Design of Heads

4.3 Designing Heads subject

to Internal Pressure

Stresses in Heads

Heads are one of the important parts in pressure vessels

and refer to the parts 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 them into operation

The heads are normally made from the same material as

the shell and may be welded to the shell itself. They also

may be integral with the shell in forged or cast

construction.

The head geometrical design is dependent on the geometry

of the shell as well as other design parameters such as

operating temperature and pressure

Types of Head

•Flanged

•Ellipsoidal

•Tori - spherical

•Hemispherical

•Conical

•Tori -conical

Classification according to the shape: i. Convex heads

Semi-spherical head

Elliptical head

Dished head (spherical head with hem)

Spherical head without hem

ii. Conical heads

Conical head without hem

Conical head with hem

iii. Flat heads

1.Semi-spherical head

i. Molding of heads

Small diameter and thin wall

——

Integrally heat-pressing

molding

Large diameter

—— Spherical petal welding

球瓣式组焊

Di

ii. Calculating equation for thickness

][ 4 c

t

ic

p

DpS

][ 4

2Cp

DpS

c

t

icd

valueofround][4

S 12

CCp

Dp

c

t

icn

2.Thickness calculating equation

of elliptical head

b

a

S

pam

2max

hi (

b)

ho

Di S

Ri (a)

i. Calculating equation

for thickness:

For the elliptical heat

whose m = a / b ≤ 2

2. The maximum stress should be at the top point:

Putting m = a / b, a = D / 2 into the equation,

getting:

S

mpD

4max

t

S

mpD][

4max

t

mpDS

][ 4

Then:

Under the condition about strength:

(1)Replacing P with Pc

(2)Multiplying []t with welded joint efficiency

(3)Substituting D with Di, D = Di + S

(4) m = a / b = Di / 2 hi

Putting these conditions into the equation:

getting:

i

i

c

t

ic

ct

ic

h

D

p

Dpm

mp

DpS

45.0 ][2

2

2 ][2

m = a / b = Di / 2 hi

For the standard elliptical head whose m=2:

c

t

ic

p

DpS

5.0 ][2

For the elliptical head whose m>2:

at boundary » and m at the top point

Then introducing the stress strengthening

coefficient K to replace (Di / 4hi)

In this equation:

2

22

6

1

i

i

h

DK

c

t

ic

p

DpKS

5.0 ][ 2

For standard elliptical head: K=1

This is the common equation for calculating

the wall thickness of elliptical heads.

Beside these conditions:

for standard elliptical heads Se ≮ 0.15% Di

for common elliptical heads Se ≮ 0.30% Di

The straight side length of standard

elliptical heads should be determined

according to P103, Figure 4-11

iii. Working stress and the maximum

allowable working pressure

e

eict

S

SKDp

2

5.0

ei

e

t

SKD

Sp

5.0

][2][

3.Dished head

i. Structure

Containing three parts:

Sphere: Ri

Transition arc (hem): r

Straightedge: ho (height)

Di

r

ho

c

t

ic

p

RpMS

5.0 ][2

r

RM i3

4

1

M —— Shape factor of dished head

and


iii. Working stress and the maximum

allowable working pressure

e

eict

S

SMRp

2

5.0

ei

e

t

SMR

Sp

5.0

][2][

iv. Dished head

When Ri = 0.9 Di & r = 0.17 Di

the dished head is standard dished head

and M = 1.325

So the equation is:

c

t

ic

p

DpS

5.0 ][2

2.1

4.Conical head

i. Structure

*without hem (suitable for ≤ 30 o )

without local strength

with local strength

*with hem (suitable for > 30 o )

—— Adding a transition arc and a

straightedge between the joint

of head and cylinder


The maximum stress is in the main aspect

of conical head.

cos

1

2 maxmax

S

pD

m

t

maxmax ][ cos

1

2

S

pD

According to the strength condition:

Then

cos

1

][2

t

pDS

cos

1

][2

c

t

cc

p

DpS

Replacing P with Pc, considering , and

changing D into Dc ，D=Dc+S

This equation only contains the membrane

stress but neglects the boundary stress at the

joint of cylinder and head. Therefore the

complementary design equation should be

established:

(1)Discriminating whether the joint of

cylinder and head should be reinforced

or not.

(2)Calculation for the local reinforcement.

Conical head without hem ( ≤ 30 o )

(1)Not require reinforcing

(consistent thickness for the whole head)

main aspect:

cos

1

][2

c

t

cc

p

DpS

cos

1

][2

c

t

sic

p

DpS

small aspect:

(2)Require reinforcing

(for the thickness of joint,

the reinforcement region)

Main aspect:

c

t

icr

p

DpQS

][2

c

t

sicr

p

DpQS

][2

Small aspect:

Interpretation:

Dc —— inside diameter of main aspect

Di.s —— inside diameter of small aspect

Di —— inside diameter of cylinder

Q —— coefficient (Consulting the Figure

4-16 or 4-18 in book)

Conical head with hem ( > 30 o )

(1)Thickness of hem at the transition section

c

t

sic

p

DpKS

5.0 ][2

c

t

sic

p

DpfS

5.0 ][

K —— coefficient (Consulting Figure4-13)

f —— coefficient (Consulting Figure4-14)

(2)Thickness of conical shell at the joint with

transition section

4.Flat head

i. Structure

The geometric form of flat heads:

rotundity, ellipse, long roundness,

rectangle, square, etc.

ii. Characteristics of load

Round flat with shaft symmetry which is

subjected to uniform gas pressure

(1)There are two kinds of bending stress states,

distributing linearly along the wall.

(2)Radial bending stress r and hoop bending

stress t distributing along the radius.

▲Fastening the periphery

max = r.max

The maximum stress is

at the edge of disk.

S

R

p

0

t

r.

ma

x

r

2

max. 188.0

S

DPr

S

PD

S

R

275.0

▲ Periphery with simply supported ends

max = r.max = t.max

The maximum stress is

in the center of disk.

S

R

P

t

r.

ma

x

0

r

2

max. 31.0

S

DPr

S

PD

S

R

2 24.1

iii. Calculation equation for thickness

From the condition of strength max ≤ []t ,

getting:

Fastening the periphery

t

PDS

][

188.0

t

PDS

][

31.0

Periphery with simply supported ends

In fact, the supporting condition at boundary

of flat head is between the previous two.

After introducing the coefficient K which is

called structure characteristics coefficient and

considering the welded joint efficient , getting

the calculating equation for thickness of round disk:

t

ccp

PKDS

][

valueofroundS 12 CCSpn

5.Examples

Design the thicknesses of cylinder and heads

of a storage tank. Calculating respectively the

thickness of each heads if it’s semi-spherical,

elliptical, dished and flat head as well as

comparing and discussing the results.

Known: Di = 1200 mm Pc = 1.6Mpa

material: 20R []t = 133Mpa C2 = 1 mm

The heads can be punch formed by a complete

steel plate.

Solution:

(1)Determining the thickness of cylinder

][ 2 c

t

ic

p

DpS

mm 26.7

6.10.11332

12006.1

mmCSSd 26.80.126.72

C1 = 0.8 mm (Checking Figure 4-7)

Sd + C1 = 8.26 + 0.8 = 9.06 mm

Round it of, getting: Sn = 10 mm

= 1.0 (Double welded butt, 100% NDE)

(2)Semi-spherical head

][ 4 c

t

ic

p

DpS

mm 62.3

6.10.11334

12006.1

mmCSSd 62.40.162.32


Sd + C1 = 4.62 + 0.5 = 5.12 mm


= 1.0 (wholly punch forming)

(3)Standard elliptical head

c

t

ic

p

DpS

5.0 ][2

mm 24.7

165.00.11332

12006.1

mmCSSd 24.80.124.72


Sd + C1 = 8.24 + 0.8 = 9.04 mm



(4)Standard dished head

c

t

ic

p

DpS

5.0 ][2

2.1

mm 69.8

6.15.00.11332

12006.12.1

mmCSSd 69.90.169.82


Sd + C1 = 9.69 + 0.8 = 10.49 mm



(5)Flat head

K = 0.25; Dc = Di = 1200 mm; []t = 110 Mpa

t

ccp

PKDS

][

mm 36.72

0.1110

6.125.01200

mmCSSd 36.730.136.722


Sd + C1 =73.36 + 1.8 = 75.16 mm



Comparison:

Head-form

Sn mm

kg

Semi-sphe. Elliptical Dished Flat

6

106

10 12 80

137 163 662

Selection:

It’s better to use the standard elliptical

head whose thickness is the same to that of

cylinder.

Design of Heads

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