as is shown in the middle row of figure 7A.a

Fig. 7.A.a Rectangular beam with various edge supports exposed to a uniformly distributed load q


From symmetry R1 = R2

Vertical force equilibrium gives 2 R1 = pressure x surface area = q2abTherefore R1 = q2ab/2 = qab

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Taking an imaginary cut in the middle of the span as shown in Fig 7.A.b, andrepresenting the moment effect of the distributed load on the left half of the beamas a concentrated load of magnitude qab acting at a distance a/2 from the end it canbe seen that the bending moment at the middle of the beam has a value:

M = (R1 x a) – (qab x a/2 )∴ M = (qab x a)- (qab x a/2 )

∴ M = qa2b/2

Fig. 7.A.b Freebody diagram of left half of the beam ©RMIT University, 2013, (Marchiori G., Dixon C.)

maxσ =

I

y M max

×

Where ymax is the distance from the neutral axis of the beam to its outer “fibre”and I is the second moment of area of the transverse cross section of the beam.

Fig. 7.A.c Transverse cross section of the beam © RMIT University, 2013, (Marchiori G., Dixon C.)

∴ max

σ =

×

12

2

3bh

h M

=2

6

bh

M ×

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Substituting M = (qa2b/ 2)

givesmax

σ = (qa2b/ 2) x (6/ [bh2 ])

maxσ = 3 qa2 / h2

or alternatively maxσ =

2

2

412

hqa

Consider now a beam with a uniformly distributed pressure load “q” on its uppersurface and with built in ends, such as is shown in the bottom row of figure 7A.a .

It can be shown (pg188 “Strength of Materials, Parts I”, by Timoshenko, S., 3rd edition, 1956, published by Van Nostrand Reinhold) for this staticallyindeterminate problem, from considerations of downward angle of deflection at the

end caused by “q” being brought back to zero by upward angle of deflection causedby M 1, that the peak bending moment is M 1 and it has a magnitude qa2b/ 3 . Fromsymmetry it can be seen that M 1 = M 2.. Substituting the magnitude of M 1 into theequation for bending stress:

maxσ =

2

6

bh

M ×

givesmax

σ = (qa2b/ 3) x (6/ [bh2 ])

∴ max

σ = 2 qa2 / h2

or alternativelymax

σ =2

2

48

hqa

Summarizing the results of the formula development in this appendix 7A, it hasbeen shown that the equation for maximum bending stress in a beam of rectangularcross section subjected to a distributed pressure “q” is of the form:

2

2

h

qak mzx =σ where: “a” is half the span, “ h” is the thickness, and “k” is a co-

efficient whose value depends on the edge support.

For the case of a beam with simply supported ends : k = 12/4

For the case of a beam with a built in ends : k = 8/4

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Appendix 7B - Optional extra reading about bending of ci rcular plates

Additional reading: Bending of circular plates.

Refer to the detailed development of formulae to predict the bending stresswithin circular plates as given on pages 92-99 of Strength of Materials part II (PDF 316KB). 3

Summarizing the results of the formula development in this appendix, it has beenshown that the equation for maximum bending stress in circular plates subjected to

a distributed pressure “q” is of the form:2

2

h

qak mzx =σ where: “a” is half the span, “

h” is the thickness, and “k” is a co-efficient whose value depends on the edgesupport.

For the case of a circular plate with a simply supported edge: k = 4.95/4(for metal with Poissons ratio of 0.3 )

For the case of a circular plate with a built in edge: k = 3/4

Appendix 7C - Optional extra reading about bending of ci rcular plates

with various edge and central supports

Additional reading: Bending stress in circular plates with various edge andcentral support systems.

Refer to Pages 113 to 114 of Strength of Materials part II which gives summaryresults for bending stress in circular plates with various edge and centralsupport systems of interest to mechanical engineers. (PDF 105KB). 4

3 Timoshenko S 1956, Strength of materials. Part II 3rd ed., Van Nostrand Reinhold co., Huntington, New York,p. 92-99. viewed 27th August 2013. <https://equella.rmit.edu.au/rmit/file/08c7214f-86f9-4ea3-bdfc-27df0daa969e/1/130821_3_032.pdf>

4 Timoshenko S 1956, Strength of materials. Part II 3rd ed., Van Nostrand Reinhold co., Huntington, New York,p. 113-114. viewed 27th August 2013. < https://equella.rmit.edu.au/rmit/file/24276c12-0cbb-42df-a798-

b53e37b886a7/1/130821_3_033.pdf>

https://equella.rmit.edu.au/rmit/file/08c7214f-86f9-4ea3-bdfc-27df0daa969e/1/130821_3_032.pdf




https://equella.rmit.edu.au/rmit/file/24276c12-0cbb-42df-a798-b53e37b886a7/1/130821_3_033.pdf






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Acti vi ty 7B - Readings, Reflect ions and Calculations

Step 1: Refer to Project Part B in Assessment

Refer to Project Part B in the Assessment Section. Read Question 5 (h) and (i).

Step 2: Read relevant parts of AS1210

Refer to the clauses in AS1210 ‘Pressure Vessels’ listed in Question 5 (h)

The Australian Standards can be accessed via the RMIT Library - SAI Globallink to the Australian Standards.5


Step 3: Plan.

Discuss with your partner the strategy for tackling Question 5 (h) and (i).

Step 4: If you are the partner responsible for completing Question 5 (h) and (i),do so, consulting with your partner where appropriate.

Feedback:Feedback will be provided on your submitted project documentation by theengineering lecturer/tutor responsible for marking it.








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Summary and Outcome Checklis t

Circular flat plates provide a common convenient means of sealing flangedopenings such as inspection openings in pressure vessels. When exposed topressure, flat plates experience significant bending stresses. The pressure vesselstandard provides equations for determining the thickness of such plates that willkeep these stresses at a safe level.

Tick the box for this statement if you agree with it:

I can determine the thickness of circular flat cover plates on pressure vessels.

Assessment

This topic will be assessed as part of the Project Part B and the end of semesterexamination (see: Assessment section of the Course Introduction for more detail).

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