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Thick Walled CylindersThick Walled Cylinders
Lecture 13Lecture 13
Engineering 473Engineering 473Machine DesignMachine Design
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AxisymmetricAxisymmetric Equation of Equation of
EquilibriumEquilibrium
(Geometry)(Geometry)
direction-in nt displacemevdirection-rin nt displacemeu
coordinateposition radialrcoordinateposition angular
pressure internalpi
0
direction.-in the variesNothingicAxisymmetr
=
Ugural, Fig. 8.1(a)
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r
d rr
+
Axisymmetric Axisymmetric Equation of Equation of
EquilibriumEquilibrium
(Differential Element)(Differential Element)
Ugural, Fig. 8.1(b)itycompatibil stress todue ,0
constraint icaxisymmetr todue ,0
r
r
=
=
-
Axisymmetric Axisymmetric Equation of Equation of
EquilibriumEquilibrium
eunit volumper forcebody radialFr
( ) 0dzdrrdFdzrd-drdz2
dsin2-dzddrrdrr rrrr =+
+
+
0Fr
drd
0rFdr
dr
rrr
rrr
=+
+
=++
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Strain Displacement EquationsStrain Displacement Equations
( )ru
rdrd-dur
drdu
dr
drdrrudr
r
=
+=
=
+=
ru ,
drdu r ==
-
Constitutive EquationsConstitutive Equations
( )
( )r
rr
E1
E1
=
=
Hookes Hookes LawLawStress-Strain equations are often referred
to as constitutiveequations, because they depend on what the part
is made of. The equilibrium and strain-displacement equations are
independent of the material.
Webster, constitutive - making a thing what it is, essential
( )
( )r2
r2r
1
E
1
E
+
=
+
=
-
Summary of Summary of AxisymmetricAxisymmetric
EquationsEquations
ru ,
drdu r ==
( )
( )r
rr
E1
E1
=
=0Fr
drd
rrr
=+
+
Equilibrium EquationEquilibrium Equation
StrainStrain--Displacement EquationsDisplacement Equations
Constitutive EquationsConstitutive Equations
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Thick Walled CylindersThick Walled Cylinders(Displacement
Differential Equation)(Displacement Differential Equation)
pressure externalppressure internalp
radius outsidebradius insidea
o
i
( )( )
0ru
drdu
r1
drud
drdu
ru
1E
ru
drdu
1E
1
E
1
E
22
2
2
2r
r2
r2r
=+
+
=
+
=
+
=
+
=
Ugural, Fig. 8.2
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Thick Walled CylindersThick Walled Cylinders(General Solution
& Boundary Conditions)(General Solution & Boundary
Conditions)
rCrCu
Solution General
0ru
drdu
r1
drud
21
22
2
+=
=+ ( )
( )
++
=
+
=
+
=
+
=
2212
2212r
2
2r
r1C1C
1E
r1C1C
1E
drdu
ru
1E
ru
drdu
1E
Ugural, Fig. 8.2
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Thick Walled CylindersThick Walled Cylinders(Boundary
Conditions)(Boundary Conditions)
obrr
iarr
p
p
=
=
=
=
Boundary ConditionsBoundary Conditions
( )
( )
( )
( )
+=
=
+
=
+
=
+
=
22oi
22
2
22o
2i
2
1
2212o
2212i
2212r
abppba
E1C
abpbpa
E1C
b1C1C
1Ep
a1C1C
1Ep
r1C1C
1E
Ugural, Fig. 8.2
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Thick Walled CylindersThick Walled Cylinders(Lame
Equations)(Lame Equations)
Ugural, Fig. 8.2
( )( )( )( )
( ) ( )( )rab
bappE1
abrpbpa
E1u
rabbapp
abpbpa
rabbapp
abpbpa
22
22oi
22o
2i
2
222
22oi
22o
2i
2
222
22oi
22o
2i
2
r
++
=
+
=
=
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Longitudinal StrainLongitudinal Strain(Unconstrained and Open
Ends)(Unconstrained and Open Ends)
( )
( )rz
z
rzz
E
0 open, and nedunconstrai are Ends
E1
+=
=
=
( )
( )
++
=
+
=
2212
2212r
r1C1C
1E
r1C1C
1E
( )[ ]
=
=
+
=+
22o
2i
2
z
1z
12r
abpbpa
E2
1C2
1C1
2E
Constant that Note r =+
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Longitudinal StressLongitudinal Stress(Constrained
Ends)(Constrained Ends)
( )
( )rz
rzz
E10
+=
==
( )
( )
++
=
+
=
2212
2212r
r1C1C
1E
r1C1C
1E
( )[ ]
=
=
+
=+
22o
2i
2
z
1z
12r
abpbpa2
12EC
1C1
2E
Constant that Note z =
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Longitudinal StressLongitudinal Stress(Closed and Unconstrained
Ends)(Closed and Unconstrained Ends)
op ipz
z
( )
22
2o
2i
z
2i
2o
22z
abbpap
0apbpab
=
=+
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Special CasesSpecial Cases
nedunconstrai and closed ,ab
pa
dconstraine ,abpa2
nedunconstrai ,0
rb1
abpa
rb1
abpa
22i
2
z
22i
2
z
z
2
2
22i
2
2
2
22i
2
r
=
=
=
+
=
=
Internal Pressure OnlyInternal Pressure Only External Pressure
OnlyExternal Pressure Only
nedunconstrai & closed ,ab
pb
dconstraine ,abpb2
nedunconstrai ,0
ra1
abpb
ra1
abpb
22o
2
z
22o
2
z
z
2
2
22o
2
2
2
22o
2
r
=
=
=
+
=
=
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Stress VariationStress Variation
Internal Pressure OnlyInternal Pressure Only External Pressure
OnlyExternal Pressure Only
b/a=4
Ugural, Fig. 8.3
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AssignmentAssignment
1. Show that the Lame equations for the case of internal
pressure reduce to the equations for a thin walled cylinder when
the ratio b/a approaches 1.
2. A thick walled cylinder with 12 and 16 inch internal and
external diameters is fabricated of a material whose tensile yield
strength is 36 ksi and Poissons ratio is 0.3. Calculate the von
Mises stress when the internal pressure is 10 ksi. The cylinder has
closed but unconstrained ends. Will the material yield?
