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Mechanics of Material-IUET LAHORE B.Sc Mechanical EngineeringSemester # 3
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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*
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© 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"
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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
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
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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
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 0
Stress-Strain Test
MECHANICS OF MATERIALST
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 1
Stress-Strain Diagram: Ductile Materials
MECHANICS OF MATERIALST
7/17/2019 2 Axial Loading
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© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 2
Stress-Strain Diagram: rittle Materials
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 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%
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 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 %
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 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$els signi!icantl#
elo t'e ultimate strengt' i!
su>ected to man# loading c#cles%
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 55
De$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' $ariations in loading3 cross-section or
material *ro*erties3
∑=i ii
ii
E A
L P δ
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 5
Exam+le ,'.
Determine t'e de!ormation o!
t'e steel rod s'on under t'e
gi$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%
MECHANICS OF MATERIALST
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MECHANICS OF MATERIALSTh i r d Beer + Johnston + e!ol"
- 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 MATERIALST
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MECHANICS OF MATERIALSThi r d Beer + Johnston + e!ol"
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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
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MECHANICS OF MATERIALSThi r d Beer + Johnston + e!ol"
- 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 ,'.
MECHANICS OF MATERIALST h
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MECHANICS OF MATERIALSThi r d Beer + Johnston + e!ol"
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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 MATERIALST h
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MECHANICS OF MATERIALShi r d Beer + Johnston + e!ol"
Pro/lem ,'.-,'0
MECHANICS OF MATERIALST h
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MECHANICS OF MATERIALShi r d Beer + Johnston + e!ol"
Pro/lem ,'1-,'2
MECHANICS OF MATERIALST h
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Pro/lem ,'3-,'4
MECHANICS OF MATERIALST h