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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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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.)
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
"� 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.
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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.
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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.
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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.
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
Figure 6.5 Cover plate designs. (Modified from ASME Boiler and Pressure Vessel Code, ASME, New York.)
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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
Copyright 2005 by CRC Press, Inc. All Rights Reserved.
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.
Copyright 2005 by CRC Press, Inc. All Rights Reserved.