2 axial loading

7/17/2019 2 Axial Loading

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MECHANICS OF

MATERIALS

Third Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

John T. e!ol"

Le#ture Notes$

J. !alt Oler

Te%as Te#h &ni'ersit(

CHAPTER

© 2002 The McGraw-Hill Companies, Inc. All rights

2Stress and Strain

) A%ial Loadin*



© 2002 The McGraw-Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALST h i r d Beer + Johnston + e!ol"

-

Contents

Stress & Strain: Axial Loading

Normal Strain

Stress-Strain Test

Stress-Strain Diagram: Ductile Materials

Stress-Strain Diagram: Brittle Materials

Hooke’s La: Modulus o! "lasticit#

"lastic $s% lastic Be'a$ior

(atigue

De!ormations )nder Axial Loading

"xam*le 2%+,

Sam*le rolem 2%,Static .ndeterminac#

"xam*le 2%+/

T'ermal Stresses

oisson’s 0atio

1eneralied Hooke’s La

Dilatation: Bulk Modulus

S'earing Strain

"xam*le 2%,+

0elation Among "3 ν3 and 1

Sam*le rolem 2%4

5om*osite Materials

Saint-6enant’s rinci*le

Stress 5oncentration: Hole

Stress 5oncentration: (illet

"xam*le 2%,2"lasto*lastic Materials

lastic De!ormations

0esidual Stresses

"xam*le 2%,/3 2%,43 2%,7

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MECHANICS OF MATERIALST h i r d Beer + Johnston + e!ol"

-

Stress & Strain: Axial Loading

8 Suitailit# o! a structure or mac'ine ma# de*end on t'e de!ormations int'e structure as ell as t'e stresses induced under loading% Statics

anal#ses alone are not su!!icient%

8 5onsidering structures as de!ormale allos determination o! memer

!orces and reactions 'ic' are staticall# indeterminate%

8 Determination o! t'e stress distriution it'in a memer also re9uires

consideration o! de!ormations in t'e memer%

8 5'a*ter 2 is concerned it' de!ormation o! a structural memer underaxial loading% Later c'a*ters ill deal it' torsional and *ure endingloads%

T




MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"

- /

Normal Strain

strainnormal

stress

==

==

L

A

P

δ ε

σ

L

A

P

A

P

δ ε

σ

=

==2

2

L L

A

P

δ δ ε

σ

==

=

2

2

MECHANICS OF MATERIA ST





- 0

Stress-Strain Test

MECHANICS OF MATERIALST





- 1

Stress-Strain Diagram: Ductile Materials






- 2

Stress-Strain Diagram: rittle Materials



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- 3

Hoo!e"s La#: Modulus o$ Elasticit%

8 Belo t'e #ield stress

"lasticit#o! Modulus

or Modulusoungs==

E

E ε σ

8 Strengt' is a!!ected # allo#ing3

'eat treating3 and manu!acturing

*rocess ut sti!!ness ;Modulus o!

"lasticit#< is not%





- 4

Elastic s' Plastic e(aior

8 .! t'e strain disa**ears 'en t'e

stress is remo$ed3 t'e material is

said to e'a$e elastically%

8 ='en t'e strain does not return

to ero a!ter t'e stress is

remo$ed3 t'e material is said to e'a$e plastically%

8 T'e largest stress !or 'ic' t'is

occurs is called t'e elastic limit %





- 56

)atigue

8 (atigue *ro*erties are s'on onS-N diagrams%

8 ='en t'e stress is reduced elo

t'e endurance limit 3 !atigue!ailures do not occur !or an#

numer o! c#cles%

8 A memer ma# !ail due to fatigue

at stress le$els signi!icantl#

elo t'e ultimate strengt' i!

su>ected to man# loading c#cles%





- 55

De$ormations *nder Axial Loading

AE

P

E E === σ

ε ε σ

8 (rom Hooke’s La:

8 (rom t'e de!inition o! strain:

L

δ ε =

8 "9uating and sol$ing !or t'e de!ormation3

AE

PL=δ

8 =it' $ariations in loading3 cross-section or

material *ro*erties3

∑=i ii

ii

E A

L P δ





- 5

Exam+le ,'.

Determine t'e de!ormation o!

t'e steel rod s'on under t'e

gi$en loads%

in%7,?%+ in%+@%, *si,+2

7

==×= −

d D E

SL)T.N:

8 Di$ide t'e rod into com*onents at

t'e load a**lication *oints%

8 A**l# a !ree-od# anal#sis on

eac' com*onent to determine t'einternal !orce

8 "$aluate t'e total o! t'e com*onent

de!lections%





- 5

SL)T.N:

8 Di$ide t'e rod into t'ree

com*onents:

22,

2,

in%+

in%,2

==

==

A A

L L

2C

C

inC%+

in%,7

=

=

A

L

8 A**l# !ree-od# anal#sis to eac'

com*onent to determine internal !orces3

l,+C+

l,+,4

l,+7+

CC

C2

C,

×=

×−=

×=

P

P

P

8 "$aluate total de!lection3

( ) ( ) ( )

in%,+%@4

C%+

,7,+C+

%+

,2,+,4

%+

,2,+7+

,+2

,

,

C

CCC

7

C

CC

2

22

,

,,

−×=

×+×−+×

×

=

++=∑=

A

L P

A

L P

A

L P

E E A

L P

i ii

iiδ

in%,+%@4 C−×=δ




MECHANICS OF MATERIALSThi r d Beer + Johnston + e!ol"

- 5/

Sam+le Pro/lem ,'.

T'e rigid ar BDE is su**orted # tolinks AB and CD%

Link AB is made o! aluminum ; E @+

1a< and 'as a cross-sectional area o! 4++

mm2% Link CD is made o! steel ; E 2++

1a< and 'as a cross-sectional area o!;7++ mm2<%

(or t'e C+-kN !orce s'on3 determine t'e

de!lection a< o! B3 < o! D3 and c< o! E %

SL)T.N:

8 A**l# a !ree-od# anal#sis to t'e ar

BDE to !ind t'e !orces exerted #

links AB and DC %

8 "$aluate t'e de!ormation o! links

AB and DC or t'e dis*lacements o! B and D%

8 =ork out t'e geometr# to !ind t'e

de!lection at " gi$en t'e de!lections

at B and D%

MECHANICS OF MATERIALST h




- 50

Dis*lacement o! B:

( )( )

( )( )

m,+4,/

a,+@+m,+4++

mC%+ N,+7+

7

27-

C

−×−=

××

×−=

= AE

PL Bδ

↑= mm4,/%+ Bδ

Dis*lacement o! D:

( )( )( )( )

m,+C++

a,+2++m,+7++m/%+ N,++

7

27-

C

−×=

×××=

= AE

PL Dδ

↓= mmC++%+ Dδ

(ree od#: Bar BDE

( )

( )

ncompressio F

F

tension F

F

M

AB

AB

CD

CD

B

kN7+

m2%+m/%+kNC+++M

kN+

m2%+m7%+kNC++

+

D

−=

×−×−==

+=

×+×−=

=

∑

∑

SL)T.N:

Sam+le Pro/lem ,'.





- 51

Dis*lacement o! D:

( )

mm@%@C

mm2++

mm+%C++

mm4,/%+

=

−=

=′′

x

x

x

HD

BH

D D

B B

↓= mm2?%, E δ

( )

mm2?%,mm@%@C

mm@%@C/++

mmC++%+=

+=

=′′

E

E

HD

HE

D D

E E

δ

δ

Sam+le Pro/lem ,'.




MECHANICS OF MATERIALShi r d Beer + Johnston + e!ol"

Pro/lem ,'.-,'0





Pro/lem ,'1-,'2





Pro/lem ,'3-,'4



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Pro/lem ,'.-,'.,

2 axial loading

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