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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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