This document provides a tutorial on pressure vessel design, focusing on thick-walled and thin-walled cylinders' stress analysis under internal and external pressures. It discusses relevant stress theories, the relationship between geometry and stress, and methodologies for calculating stress states, including practical examples. The tutorial emphasizes the conditions under which thick-wall and thin-wall theories are applicable, demonstrating mathematical components to compute principal stresses and maximum shear stresses.

1. MECHANICAL ENGINEERING DESIGN
TUTORIAL 4 –15: PRESSURE VESSEL DESIGN
PRESSURE VESSEL DESIGN MODELS FOR CYLINDERS:
1. Thick-walled Cylinders
2. Thin-walled Cylinders
THICK-WALL THEORY
• Thick-wall theory is developed from the Theory of Elasticity which yields the state of
stress as a continuous function of radius over the pressure vessel wall. The state of
stress is defined relative to a convenient cylindrical coordinate system:
1. σ t — Tangential Stress
2. σ r — Radial Stress
3. σ l — Longitudinal Stress
• Stresses in a cylindrical pressure vessel depend upon the ratio of the inner radius to
the outer radius ( ro / ri ) rather than the size of the cylinder.
• Principal Stresses (σ 1,σ 2 ,σ 3 )
1. Determined without computation of Mohr’s Circle;
2. Equivalent to cylindrical stresses (σ t ,σ r ,σ l )
• Applicable for any wall thickness-to-radius ratio.
Cylinder under Pressure
Consider a cylinder, with capped ends, subjected to an internal pressure, pi, and an
external pressure, po,
σ r
σ l
FIGURE T4-15-1
po
ro
ri
pi
σ t
σ l
σ r
σ t
† Text Eq. refers to Mechanical Engineering Design, 7th edition text by Joseph Edward Shigley, Charles
R. Mischke and Richard G. Budynas; equations and figures with the prefix T refer to the present tutorial.

2. The cylinder geometry is defined by the inside radius, ri , the outside radius, ro , and the
cylinder length, l. In general, the stresses in the cylindrical pressure vessel (σ t , σ r ,σ l )
can be computed at any radial coordinate value, r, within the wall thickness bounded by
ri and ro , and will be characterized by the ratio of radii, ζ = ro / ri . These cylindrical
stresses represent the principal stresses and can be computed directly using Eq. 4-50 and
4-52. Thus we do not need to use Mohr’s circle to assess the principal stresses.
Tangential Stress:
2 2 2 2 ( ) / 2
p r − p r − r r p −
p r
i i o o i o o i
σ = for ri ≤ r ≤ ro (Text Eq. 4-50)
t r r
2 2
−
o i
Radial Stress:
2 2 2 2 ( ) / 2
p r − p r + r r p −
p r
i i o o i o o i
σ = for ri ≤ r ≤ ro (Text Eq. 4-50)
r r r
2 2
−
o i
Longitudinal Stress:
• Applicable to cases where the cylinder carries the longitudinal load, such as
capped ends.
• Only valid far away from end caps where bending, nonlinearities and stress
concentrations are not significant.
2 2
p r −
p r
i i o o
σ = for ri ≤ r ≤ ro (Modified Text Eq. 4-52)
l r r
2 2
−
o i
Two Mechanical Design Cases
1. Internal Pressure Only ( po = 0 )
2. External Pressure Only ( pi = 0 )
Design Case 1: Internal Pressure Only
• Only one case to consider — the critical section which exists at r = ri .
• Substituting po = 0 into Eqs. (4-50) and incorporating ζ = ro / ri , the
largest value of each stress component is found at the inner surface:
2 2 2
r = r = = p r + r = p + =
p C
σ σ ζ
( ) o i
1
t i t i i i ti
,max 2 2 2
1
r r
− −
o i
ζ
(T-1)
Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 2/10

3. where
2 2 2
2 2 2
C = + = r +
r
1
1
o i
ti
r r
− −
o i
ζ
ζ
is a function of cylinder geometry only.
σ r (r = ri ) =σ r,max = − pi Natural Boundary Condition (T-2)
• Longitudinal stress depends upon end conditions:
piCli Capped Ends (T-3a)
σ l =
0 Uncapped Ends (T-3b)
Cli
=
1
1 where 2
ζ
−
.
Design Case 2: External Pressure Only
• The critical section is identified by considering the state of stress at two
points on the cylinder: r = ri and r = ro. Substituting pi = 0 into Text
Eqs. (4-50) for each case:
r = ri σ r (r = ri ) = 0 Natural Boundary Condition (T-4a)
r r p r p p C
σ σ ζ
( )
2 2
2 2
o
= = = − = − = −
t i t o o o to
,max 2 2 2
1
r r
− −
o i
ζ
(T-4b)
where,
2 2
C 2 2
r
o
ζ
ζ
= =
2 2 2
− −
1
to
r r
o i
.
r = ro σ r (r = ro ) =σ r,max = − po Natural Boundary Condition (T-5a)
2 2 2
2 2 2
r = r = − p r + r = − p + = −
p C
σ ζ
( ) 1
t o o o o ti
1
o i
r r
− −
o i
ζ
(T-5b)
• Longitudinal stress for a closed cylinder now depends upon external
pressure and radius while that of an open-ended cylinder remains zero:
− poClo Capped Ends (T-6a)
σ l =
0 Uncapped Ends (T-6b)
Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 3/10

4. where
ζ
2
Clo =
ζ
2 −
1 .
Example T4.15.1: Thick-wall Cylinder Analysis
Problem Statement: Consider a cylinder subjected to an external pressure of
150 MPa and an internal pressure of zero. The cylinder has a 25 mm ID and a 50
mm OD, respectively. Assume the cylinder is capped.
Find:
1. the state of stress (σ r , σ t , σ l ) at the inner and outer cylinder
surfaces;
2. the Mohr’s Circle plot for the inside and outside cylinder surfaces;
3. the critical section based upon the estimate of τ max .
Solution Methodology:
Since we have an external pressure case, we need to compute the state of
stress ( , r σ , t σ
σ l ) at both the inside and outside radius in order to determine
the critical section.
1. As the cylinder is closed and exposed to external pressure only,
Eq. (T-6a) may be applied to calculate the longitudinal stress
developed. This result represents the average stress across the wall
of the pressure vessel and thus may be used for both the inner and
outer radii analyses.
2. Assess the radial and tangential stresses using Eqs. (T-4) and (T-5)
for the inner and outer radii, respectively.
3. Assess the principal stresses for the inner and outer radii based
upon the magnitudes of ( , r σ , t σ
σ l ) at each radius.
4. Use the principal stresses to calculate the maximum shear stress at
each radius.
5. Draw Mohr’s Circle for both states of stress and determine which
provides the critical section.
Solution:
1. Longitudinal Stress Calculation:
OD 50mm 25mm; ID 25mm 12.5mm
o 2 2 i 2 2 r = = = r = = =
Compute the radius ratio, ζ
25 mm 2.0
12.5 mm
r
r
o
i
ζ = = =
Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 4/10

5. Then,
2 2
2 2
1.3333 mm2
2
2
lo
ζ
ζ
= = =
− −
σ σ ζ
r r r r p pC
= = = =− =− = −
( ) ( ) ( 150MPa)(1.3333 mm )
l i l o o o lo
2
(2)
1 (2) 1
1
C
ζ
−
σ l = −200MPa
2. Radial & Tangential Stress Calculations:
Inner Radius (r = ri)
2 2
ζ
ζ
2 2(2)
= = =
2 2
2.6667
2
− −
1 (2) 1
C
to
r r p r p C
( = ) = =− 2 o
=− = ( −
150MPa)(2.6667)
σ σ
t i t o o to
,max 2 2
r −
r
o i
σt (r = ri ) = −400MPa Compressive
σr (r = ri ) = 0 Natural Boundary Condition for pi = 0
Outer Radius (r = ro)
2 2
2 2
1.6667
2 2
= + = + =
1 (2) 1
1 (2) 1
C
ti
ζ
ζ
− −
r r p r r p C
= = = − + = − = −
σ σ
( ) o i
( 150MPa)(1.6667)
t o t o o ti
,min 2 2
r −
r
o i
σt (r = ro ) = −250MPa Compressive
σr (r = ri ) = − po = −150MPa Natural Boundary Condition
3. Define Principal Stresses:
Inner Radius (r = ri ) Outer Radius (r = ro )
0 MPa
200 MPa
= =
σ σ
r
σ σ
= = −
400 MPa
1
2
σ σ
3
l
= = −
t
150 MPa
200 MPa
= = −
σ σ
r
σ σ
= = −
250 MPa
1
2
σ σ
3
l
= = −
t
4. Maximum Shear Stress Calculations:
τ ( r = r ) =σ −σ = 0 − ( − 400)
= 200MPa
i 2 2 Inner Radius (r = ri ) 1 3
max
Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 5/10

6. τ ( r = r ) =σ −σ = ( − 150) − ( − 250)
= 50MPa
o 2 2 Outer Radius (r = ro ) 1 3
max
5. Mohr’s Circles:
Inner Radius (r = ri )
FIGURE T4-15-2
σ 3 = -400 MPa
Outer Radius (r = ro )
τ
τ
σ 2 = -200 MPa
τ max = 200 MPa
σ1 = 0 MPa
FIGURE T4-15-3
σ 3 = -250 MPa σ1 = −150 MPa
σ
σ
•
σ 2 = -200 MPa
τmax = 50 MPa
Critical Section
τ max (r = ri ) = 200MPa ⇐ Critical Section is at Inside Radius!
Shigley, Mischke & Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 6/10

7. THIN-WALL THEORY
• Thin-wall theory is developed from a Strength of Materials solution which yields the
state of stress as an average over the pressure vessel wall.
• Use restricted by wall thickness-to-radius ratio:
t
r
According to theory, Thin-wall Theory is justified for 1
20
≤
t
r
In practice, typically use a less conservative rule, 1
10
≤
• State of Stress Definition:
1. Hoop Stress, t σ
, assumed to be uniform across wall thickness.
2. Radial Stress is insignificant compared to tangential stress, thus, σ r 0.
3. Longitudinal Stress, l σ
Exists for cylinders with capped ends;
Assumed to be uniformly distributed across wall thickness;
This approximation for the longitudinal stress is only valid far away
from the end-caps.
4. These cylindrical stresses (σ t ,σ r ,σ l ) are principal stresses (σ t ,σ r ,σ l )which
can be determined without computation of Mohr’s circle plot.
• Analysis of Cylinder Section
1
di
FIGURE T4-15-4
t
FV
FHoop FHoop
Pressure Acting over
Projected Vertical Area
Shigley, Mischke Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 7/10

8. The internal pressure exerts a vertical force, FV, on the cylinder wall which is
balanced by the tangential hoop stress, FHoop.
F = pA = p {( d )(1)}
=
pd
V proj i i
F = σ A = σ {( t )(1)}
=
σ
t
Hoop t stressed t t
F F F pd σ
t
= = − = −
0 2 2
y V Hoop i t
Solving for the tangential stress,
σ = Hoop Stress
(Text Eq. 4-53)
pd
2
t
i
t
• Comparison of state of stress for cylinder under internal pressure verses external
pressure:
Internal Pressure Only
pd Hoop Stress
t
σ
=
2
σ
=
0
σ = =
σ
(Text Eq.4-55)
i
pd i t
Capped Case
t
4 2
t
r
l
By Definition
External Pressure Only
pd Hoop Stress
t
σ
=
2
σ
=
0
σ = =
σ
o
pd o t
Capped Case
t
4 2
t
r
l
By Definition
Example T4.15.2: Thin-wall Theory Applied to Cylinder Analysis
Problem Statement: Repeat Example T1.1 using the Thin-wall Theory
(po = 150 MPa, pi = 0, ID = 25 mm, OD = 50 mm).
Find: The percent difference of the maximum shear stress estimates found using
the Thick-wall and Thin-wall Theories.
Shigley, Mischke Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 8/10

9. Solution Methodology:
1. Check t/r ratio to determine if Thin-wall Theory is applicable.
2. Use the Thin-wall Theory to compute the state of stress
3. Identify the principal stresses based upon the stress magnitudes.
4. Use the principal stresses to assess the maximum shear stress.
5. Calculate the percent difference between the maximum shear stresses
derived using the Thick-wall and Thin-wall Theories.
Solution:
1. Check t/r Ratio:
1
10
t
= =
12.5 mm 1
or
20
1
2
25 mm
r
The application of Thin-wall Theory to estimate the stress state of this
cylinder is thus not justified.
2. Compute stresses using the Thin-wall Theory to compare with Thick-wall
theory estimates.
a. Hoop Stress (average stress, uniform across wall)
podo
= − = − (150MPa)(50 mm)
= −
2(12.5mm)
t
2
r =
0
σ
σ
t
b. RadialStress by definition
300MPa
c. Longitudinal Stress (average stress, uniform across wall)
p d
t
σ = − σ o o =t
= −150MPa
4 2
l
3. Identify Principal Stresses in terms of “Average” Stresses:
0 MPa
150 MPa
= =
σ σ
r
σ σ
= = −
300 MPa
1
2
σ σ
3
l
= = −
t
4. Maximum Shear Stress Calculation:
150 MPa
σ −
σ
= 0 − ( −
300 MPa)
max = + 2
1 3
2
=
τ
5. Percent Difference between Thin- and Thick-wall Estimates for the
Critical Section:
Shigley, Mischke Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 9/10

10. τ −
τ
max,Thin max,Thick
Difference
= ∗
% 100%
τ
max,Thick
= + − + ∗ = −
( 150) ( 200) (100%)
+
( 200)
25%
Thin -wall estimate is 25% low! ⇐
Shigley, Mischke Budynas Machine Design Tutorial 4–15: Pressure Vessel Design 10/10