Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

22
Week 8 § 3-9 Strain Energy in Torsion and Pure Shear Fig. 3-34 Prismatic bar in pure torsion. P GI TL = φ Fig. 3-35 Torque-rotation diagram for a bar in pure torsion (linearly elastic material). 2 T U = (類似軸向受力元件之 2 δ T W U = = P GI TL = φ L GI GI L T U P P 2 2 2 2 φ = = (單位 JouleN.m ftlb1

Transcript of Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Page 1: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

§ 3-9 Strain Energy in Torsion and Pure Shear

Fig. 3-34 Prismatic bar in pure torsion.

PGI

TL=φ

Fig. 3-35 Torque-rotation diagram for a bar in pure torsion (linearly elastic material).

PGITL

L

GIGI

LTU P

P 22

22 φ==

(單位 Joule=N.m 或 ft-lb)

1

φW= 2

TU =

(類似軸向受力元件之

2δTWU == )

Page 2: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Non-uniform Torsion

( )∑∑==

==n

i iPi

iin

ii IG

LTUU

1

2

1 2

( )[ ]( )xGI

xTdUP2

2

=

( )[ ]( )∫=

L

P xGIdxxTU

0

2

2

重要假設:linearly elastic material

Strain Energy Density in Pure Shear

Fig. 3-36 Element in pure shear.

Shear Force

V=τht

©

2

0

0

1

2

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Week 8

δ=γh

the Strain energy

2δVWU ==

2

2thU τγ=

volume of the element=h2t

Strain-energy density

2τγ

=U

代入τ=Gγ

G2

2

u τ= 或

2

2γGu =

(單位 J/m3 或 psi) (pa)

例 3-10

Fig. 3-37 Example 3-10. Strain energy produced by two loads.

©

2

0

0

1

3

Page 4: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

TLI

(a)Ta (b) Tb (c)Ta & Tb (d)Ta=100 N.m,

b=150N.m =1.6m,G=80Gpa

P=79.52×103 mm4

a)P

aa GI

LT2

2

=U

b)P

b

P

b

b GILT

GI

LT

422 2

2

=

=U

c)

( )

P

b

P

ba

P

a

pP

b

iP

iin

ic

GILT

GILTT

GILT

GI

LTbTa

GI

LT

IGLT

U

422

22

22

)(222

222

1

++=

+

+

== ∑=

d)將數值代入

Ua=1.26 J Ub=1.41 J

Uc=4.56 J

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Page 5: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 3-11

Fig. 3-38 Example 3-11. Strain energy produced by

a distributed torque.

(a) T(x)=tx

( )[ ]( ) ( )∫ ∫ ===

L

P

L

PP GILtdxtx

GIxGIdxxTU

0

32

0

22

621

2

(b) 數值代入

U=580 in-lb

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Page 6: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 3-12

Fig. 3-39 Example 3-12. Tapered bar in torsion.

(work by the applied torque)=(Strain energy of the bar)

W=U

( ) ( )[ ]

( )

( )

( )[ ]( )

( )

−=

+

==

++=⇒

⋅+

+=

=

∫∫

33

2

0 4

2

0

2

4

4

113

16

)(

162

32

32

BAAB

L

ABA

L

P

ABAP

ABA

P

ddddGLT

AppendixCL

ddd

dxGT

xGIdxxTU

XL

dddXI

XL

dddxd

xdxI

π

π

π

π

積分查表

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Page 7: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

§ 3-10 Thin-Walled Tubes

Shear Stress & Shear Flow

Fig. 3-40 Thin-walled tube of arbitrary cross-sectional shape.

Fb=τbtbdx Fc=τctcdx Fb=Fc

τbtb=τctc Shear Flow f=τt=const

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Page 8: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Torsion Formula for Thin-Walled Tubes Dash line median line

Fig. 3-41 Cross section of thin-walled tube.

Total shear force on the element f ds The moment

dT=rfds

the total torque

∫=Lm

rdsfT0

其中 Lm為中線總長度

中線(虛線)所夾面積 Am

fAmT

AmrdsLm

2

20

=⇒

=∫

Shear flow

tAmT

tAmTf

2

2

=⇒

==

τ

τ

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Page 9: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Thin-walled circular tube

Am=πγ2

tT

22πγτ =

Fig. 3-42 Thin-walled circular tube.

©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark

Thin-walled rectangular tube

Am=bh

bht

Tvert

12=τ

bht

Thoriz

22=τ

Fig. 3-43 Thin-walled rectangular tube.

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Page 10: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Strain Energy & Torsion Constant

Fig. 3-41 Cross section of thin-walled tube.

Fig. 3-40

Thin-walled tube of arbitrary cross-sectional shape.

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Page 11: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

The volume of the element abcd(Fig 3-40)

tdsdx

the strain-energy density (Eq.3-55a)

G2

the total strain energy of the element

dxtds

Gf

dxtds

Gt

tdsdxG

dU

2

2

2

2

22

2

=

=

=

τ

τ

the total strain energy

∫∫∫

=

=

=

==

Lm

m

Lm

LLm

tds

GALT

AmTf

tds

Gt

dxtds

GdUU

02

2

0

22

00

2

8

2

2

τ

定義 torsion constant

∫=

Lm

tds

J

0

4

GJ

LTU2

2

=

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Page 12: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Special case t=const

LmtA

J m24

=

Thin-walled circular tube Lm=2πr Am=πr2

J=2πr3t Rectangular tube

Am=bh

21

2122

210

20

10

2

222

htbttthbJ

tb

th

tds

tds

tds bhLm

+=

+=+= ∫∫∫

Angle of Twist

Fig. 3-44 Angle of twist φ for a thin-walled tube.

GJTL

GJLTUWT

=⇒

===

φ

φ22

2

(GJ:torsion rigidity)

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Page 13: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 3-13

Fig. 3-45 Example 3-13. Comparison of approximate and exact theories of torsion.

©

2

0

0

1

B

r

o

Thin-Wall approximation

2321 22 βππγτ

tT

tT

==

tr

Torsion formula

PI

trT

+

= 22τ

( )

( )( )

( )( )

( )12214

1412

42

42222

2

2

1

23222

2244

++

=

++

=++

=⇒

+=

−−

+=

βββ

ττ

ββπβ

πτ

ππ

tT

trrttrT

trrttrtrI P

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Page 14: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 3-14

Fig. 3-46 Example 3-14. Comparison of circular and square tubes.

Same material Same length Same wall thickness Same cross-sectional area Circular tube Am1=πr2

J1=2πr3t , A1=2πrt

Square tube

4

222

2rbAm π

==

62.0162

8

79.04

4

4,8

2

3

33

1

2

2

1

2

22

1

2

2

1

2

333

2

====

====⇒

===

ππ

πφφ

ππ

π

ττ

π

trtr

JJ

r

r

AmAm

btAtrTbJ

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Page 15: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

§ 4-1 Introduction

©

2

0

0

1

B

r

o

Fig. 4-1 Examples of beams subjected to lateral loads.

Beams

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Page 16: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

§ 4-2 Types of Beams,Loads,and Reaction

Simply supported beam Cantilever beam Beam with a overhang

Fig. 4-2 Types of beams: (a) simple beam, (b) cantilever beam, and (c) beam with an overhang.

©

20

01

Br

oo

ks

/C

ol

e,

Types of Loads Concentrated load Distributed load Linearly varying load Simple beam Fig 4-2a

ΣFhoriz=0 HA-P1cosα=0

HA=P1cosα

ΣMB=0 , -RAL+(P1sinα)(L-a)+P2(L-P)+qc2 / 2=0

ΣMA=0 , RBL-(P1sinα)(a)-P2b+qc(L-c / 2)=0

( )( )L

cLqc

LbP

LaLPRB

++−

= 2sin 21 α

( )( )L

qcL

bLPL

aLPRA 2)(sin 21 +

−+

−=

α

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Page 17: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

Cantilever beam Fig 4-2b

ΣFhoriz=0

135 3P

H A =

ΣFvert=0

bqqPRA

+

+=213

12 213

ΣMA=0

−+

−+=⇒

=

−−

−−

3232

21312

0323

2213

12

213

213

bLbqbLbqaPM

bLbqbL

bqa

PM

A

A

Beam with an overhang Fig 4-2c

ΣMB=0 , -RAL+P4(L-a)+M1=0

ΣMA=0 , -R4a+PBL+M1=0

( )L

ML

aLPRA14 +

−=

LM

LaPRB

14 −=

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Page 18: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

§ 4-3 Shear Forces &Bending Moments

Fig. 4-4 Shear force V and bending moment M in a beam.

©

2

0

0

1

B

ΣFvert=0,P-V=0 或 P=V

ΣM=0,M-PX=0 或 M=PX

Sign Convention Fig. 4-5 Sign

conventions for shear force V and bending moment M.

+ in the positive direction

Fig. 4-6 Deformations (highly exaggerated) of a beam element caused by (a) shear forces, and (b) bending moments.

©

2

0

0

1

B

r

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Page 19: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 4-1

LMPRA

0

43

+=

LMPRB

0

4+=

Fig. 4-7 Example 4-1.

Shear forces and bending moments in a simple beam.

©

2

0

0

1

B

r

(a) To the left of the midpoint (Fig.4-7b)

ΣFvert=0,RA-P-V=0

V=RA-P= L

MP 0

4−−

ΣM=0, 0)4

()2

( =++ MLPLRA−

28

)4

()2

( 0MPLLPLRM A −=−=

(b)To the right of the midpoint (Fig.4-7c)

LMPV 0

4−−=

280MPLM +=

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Page 20: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 4-2

Fig. 4-8 Example 4-2.

Shear force and bending moment in a cantilever beam.

©

2

0

0

1

B

Shear force

Lxq

q 0=

Form Fig.4-8b

( ) VLxq

xLxq

−==

221 2

00

when X=0,V=0

X=L,20

maxLq

−=V

Bending moment

ΣM=0, ( ) 0)3

(2

0 =

+

XXLxqLM

LXq

M6

30−

=

when X=0,M=0

X=L,LLq

M6

20

max−

=

20

Page 21: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

例 4-3

©

2

0

0

1

B

r

o

Fig. 4-9 Example 4-3. Shear force and bending moment in a beam with an overhang.

q=20 lb/ft

P=14K

RA=11K,RB=9K,

Form Fig.4-9b

ΣFvert=0,11K-14K-(0.2K/ft)(15ft)-V=0

V=-6K

ΣMD=0,-(11K)(15ft)+(14K)(6ft)+(0.2K/ft)(15ft)(7.5ft)+M=0

M=58.5 K-ft

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Page 22: Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Week 8

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Homework

3.9-2 3.10-2 4.3-2

3.9-4 3.10-4 4.3-4

3.9-6 3.10-6 4.3-6

3.9-8 3.10-10 4.3-8

4.3-10

4.3-12