Ch03 Section14 Pressure Vessel Design

Prévia do material em texto

† Text Eq. refers to Shigley
 Nisbett; equations and fig
Shigley's Mechanical Engineering Design 
 Tutorial 3 –14: Pressure Vessel Design 
 
 
PRESSURE VESSEL DESIGN MODELS FOR CYLINDERS: 
 
1. Thick-walled
2. Thin-walled 
 
THICK-WALL THEORY 
 
• Thick-wall th
stress as a co
stress is defin
 
1. σ
2. σ
3. σ
 
• Stresses in a 
the outer rad
• Principal Str
 
1. D
2. E
 
• Applicable fo
 
Cylinder under Pressu
 
Consider a cylin
external pressure
 
 
 
 
 
 
 
 
 
 
 
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yl
ineering Design, 8th edition text by Richard G. Budynas and J. Keith; 
 T refer to the present tutorial. 
 from the Theory of Elasticity which yields the state of 
 of radius over the pressure vessel wall. The state of 
onvenient cylindrical coordinate system: 
tress 
s 
l Stress 
re vessel depend upon the ratio of the inner radius to 
than the size of the cylinder. 
) 
t computation of Mohr’s Circle; 
drical stresses ( , ,t r lσ σ σ ) 
ess-to-radius ratio. 
nds, subjected to an internal pressure, pi, and an 
GURE T3-14-1 
op
or
ir
ip 
lσ
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 2/10 
 
The cylinder geometry is defined by the inside radius, ,ir the outside radius, ,or and the 
cylinder length, l. In general, the stresses in the cylindrical pressure vessel ( , ,t r lσ σ σ ) 
can be computed at any radial coordinate value, r, within the wall thickness bounded by 
ir and ,or and will be characterized by the ratio of radii, / .o ir rζ = These cylindrical 
stresses represent the principal stresses and can be computed directly using Eq. 3-49 and 
3-51. Thus we do not need to use Mohr’s circle to assess the principal stresses. 
 
Tangential Stress: 
 
22
22222 /)(
io
iooiooii
t rr
rpprrrprp
−
−−−
=σ for i or r r≤ ≤ (Text Eq. 3-49) 
 
Radial Stress: 
 
22
22222 /)(
io
iooiooii
r rr
rpprrrprp
−
−+−
=σ for i or r r≤ ≤ (Text Eq. 3-49) 
 
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. 
 
22
22
io
ooii
l rr
rprp
−
−
=σ for i or r r≤ ≤ (Modified Text Eq. 3-51) 
 
Two Mechanical Design Cases 
 
1. Internal Pressure Only ( 0=op ) 
2. External Pressure Only ( 0=ip ) 
 
Design Case 1: Internal Pressure Only 
 
• Only one case to consider — the critical section which exists at irr = . 
• Substituting 0=op into Eqs. (3-49) and incorporating / ,o ir rζ = the 
largest value of each stress component is found at the inner surface: 
 
2 2 2
,max 2 2 2
1( )
1
o i
t i t i i i ti
o i
r rr r p p p C
r r
ζ
σ σ ζ
+ +
= = = = =
− −
 (T-1) 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 3/10 
 
 where 
2 22
2 2 2
1
1
o i
ti
o i
r rC
r r
ζ
ζ
++
= =
− −
 is a function of cylinder geometry only. 
irir prr −=== max,)( σσ Natural Boundary Condition (T-2) 
 
• Longitudinal stress depends upon end conditions: 
 
 
 i lip C Capped Ends (T-3a) 
lσ = 
0 Uncapped Ends (T-3b) 
 
 
where 2
1
1li
C 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. (3-49) for each case: 
 
r = ri 0)( == ir rrσ Natural Boundary Condition (T-4a) 
 
( )
2 2
,max 2 2 2
2 2
1
o
t i t o o o to
o i
rr r p p p C
r r
ζ
σ σ ζ= = = − = − = −− − (T-4b) 
where, 
22
2 2 2
22
1
o
to
o i
rC
r r
ζ
ζ= =− − . 
r = ro oror prr −=== max,)( σσ Natural Boundary Condition (T-5a) 
2 2 2
2 2 2
1( )
1
o i
t o o o o ti
o i
r rr r p p p C
r r
ζ
σ 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: 
 
 o lop C− Capped Ends (T-6a) 
lσ = 
 0 Uncapped Ends (T-6b) 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 4/10 
 
where 
2
2 1lo
C ζζ= − . 
 
Example T3.14.1: 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 ( rσ , tσ , lσ ) 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 maxτ . 
 
Solution Methodology: 
Since we have an external pressure case, we need to compute the state of 
stress ( ,rσ ,tσ lσ ) 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 ( ,rσ ,tσ lσ ) 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 50 mm ID 25 mm25 mm ; 12.5 mm
2 2 2 2o i
r r= = = = = = 
 
 Compute the radius ratio, ζ 
25 mm 2.0
12.5 mm
o
i
r
r
ζ = = = 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 5/10 
 
Then, 
2 2
2 2
2
2
2
(2)
1 (2) 1
( ) ( ) ( 150MPa)(1.3333 mm )
1
lo
l i l o o o lo
C
r r r r p p C
ζ
ζ
ζ
σ σ ζ
= = =
− −
= = = = − = − = −
−
21.3333 mm
MPa200−−−−====lσσσσ 
 
2. Radial & Tangential Stress Calculations: 
 
Inner Radius (r = ri) 
2 2
2 2
2
,max 2 2
2 2(2)
1 (2) 1
2( ) ( 150 MPa)(2.6667)
to
o
t i t o o to
o i
C
rr r p p C
r r
ζ
ζ
σ σ
= = =
− −
= = = − = − = −
−
2.6667
 
400 MPat iσ (r r ) Compressive= = − 
 
0pforConditionBoundaryNatural i === 0)r(rσ ir 
 
Outer Radius (r = ro) 
2 2
2 2
2 2
,min 2 2
1 (2) 1
1 (2) 1
( ) ( 150 MPa)(1.6667)
ti
o i
t o t o o ti
o i
C
r rr r p p C
r r
ζ
ζ
σ σ
+ +
= = =
− −
+
= = = − = − = −
−
1.6667
 
eCompressivMPa250−−−−======== )r(rσ ot 
ConditionBoundaryNaturalMPa150−−−−====−−−−======== oir p)r(rσ 
 
3. Define Principal Stresses: 
 
Inner Radius (r = ri ) Outer Radius (r = ro ) 
 
MPa400
MPa200
MPa0
3
2
1
−==
−====
t
l
r
σσ
σσ
σσ
 
MPa250
MPa200
MPa150
3
2
1
−==
−==
−==
t
l
r
σσ
σσ
σσ
 
 
4. Maximum Shear Stress Calculations: 
 
Inner Radius (r = ri ) 1 3max
0 ( 400)( )
2 2i
r r σ στ − − −= = = = 200 MPa 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel
 
Outer Radius (r = ro ) 1 3max
( 150) ( 250)( )
2 2o
r r σ στ − − − −= = = = 50 MPa 
 
5. Mohr’s Circles: 
 
Inner Radius (r = ri ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Outer Radius (r = ro ) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Critical Section 
 
SectionCriticalrr i ⇐⇐⇐⇐======== MPa200)(maxττττ
3σ = -250 MPa
τ 
2σ = -200 MPa
τ
σ 
2σ = -200 MPa
3σ = -400 MPa
τ 
1 0 MPaσ = 
maxτ = 200 MPa 
FIGURE T3-14-2 
FIGURE T3-14-3
•
 Design 6/10 
!RadiusInsideatis 
1 150 MPaσ = −
σ 
max = 50 MPa 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 7/10 
 
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: 
��
1According to theory, Thin-wall Theory is justified for
20
t
r
≤ 
��
1In practice, typically use a less conservative rule, 
10
t
r
≤ 
 
• State of Stress Definition: 
 
1. Hoop Stress, tσ , assumed to be uniform across wall thickness. 
2. Radial Stress is insignificant compared to tangential stress, thus, 0.rσ � 
3. Longitudinal Stress, lσ 
��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 ( , , )t r lσ σ σ are principal stresses ( , , )t r lσ σ σ which 
can be determined without computation of Mohr’s circle plot. 
 
• Analysis of Cylinder Section 
 
1 
t 
FV 
FHoop FHoop 
Pressure Acting over 
Projected Vertical Area
di 
FIGURE T3-14-4 
 
Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 8/10 
 
The internal pressure exerts a vertical force, FV, on the cylinder wall which is 
balanced by the tangential hoop stress, FHoop. 
 
tpdFFF
ttAF
pddppAF
tiHoopVy
ttstressedtHoop
iiprojV
σ
σσσ
� −=−==
===
===
220
)}1)({(
)}1)({(
 
 
Solving for the tangential stress, 
Hoop Stress
2
i
t
pd
t
σ = (Text Eq.3-52) 
• Comparison of state of stress for cylinder under internal pressure verses external 
pressure: 
Internal Pressure Only 
2
0
(Text Eq.3-54)
4 2
i
t
r
i t
l
pd Hoop Stress
t
By Definition
pd Capped Case
t
σ
σ
σ
σ
=
=
= =
 
External Pressure Only 
2
0
4 2
o
t
r
o t
l
pd Hoop Stress
t
By Definition
pd Capped Case
t
σ
σ
σ
σ
=
=
= =
 
 
 
Example T3.14.2: 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. 
 
 
 
 Budynas & Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 9/10 
 
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: 
10
1
20
1
2
1
mm25
mm5.12 or
r
t
�== 
 
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. 
 
definitionbyStress Radialb.
mm)2(12.5
)mm50)(MPa150(
2
wall)acrossuniformstress,(averageStressHoopa.
r 0
MPa300
=
−=
−
=
−
=
σ
σ
t
dp oo
t
 
c. Longitudinal Stress (average stress, uniform across wall)
4 2
o o t
l
p d
t
σ
σ
−
= = = −150 MPa
 
3. Identify Principal Stresses in terms of “Average” Stresses: 
 
MPa300
MPa150
MPa0
3
2
1
−==
−==
==
t
l
r
σσ
σσ
σσ
 
 
4. Maximum Shear Stress Calculation: 
 
MPa150
2
)MPa300(0
2
31
max +=
−−
=
−
=
σσ
τ 
 
5. Percent Difference between Thin- and Thick-wall Estimates for the 
Critical Section: 
 
 
 Budynas &Nisbett Machine Design Tutorial 3–14: Pressure Vessel Design 10/10 
 
max,Thin max,Thick
max,Thick
% 100%
( 150) ( 200) (100%)
( 200)
Difference
τ τ
τ
−
= ∗
+ − +
= ∗ = −
+
25%
 
 
���� Thin -wall estimate is 25% low! ⇐⇐⇐⇐