1369_ch06 asme

chapter six

Design of heads and covers

Contents

6.1 Introduction ...............................................................................................6.2 Hemispherical heads under internal pressure .....................................6.3 ASME equation for hemispherical heads ..............................................6.4 Example problem 1....................................................................................

6.4.1 Thin-shell theory ............................................................................6.4.2 ‘‘Exact’’ theory ................................................................................6.4.3 ASME equation (assuming E ¼ 1) ..............................................

6.5 ASME design equation for ellipsoidal heads .......................................6.6 ASME equation for torispherical heads.................................................6.7 Example problem 2....................................................................................

6.7.1 Solution for ASME head using Eq. (6.15)..................................6.8 ASME design equations for conical heads............................................6.9 ASME design equations for toriconical heads......................................6.10 Flat heads and covers................................................................................

6.10.1 Case 1 ...............................................................................................6.10.2 Case 2 ...............................................................................................

6.11 ASME equation for unstayed flat heads and covers...........................6.12 Example problem 3....................................................................................

6.12.1 Considering simply supported edges ........................................6.12.2 Considering clamped edges.........................................................6.12.3 Considering unstayed plates and covers...................................

References .............................................................................................................

6.1 Introduction

Heads are one of the important parts in pressure vessels and refer to theparts of the vessel that confine the shell from below, above, and the sides.The ends of the vessels are closed by means of heads before putting theminto operation.

Copyright 2005 by CRC Press, Inc. All Rights Reserved.

The heads are normally made from the same material as the shell andmay be welded to the shell itself. They also may be integral with the shell inforged or cast construction. The head geometrical design is dependent onthe geometry of the shell as well as other design parameters such asoperating temperature and pressure.

The heads may be of various types such as:

� Flanged� Ellipsoidal� Torispherical� Hemispherical� Conical� Toriconical

The different types of heads are shown in Figure 6.1.The geometry of the head is selected based on the function as well as on

economic considerations, and methods of forming and space requirements.The elliptical and torispherical heads are most commonly used. The carbonsteel hemispherical heads are not so economical because of the highmanufacturing costs associated with them. They are thinner than thecylindrical shell to which they are attached, and require a smooth transitionbetween the two to avoid stress concentration effects.

The thickness values of the elliptical and torispherical heads aretypically the same as the cylindrical shell sections to which they areattached. Conical and toriconical heads are used in hoppers and towers.

6.2 Hemispherical heads under internal pressure

The force due to internal pressure is resisted by the membrane stress in theshell (see Figure 6.2). Because of the geometrical symmetry, the membranestresses in the circumferential and the meridional directions are the same,and are denoted by S. We have

P�R2¼ 2�RSt

and

S ¼PR

2tð6:1Þ

where S is the membrane stress, ðS� ¼ S�), from symmetry.The hoop and meridional strains are indicated by "� and "�

"� ¼w

R¼ "� ð6:2Þ


where w is the radial displacement, and

"R ¼dw

dRð6:3Þ

The stress–strain relationship is given by

"R ¼1

ESR � �ðS� þ S�Þ� �

¼1

ESR � 2�S�½ � ð6:4Þ

Figure 6.1 Different types of heads.(Modified from ASME Boiler and Pressure Vessel Code, ASME, New York.)


"� ¼1

ES� � �ðSR þ S�

� �¼

1

Eð1� �ÞS� � �SR½ � ð6:5Þ

"R ¼dw

dR; "� ¼

w

R

w ¼ R"�

"R ¼dw

dR¼

d

drR"�ð Þ

1

"SR � 2�S�½ � ¼

ð1� �Þ

E

d

dRRS�ð Þ �

�

E

d

dRRSRð Þ

or

ð1� �Þd

dRðRS�Þ � �

d

dRðRSRÞ � SR þ 2�S� ¼ 0

From Figure 6.3 the force equilibrium gives

2S�Rd�

2dR þ 2S�R

d�

2dR þ SR þ

dSR

dRdR

� �ðR þ dRð Þd�ðR þ dRÞd�

� SRdRðRd�ÞðRd�Þ ¼ 0

With S� ¼ S�; d� ¼ d�

Figure 6.2 Hemispherical head.


2RS�dRd�2 þ SR þdSR

dRdR

� �R2

þ 2RDRÞd�2� �

� SRR2d�2 ¼ 0

2RS�dRd�2 þ SRR2d�2 þ 2RSRdRd�2 þ R2 dSR

dRdRd�2 þ 2R

dSR

dR

ðdRÞ2

d�2

� SRR2d�2 ¼ 0

2S� ¼ �2RSR � RdSR

dR

¼�1

R

d

dRðR2SRÞ

S� ¼ �1

2R

d

dRR2SR

� �

Substituting, we have

�ð1� �Þ

2

d

dR

1

R2

d

dRR2SRð Þ

� �

d

dRðRSRÞ � SR �

�

R

d

dRR2SR

� �¼ 0

Figure 6.3 Equilibrium of a hemispherical element.


which gives

Rd

dR

1

R2

d

dRR3SR

� � ¼ 0

SR ¼A

3þ

B

R3ð6:6Þ

With the boundary conditions specified as

SR ¼ �P at R ¼ Ri

SR ¼ 0 at R ¼ Ro ð6:7Þ

0 ¼A

3þ

B

R3o

or B ¼�AR3

o

3

Therefore

SR ¼A

31�

R3o

R3i

!

and

�P ¼A

31�

R3o

R3i

!

This gives

A ¼3R3

i P

R3o � R3

i

; B ¼�AR3

o

3

SR ¼PR3

i

R3o � R3

i

1�R3o

R3

!ð6:8Þ

and

S� ¼ S� ¼PRi

R3o � R3

i

1þR3o

2R3

!ð6:9Þ


6.3 ASME equation for hemispherical headsASME Section VIII Division 1 provides the following equation for internalpressure.1 This is a compromise between a thin-shell equation and ‘‘exact’’equation.

The design thickness of a hemispherical head is given by

t ¼PR

25E � 0:2Pð6:10Þ

where R is the inside radius, S is the allowable shear, and E ¼ is the jointefficiency.

6.4 Example problem 1A hemispherical head having an inside radius of 380 mm is subjected to aninternal pressure of 28 Megapascals (MPa). This allowable stress is160 MPa. What is the required thickness using the shell theory and‘‘exact’’ theory, and the ASME equation (assume joint efficiency, E ¼ 1)?

6.4.1 Thin-shell theory

From Eq. (6.1) membrane stress

S ¼PR

2t

or

t ¼PR

2s¼

28 � 380

320¼ 33:25 mm

taking the radius as the inside radius.

6.4.2 ‘‘Exact’’ theory

Using Eq. (6.9)

S ¼ ðS� ¼ S�Þ ¼PR3

i

R3o � R3

i

1þR3o

2R3i

!

which simplifies to

Ro ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðS þ PÞR3

i

2S � P

s


Thus

Ro ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð160þ 28Þ � 3803

2ð160Þ � 28

s¼ 413:4 mm

Therefore

t ¼ Ro � Ri ¼ 413:4� 380 ¼ 33:4 mm

Therefore the assumption of this shell theory is valid here.

6.4.3 ASME equation (assuming E ¼ 1)

Using Eq. (6.10)

t ¼28� 380

2ð160Þð1Þ � 0:2ð28Þ¼ 33:8 mm

The ASME estimate is conservative in this case.

6.5 ASME design equation for ellipsoidal headsFor an internal pressure P, the thickness t of the ellipsoidal head is given by

t ¼PDK

2SE � 0:2Pð6:11Þ1

where D ¼ diameter of the shell to which the head is attached, E ¼ jointefficiency, S ¼ allowable stress, and K ¼ stress intensity factor.

K is given by the following expression:

K ¼1

62þ

a

b

� �2 ð6:12Þ

where a and b are the semi-major and semi-minor axes of the ellipse.

6.6 ASME equation for torispherical headsFor an internal pressure P, the thickness of the torispherical head is given by

t ¼PLM

2SE � 0:2Pð6:13Þ1


where L ¼ spherical cross radius, S ¼ allowable stress, E ¼ joint efficiency,and M ¼ shear intensity factor. M, the stress intensity factor

M ¼1

43þ

ffiffiffiL

r

r !ð6:14Þ

where r is the knuckle radius. The special case when the knuckle radius is 6percent of the spherical crown radius, or r ¼ 0.06L is known as ASME head.

For the ASME head, M ¼ 1.77 (from Eq. (6.14)) and the thickness t is thengiven by

t ¼0:885PL

SE � 0:1Pð6:15Þ

It turns out that for large ratios of R/t, the knuckle region of the head isprone to buckling under internal pressure. Based on plastic analysis,1 thefollowing expression is used for t:

lnt

l¼ �1:26177� 4:55246

r

D

� �þ 28:9133

r

D

� �2þ 0:66299� 2:24709

r

D

� �2 þ

0:66299� 2:24709r

D

� �þ 15:62899

r

D

� �2 ln

P

Sþ :

0:26879� 10�4� 0:44262

r

D

� �þ 1:88783

r

D

� �2 ln

P

S

� �2

where L ¼ crown radius, r ¼ knuckle radius, D ¼ diameter of the shell towhich the head is attached, and S ¼ allowable stress.

6.7 Example problem 2

What is the required thickness of a torispherical head attached to a shell ofdiameter 6 mm, to have a crown radius of 6 mm and a knuckle radius of360 mm? (ASME head r/L ¼ 0.06). The allowable stress is 120 MPa and theinternal pressure is 345 KPa.

6.7.1 Solution for ASME head using Eq. (6.15)

t ¼0:885PL

SE � 0:1P


assuming E ¼ 1. With S ¼ 120 MPa, we have

t ¼0:885ð0:345Þ6

ð120Þð1Þ � ð0:1Þð0:345Þ¼ 0:0153 m ¼ 15:3mm

The thickness is small compared to the diameter of the head and shouldbe checked for buckling at the knuckle region of the head.

We also have

r

D¼

0:36

6¼ 0:06

P

S¼

0:345

120¼ 0:002875

lnP

S¼ �5:8517

lnP

S

� �2

¼ 34:2424 ð6:16Þ

We have using Eq. (6.16)

lnt

l

� �¼ �1:26177 � 4:55246ð0:06Þ þ 28:93316ð0:06Þ2þ

0:66299 � 2:4709ð0:06Þ þ 15:68299ð0:06Þ2� �

ð� 5:8517Þ

þ 0:26879 � 10�4� 0:44262ð0:06Þ þ 1:88783ð0:06Þ2

� �ð34:2424Þ

¼ �1:26177� 0:27315þ 0:10416 � 5:8517 0:66299 � 0:13483 þ 0:05646½ �

þ 34:824½0:26879 � 10�4� 0:02656 þ 0:0680� ¼ � 5:53897

This gives t/L ¼ 0.00393, or t ¼ (60000)(0.00393) ¼ 23.6 mm. Hence aminimum thickness 23.6 mm is required. The design is therefore dictatedby stability of the knuckle region of the head.

6.8 ASME design equations for conical headsASME Code Section VIII Division I provides the following equation forthickness t of conical heads subjected to an internal pressure P.1 With � asthe semi-apex angle of the cone

t ¼PD

2 cos�ðSE � 0:6PÞð6:17Þ


where D is the inside diameter of cone measured perpendicular tolongitudinal axis, S is the allowable stress, and E is the joint efficiency

6.9 ASME design equations for toriconical headsA toriconical head is a blend of conical and torispherical heads.Accordingly, the thickness, tc in the cone region is calculated using conicalhead equations and that in the head transition section is calculated usingtorispherical head equations.

Referring to Figure 6.4 for the conical region we have, using Eq. (6.17),

tc ¼PD1

2 cos�ðSE � 0:6PÞð6:18Þ

and for the torispherical region using Eq. (6.13)

tk ¼PLM

2SE � 0:2Pð6:19Þ

where

L ¼D1

2 cos�

and

M ¼1

43þ

ffiffiffiL

r

r !

from Eq. (6.14)A pressure vessel designer generally has flexibility in selecting head

geometry. Most common is of course the torispherical head, which ischaracterized by inside diameter, crown radius, and knuckle radius. Thedesigner selects a head configuration that minimizes the total cost of theplate material and its formation.

Figure 6.4 Toriconical head.


6.10 Flat heads and coversFlat heads or covers are used widely as closures to pressure vessels. Theyare either integrally formed with the shell, or may be attached by bolts.Figure 6.5 shows some typical designs of covers.

6.10.1 Case 1

A simply supported circular plate of radius R and thickness t subjected touniform pressure P. The deflection at the center of this plate is a maximumand this value is given by2,3

�max ¼5þ �

1þ ��

PR4

64Dð6:20Þ

where

D ¼Et3

12ð1� �2Þð6:21Þ

where t ¼ plate thickness.The stress is a maximum at the bottom surface2,3

ðSrÞmax ¼ ðS�Þmax ¼3ð3þ �Þ

8

PR2

t2ð6:22Þ

6.10.2 Case 2

A circular plate is clamped around outer periphery and subjected touniform pressure P. The maximum deflection occurs at the center of theplate where the value is2,3

�max ¼PR4

64Dð6:23Þ

The maximum radial and tangential stresses are given by2,3

ðSrÞmax ¼3PR2

4t2ð6:24Þ

occurring at the edge and at the top surface, and

ðS�Þmax ¼3ð1þ �Þ

8

PR2

t2ð6:25Þ

occurring at the center and the top surface of the plate.


Figure 6.5 Cover plate designs. (Modified from ASME Boiler and Pressure Vessel Code, ASME, New York.)


6.11 ASME equation for unstayed flat heads andcovers

The thickness of unstayed flat heads and covers subjected to a pressure P,and an allowable stress S with a joint efficiency E, for a variety of casescharacterized by the constant C, is given by

t ¼ d

ffiffiffiffiffiffiffiCP

SE

rð6:26Þ1

The cases are shown in Figure 6.5 each with a typical value of C. Thevalue of C could range anywhere from 0.10 to 0.33.

6.12 Example problem 3

A circular plate of diameter 1 m, forms the cover for a cylindrical pressurevessel subjected to a pressure of 0.04 MPa. We wish to determine thethickness of the head if the allowable stress in the material is limited to120 MPa.

6.12.1 Considering simply supported edges

Using Eq. (6.22) we have

Smax ¼3ð3þ �Þ

8

PR2

t2

or

120 ¼9:9

8ð0:04Þ

500

t

� �2

500

t¼

120ð8Þ

ð9:9Þð0:04

:ð1=2Þ¼ 49:24

t ¼ 10:16 mm


6.12.2 Considering clamped edges

Srjmax > S�jmax

500

t¼

120ð4Þ

0:12

ð1=2Þ¼ 63:25

t ¼ 7:91 mm

6.12.3 Considering unstayed plates and covers

See Figure 6.5. We have from Eq. (6.26)

t ¼ d

ffiffiffiffiffiffiffiCP

SE

r

where C ¼ 0.10–0.33 depending on construction, d ¼ diameter of the head, P¼ design pressure, S ¼ allowable tensile stress, and E ¼ butt weld jointefficiency. Assuming E ¼ 1, then S ¼ 120 MPa.

For C ¼ 0.10

t ¼ 1000

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:10ð0:04Þ

ð1Þð120Þ

s¼ 5:77 mm

For C ¼ 0.33

t ¼ 1000

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:33ð0:04Þ

ð1Þð120Þ

s¼ 10:49 mm

References1. American Society of Mechanical Engineers, Boiler and Pressure Vessel Code,

ASME, New York.2. Timoshenko, S.P., and Woinowsky-Kreiger, S., Theory of Plates and Shells, 2nd

ed., McGraw-Hill, 1959.3. Roark, R.J. and Young, W.C., Formulas for Stress and Strain, 5th ed., McGraw-

Hill, New York, 1975.


1369_ch06 asme

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