Formula Sheet: Pressure Vessels | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Pressure V essels
Introduction to Pressure V essels
• Definition : Pressure vessels are containers designed to hold gases or liq-
uids at a press ure substantially different from the ambient pressure.
• Classification :
– Thin-walled:
D
t
= 20 , whereD = diameter ,t = wall thickness.
– Thick-walled:
D
t
< 20 .
• Assumptions : Linear elastic material, uniform pressure, axisymmetric ge-
ometry for cylind rical and spherical vessels.
Thin-W alled Pressure V essels
• Hoop Str ess (Circumferential Stress) :
s
h
=
pD
2t
wherep = internal pressure,D = mean diameter ,t = wall thickness.
• Longitudinal S tress (Axial Stress) :
s
l
=
pD
4t
• Maximum Shea r Stress :
t
max
=
s
h
-s
l
2
=
pD
8t
• Spherical V essel Stress :
s =
pD
4t
(Hoop and lon gitudinal stresses are equal in spherical vessels.)
Thick-W alled Pressure V essels
• Lamé’ s Eq uations (Cylindrical V essel) :
– Radial Stress:
s
r
=A-
B
r
2
1
Page 2


F ormula Sheet: Pressure V essels
Introduction to Pressure V essels
• Definition : Pressure vessels are containers designed to hold gases or liq-
uids at a press ure substantially different from the ambient pressure.
• Classification :
– Thin-walled:
D
t
= 20 , whereD = diameter ,t = wall thickness.
– Thick-walled:
D
t
< 20 .
• Assumptions : Linear elastic material, uniform pressure, axisymmetric ge-
ometry for cylind rical and spherical vessels.
Thin-W alled Pressure V essels
• Hoop Str ess (Circumferential Stress) :
s
h
=
pD
2t
wherep = internal pressure,D = mean diameter ,t = wall thickness.
• Longitudinal S tress (Axial Stress) :
s
l
=
pD
4t
• Maximum Shea r Stress :
t
max
=
s
h
-s
l
2
=
pD
8t
• Spherical V essel Stress :
s =
pD
4t
(Hoop and lon gitudinal stresses are equal in spherical vessels.)
Thick-W alled Pressure V essels
• Lamé’ s Eq uations (Cylindrical V essel) :
– Radial Stress:
s
r
=A-
B
r
2
1
– Hoop Stress:
s
h
=A+
B
r
2
whereA andB are constants determined b y boundary conditions,r =
r adial distance.
• F or Inter nal Pressure Only (p
i
,p
o
= 0 ) :
s
r
=
p
i
R
2
i
R
2
o
-R
2
i
(
1-
R
2
o
r
2
)
s
h
=
p
i
R
2
i
R
2
o
-R
2
i
(
1+
R
2
o
r
2
)
whereR
i
= inner r adius,R
o
= outer r adius.
• Maximum Ho op Stress (atr =R
i
) :
s
h, max
=
p
i
(R
2
o
+R
2
i
)
R
2
o
-R
2
i
• Maximum Shea r Stress :
t
max
=
s
h
-s
r
2
Design Consider ations
• Minimum W all T hickness (Thin-W alled Cylindrical V essel) :
t =
pD
2s
allow
wheres
allow
= allowable stress (yield strength divided b y factor of safety).
• Joint Effic iency ( E ) :
t =
pD
2s
allow
E
whereE = joint efficiency (e.g., 1 for seamless, < 1 for welded joints).
• Spherical V essel Thickness :
t =
pD
4s
allow
E
F ailure Criteria
• Maximum Pr incipal Stress Theory :
s
h
=s
yield
2
Page 3


F ormula Sheet: Pressure V essels
Introduction to Pressure V essels
• Definition : Pressure vessels are containers designed to hold gases or liq-
uids at a press ure substantially different from the ambient pressure.
• Classification :
– Thin-walled:
D
t
= 20 , whereD = diameter ,t = wall thickness.
– Thick-walled:
D
t
< 20 .
• Assumptions : Linear elastic material, uniform pressure, axisymmetric ge-
ometry for cylind rical and spherical vessels.
Thin-W alled Pressure V essels
• Hoop Str ess (Circumferential Stress) :
s
h
=
pD
2t
wherep = internal pressure,D = mean diameter ,t = wall thickness.
• Longitudinal S tress (Axial Stress) :
s
l
=
pD
4t
• Maximum Shea r Stress :
t
max
=
s
h
-s
l
2
=
pD
8t
• Spherical V essel Stress :
s =
pD
4t
(Hoop and lon gitudinal stresses are equal in spherical vessels.)
Thick-W alled Pressure V essels
• Lamé’ s Eq uations (Cylindrical V essel) :
– Radial Stress:
s
r
=A-
B
r
2
1
– Hoop Stress:
s
h
=A+
B
r
2
whereA andB are constants determined b y boundary conditions,r =
r adial distance.
• F or Inter nal Pressure Only (p
i
,p
o
= 0 ) :
s
r
=
p
i
R
2
i
R
2
o
-R
2
i
(
1-
R
2
o
r
2
)
s
h
=
p
i
R
2
i
R
2
o
-R
2
i
(
1+
R
2
o
r
2
)
whereR
i
= inner r adius,R
o
= outer r adius.
• Maximum Ho op Stress (atr =R
i
) :
s
h, max
=
p
i
(R
2
o
+R
2
i
)
R
2
o
-R
2
i
• Maximum Shea r Stress :
t
max
=
s
h
-s
r
2
Design Consider ations
• Minimum W all T hickness (Thin-W alled Cylindrical V essel) :
t =
pD
2s
allow
wheres
allow
= allowable stress (yield strength divided b y factor of safety).
• Joint Effic iency ( E ) :
t =
pD
2s
allow
E
whereE = joint efficiency (e.g., 1 for seamless, < 1 for welded joints).
• Spherical V essel Thickness :
t =
pD
4s
allow
E
F ailure Criteria
• Maximum Pr incipal Stress Theory :
s
h
=s
yield
2
• von Mises Stress (for combined stresses) :
s
von Mises
=
v
s
2
h
+s
2
l
-s
h
s
l
+3t
2
• Tresca C riterion (Maximum Shear Stress Theory) :
t
max
=
s
max
-s
min
2
=
s
yield
2
Str ains in Thin-W alled V essels
• Hoop Str ain :
?
h
=
1
E
(s
h
-?s
l
)
• Longitudinal S tr ain :
?
l
=
1
E
(s
l
-?s
h
)
whereE = Y oung’ s modulus,? = Poisson’ s r atio.
• V olumetric S tr ain (Cylindrical V essel) :
?
v
=?
h
+?
l
+?
r
˜
pD
4tE
(5-4?)
3