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Dec 22, 2015
Chapter 7A 1 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Chapter 7
Axi symmetrically Loaded
Members
Chapter 7A 2 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
OUTLINE
7.1 Equation of Equilibrium.
7.2 Constitutive Equation.
7.3 Strain – Displacement Relationship.
7.4 Governing Equation.
7.5 Stress Components of Loaded Members.
7.6 Stresses in Pressurized Cylinder.
Chapter 7A 3 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
OUTLINE (Cont’d)
7.7 Thin Walled Vessels.
7.8 Press and Shrink Fits.
7.9 Curved Beam.
7.10 Winkler’s Theory.
Chapter 7A 4 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Definition
A structure under load exhibiting a symmetrical
stress distribution with respect to an axis is
called as “ axisymmetrically loaded member”.
Assuming z-axis to be the symmetrical axis,
then we can conclude the stresses to be
independent of . This implies
In this chapter, we assume
0V
0 r
0z
Chapter 7A 5 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.1 Equation of Equilibrium.
01
r
rrr Frrr
02
.1
F
rrr
rr
0 0 0
0
The above equation reduces to
0
rrr F
rdr
d (7-1)
Chapter 7A 6 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.2 Constitutive Equation
r 1
E r
rE
1
(7-2) r
E
1 2r
E
1 2 r
Chapter 7A 7 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.3 Strain–Displacement Relationship
r
ur
directionrinu
directioninv
v
rr
u 1
01
u
rr
v
r
ur
r
uv
r
u (7-3)
Chapter 7A 8 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.4 Governing Equation
Combining (7-1) through (7-3), we have
011 2
22
2
rFEr
u
dr
du
rdr
ud
Assuming Fr=0, then equation (6-4) becomes
02
22 u
dr
dur
dr
udr
(7-4)
(7-4a)
This is an equi-dimensional differential
equation which has a solution of form mru
Chapter 7A 9 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Governing Equation (Cont’d)
(7-5)
2)1( mrmmu 1 mmru
mrmmur )1(2 mmrur
011 mrmmm 1m
Equation (7-4a) has a solution of
r
CrCu 2
1
Chapter 7A 10 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.5 Stress Components of Loaded Members
2
21
r
CC
dr
dur
2
21
r
CC
r
u (7-6)
Substituting (7-6) into (7-2),
2
21
2212
212
212
2
21
2212
212
212
1.
111
1
111
r
KK
r
EC
EC
r
CC
r
CC
E
r
KK
r
EC
EC
r
CC
r
CC
Er
2
21
r
KKr
2
21
r
KK , (7-7)
Chapter 7A 11 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.6 Stresses in Pressurized Cylinder
ri
ro P0
ri ro
Pi
B.C.
0Pr 0rrat
ir Pirrat
outero inneri
Chapter 7A 12 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Stresses in Pressurized Cylinder (Cont’d)
o
o
r Pr
KK
2
21
i
i
r Pr
KK
2
21
22
22
1
io
ooii
rr
rPrPK
K2 Pi Po ri
2ro2
ro2 ri
2
(7-8)
For rotating cylinder with angular velocity,
Fr r2
densitymass
Chapter 7A 13 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 1: No Rotation and No External Pressure
(7-9)
0rF 00 P&
22
2
1
io
ii
rr
rPK
1
2
22
22
2 Krrr
rrPK o
io
oii
2
2
12
21 1
r
rK
r
KK o
r
2
2
12
21 1
r
rK
r
KK o
oi rrr
Chapter 7A 14 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 1: (Cont’d)
0
0r r r
0at r r
Maximum stress occurs at ir r
P
rr
rrP
r
rK
r
i
ii
i
min
22
0
22
0
2
01max
1
max
r
or
P
ir2
1 2 22 2 i i
o i
PrK
r r
2
2
12
21 1
r
rK
r
KK o
r
2
2
12
21 1
r
rK
r
KK o
22
2
1
io
ii
rr
rPK
Chapter 7A 15 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 2: Non-Rotating Cylinder with End Cap
Force acting on the end cap
F A z ro2 ri
2 Piri2
22
2
io
iiz
rr
rP
zL
(7-10)
Chapter 7A 16 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 3: Rotating Cylinder
The governing equation gives
22
22
2 11
r
Er
u
dr
du
rdr
ud
r2d2u
dr2 rdu
dr u
1 2
E 2 r3
PH UUU
Particular Solution
Homogeneous Solution
Chapter 7A 17 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 3: Rotating Cylinder (Cont’d)
UP A r3
r2UP" rUP
' UP r2 3 2 r r 3 r2 r3
A
A 1 2
E1
8 2
UH C1 r C2 1
r
U UH UP C1 r C2 1
r A r3
2
32
21 rK
r
KKr
2
42
21 rK
r
KK
1 2
E 2 r3
Chapter 7A 18 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Case 3: Rotating Cylinder (Cont’d)
The stresses have the form
2
32
21 rK
r
KKr
2
42
21 rK
r
KK
K1 Piri
2 Poro2
ro2 ri
2K2
Pi Po ri2ro2
ro2 ri
2
2
38
3
K
2
48
31
K
(7-11)
Chapter 7A 19 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
7.7 Thin-Walled Vessels (No External Pressure)
ro
t
ri
io rrt (wall thickness)
20
1
ir
t
22
2
1
io
ii
rr
rPK
1o
i
r
r
0oP
Piri
2
ro ri ro ri Piri
2
2 ri t
t
rP ii
2
2
21
r
KKr
22
22
2
io
oii
rr
rrPK
1
2 Kro
Chapter 7A 20 Department of Mechanical & Aerospace Engineering
MAE4301/AE5339/ME5339 Spring 2015
Thin-Walled Vessels (Cont’d)
012
2
1
r
rK o
r
t
rPK
r
rK
r
KK iio
12
2
12
21 21 (7-12)
Pi sin ri d0
t 2
t
rP ii