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Mechanics of Deformable Bodies: Stress & Strain

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INTRODUCTION TO MECHANICS OF
DEFORMABLE BODIES
M E C H A N IC S M A Y B E D E F IN E D A S T H E S C IE N C E W H IC H
C O N S ID E R S T H E E F F E C T S O F F O R C E S O N R IG ID B O D IE S
M E C H A N IC S O F D E F O R M A B LE B O D IE S
D E A LS W IT H R E LA T IO N S B E T W E E N E X T E R N A LLY A P P L IE D
LO A D S A N D T H E IR IN T E R N A L E F F E C T S O F B O D IE S
S T U D IE S H O W M A T E R IA LS B E H A V E U N D E R D IF F E R E N T
T Y P E S O F LO A D
U N D E R S T A N D IN G T H E S T R A IN S T H A T D E V E LO P W IT H IN
M A T E R IA LS
T H E P R IN C IP LE S A N D C O N C E P T O F S T R ES S ES A N D
D E F LE C T IO N S S H A LL B E U S E D F O R T H E S A F E D E S IG N O F
S T R U C T U R ES T H AT A R E C AP A B LE O F S U P P OR T IN G T H E IR
IN T E N D E D LO A D S
F O U R D IF F E R E N T T Y P E S O F LO A D I N G W H IC H R E S U LT IN
D IF F E R E N T T Y P E S O F S T R E S S :
A X IA L F O R C E ( P )
M E A S U R ES T H E P U LL IN G ( T E N S I LE F O R C E ) O R P U S HIN G
( C O M P R E S IV E F O R C E ) A C T IO N O V E R T H E S E C T IO N
SHEAR FORCE ( )
C O M P O N E N TS O F T H E T O T A L R E S IS T A N C E T O S LID IN G
T H E P O R T IO N T O O N E S ID E O F T H E E X LO R A T O R Y
S E C T IO N P A R T T H E O T H E R
T H E F O R C E T H A T T R IE S T O C U T O R T E A R A M A T ER IA L
B Y M A K IN G O N E S ID E R E LA T IV E T O T H E O T H E R
B E N D IN G M O M E N T S ( M )
M E A S U R ES T H E R E S IS T A N C E T O B E N D IN G T H E M E M B E R
ABOUT THE Y OR Z AXES
S A G G IN G IS A P O S IT IV E B E N D IN G M O M E N T W H E N A
B E A M B E N D S IN A ∪ S H A P E
H A G G O IN G IS A N E G A T IV E B E N D IN G M O M E N T W H E N A
B E A M B E N D S IN A ∩ S H A P E
TORQUE (T)
M E A S U R ES T H E R E S IS T A N C E T O TW IS T IN G T H E
MEMBER
LESSON 1: SIMPLE STRESS
S IM P LE S T R E S S IS T H E IN T E R N A L R E S IS T A N C E O F A
M A T E R IA L P E R U N IT A R E A W H E N S U B J EC T E D T O AN
EXTERNAL FORCE
IT R E P R E S E N TS TH E IN T E N S IT Y O F F O R C E A C T IN G W IT H IN A
M A T E R IA L T O R E S IS T D E F O R M A T IO N . O R T H E U N IT S T R E G N T H
O F M A T E R IA L
IT C A N B E D E F IN E D A S :
WHERE:
P
=A
STRESS
P = APPLIED LOAD
A = CROSS-SECTIONAL
AREA
B A S IC T Y P E S O F S IM P LE S T R E S S :
1 . N O R M A L S T R E S S ( A X IA L F O R C E )
2 . S H E A R IN G S T R E S S ( S H E AR F O R C E )
3 . B E A R IN G S T R E S S ( A X IA L F O R C E )
NORMAL STRESS
T H E IN T E N S IT Y O F T H E F O R C E A C TIN G N O R M A L T O A N
A R E A IS D E F IN E D A S T H E N O R M A L S T R ES S
IF T H E N O R M A L F O R C E OR S TR E S S P U LLS O N A N A R E A
IT IS R E F E R R E D T O A S T E N S ILE S T R E S S
IF P U S H E S O N A N A R E A IT IS C A LL E D C O M P R E S S IV E
S T R ES S
SIMPLE STRESS
S H E A R IN G S T R E S S
IT IS C A U S E D B Y F O R C E S AC T IN G A LO N G O R P A R A LLE L
T O T H E A R E A R E S IS T IN G T H E F O R C E S
IT IS A LS O K N O W N A S T A N G E N T IA L S T R E S S A S IT A C T S
A LO N G T H E S U R F A C E
F O R M U LA C A N B E D E F IN E D A S :
WHERE:
V
=A
Z
SHEAR STRESS
V = SHEAR FORCE
A = CROSS-SECTIONAL
AREA
T Y P E S O F S H E A R IN G :
1 . S IN G LE S H E A R
F O R C E S AC T S O N A S IN G LE C R O S S -S E C T IO N A L
A R E A O F A M A T ER IA L
F O R M U LA :
V
=A
Z
2 . D O U B LE S H E A R
F O R C E S AC T S O VE R TW O C R O S S - S E C T IO N A L
AREA
F O R M U LA :
V
= 2A
Z
SIMPLE STRESS
3 . P U N C H IN G S H E A R
W H E N A T H IN M A T E R IA L IS P R E S S E D B Y A
C O N C E N T R A T E D LO A D
F O R M U LA :
WHERE:
V
= C
SHEAR STRESS
V = SHEAR FORCE
THICKNESS OF MATERIAL
C = CIRCUMFERENCE OF THE
B E A R IN G S T R E S S
A C O N T A C T P R ES S U R E B E TW E E N S E P A R A T E B O D IE S
W H E N O N E P R ES S E S A G A IN S T T H E O T H E R
F O R M U LA :
P
= Ab
WHERE:
BEARING STRESS
Pb = BEARING FORCE
Ab = CROSS-SECTIONAL
AREA
*NOTE THAT Ab = D WHERE;
THICKNESS OF THE
MATERIAL
D = DIAMETER OF THE HOLE
OR PIN
LESSON 2: SIMPLE STRAIN
S T R A IN D E S C R IB E S H O W A M A T ER IA L D E F O R M S W HE N
S U B J EC T E D T O EX T E R N A L F O R C E S . IT Q U A N T IF IE S T H E
R E LA T IV E C H A N G E IN S H A P E O R S IZE O F A B U D Y D U E T O
A P P LIE D S T R E S S
IN O R D E R T O D E S C R IB E T H E D E F O R M A T IO N O F A B O D Y B Y
C H A N G E S IN LE N G T H S A N D S H AP ES , T H E C O N C EP T O F S T R A IN
IS D E V E LO P E D
U N D E R T H E C O ND IT IO N S B E LO W , T H E S T R A IN M A Y B E
A S S U M E D C O N S TA N T A N D V A LU E C O M P U T E D:
1 . T H E M A T ER IA L H A S A C O NS T A NT C R OS S S E C T IO N A L A R E A
2 . T H E M A T ER IA L M U S T B E H O M O GE N E O U S ( S A M E )
3 . T H E LO A D M U S T B E A X IA L, P R O D U C E U N IF O R M S T R E S S
IT C A N B E D E F IN E D A S :
WHERE:
=L
STRAIN
DEFORMATION
L = ORIGINAL LENGTH
A C C O R D IN G T O R O B E R T H O O K E T H E R E LA T IO N S H IP B E T W E E N
S T R ES S A N D S T R A IN A R E D IR E C T LY P R OP O R T IO N A L
WHERE:
=E
= STRESS
= STRAIN
= YOUNG’S MODULUS OF
ELASTICITY
(MATERIAL PROPERTY THAT MEASURES
STIFFNESS)
SIMPLE STRAIN
S T R ES S S T R A IN D IA G R A M
P R O P OR T IO N A L L IM IT
T H E M A X IM U M P O IN T W H E R E S TR E S S A N D S T R A IN
R E M A IN D IR E C T LY P R OP O R T IO N A L
B E Y O N D T H IS T H E R E LA T IO N S H IP B E C O M E S N O N L IN E A R
E LA S T IC L IM IT
T H E M A X IM U M S T R E S S A M A T ER IA L C A N E D U R E
W IT H O U T P E R M A N E N T D E F O R M A T IO N
B E Y O N D T H IS T H E S T R ES S A P P LIE D W IL L R E S U LT I N T H E
M A T E R IA L N O T R E T U R N IN G T O IT S O R IG IN A L S H A P E
A N D S IZE W H E N S T R E S S IS U N LO A D E D . IT W IL L R E T A IN
A P E R M A N E N T D EF O R M A T IO N C A LLE D P E R M A N E N T S E T
Y IE LD P O IN T
P LA S T IC D E F O R M A T IO N B E G IN S
T H E R E IS A N A P P R E C IA B LE E LO N G A T IO N O R Y IE LD I N G
O F T H E M A T E R IA L W IT H O U T A N Y C O R R ES P O N D IN G
IN C R E A S E O F LO A D ( P A R T IC U LA R LY S E E N IN S T E E LS )
U LT IM A T E S T R E NG T H
T H E H IG H E S T S T R E S S A M A T E R IA L C A N W IT H S T A N D
B E F O R E N E C K IN G ( A S IG N IF IC A N T R E D U C T IO N IN C R O S S
S E C T IO N A R E A ) B E G IN S
R U P T U R E S TR E N GT H
T H E S T R ES S A T F A ILU R E C O N S ID E R IN G T H E O R IG IN A L
D IM E N S IO N S O F T H E C R OS S S E C T IO N A L A R E A O F T H E
M A T E R IA L
SIMPLE STRAIN
A C T U A L R U PT U R E S T R E N G T H
T H E S T RE S S AT FA I L U R E C O NS I DE R I N G T H E C R O S S S E C T I O N A L A R E A A T R UP T U R E P O IN T
S T A T I C A L L Y I N DE T E R M I N AT E A XI A L L O A D E D M E M B E R S
I N S T R U CT U R A L A N A L Y S IS , A X IA L LY LO A D E D M E M B E R S
A R E E L E M E NT S T H A T E X PE R I E NC E FO R C E S A L O N G T H E I R
L E N G T H , S U C H AS R O DS , B A RS , AN D C O L U M N S . WH E N
A N A L Y Z I N G T H E S E M E MB E RS , WE T Y PI C A L L Y US E
E Q U IL IB R IU M E Q U A T IO N S T O D E T E R M I N E T H E I N T E R N A L
FO R C E S A N D R E A C T I O N S .
H O W E VE R , I N S O M E C A S E S , E Q U IL I B R I U M E Q U A T IO N S
A L O N E A RE N O T E N O U G H T O FU L L Y D E T E R M I N E T H E
FO R C E S I N T H E S T R U C T U R E . S U CH C A S E S A R E C AL L E D
S T A T IC A L LY IN D E T E R M IN A T E S Y S T E M S B E C A U S E
A D D I T I O N AL I N FO R M A T I O N IS NE E D E D T O S O L VE FO R T H E
UNKNOWNS.
T H E RE A S O N E Q U I L I B RI U M E Q UA T I O NS A R E I NS U F FI C I E N T
I S T H A T T H E S T RU C T U R E H AS M O R E U N K N O W N S T H A N
E Q U A T I O N S . T H IS O FT E N H A PPE N S W H E N :
A B A R I S FI X E D A T M U L T I PL E PO I N T S (E . G . , C L A MPE D
AT BOTH ENDS).
E X T R A S U PPO R T S O R RE D U N D A N T C O N S T R A I N T S A R E
PR E S E N T .
T O S O L VE FO R T H E U N K N O W N FO R C E S , WE N E E D
A D D I T I O N AL E Q U A T I O N S B E Y O ND E Q U IL I B R I U M . T H E S E
A D D I T I O N AL E Q U A T I O N S C O M E FR O M T H E C O NC E P T O F
C O M P A T IB I L IT Y , W H I C H E N S U RE S T H A T T H E S T R UC T U R E
D E FO R M S I N A WA Y T H A T I S PH YS I C A L L Y PO S S IB L E .
C O M P A T IB I L IT Y E Q U A T IO N S E X PR E S S T H E FA C T T H A T
D I F FE R E N T PO I N T S I N T H E S T R U CT U R E M U S T M O VE
T O G E T H E R C O NS IS T E N T LY . T H E S E E Q U A T IO N S A RE D E R I V E D
FR O M LO A D - D IS P LA C E M E N T R E LA T IO N S H IP S , W H I C H
D E S C R I B E H O W FO R C E S C A U S E D E FO R M A T I O N S IN T H E
MATERIAL.
SIMPLE STRAIN
THERMAL STRESS
T H E R M A L S T R E S S IS T H E IN T E R N A L S T R E S S T H AT
D E V E L O PS W I T H I N A M A T E R I AL W H E N IT E X PE R IE N C E S
T E M P E R A T UR E C H A N G E S B U T I S R E S T RI C T E D FR O M
E X PA N D I N G O R C O N T R A C T I N G
W H E N T E M PE R A T U R E R IS E S , M AT E R I AL S N AT U R A L L Y
E X PA N D . W H E N T E M PE R A T U R E DR O PS , M A T E R I AL S
CONTRACT.
I F T H E M A T E R I AL I S A L L O W E D T O E X PA N D O R C O N T R A C T ,
N O S T RE S S IS DE V E L O PE D . H O W E V E R , I F I T S M O VE M E N T I S
R E S T RI C T E D B Y E X T E R N AL S U PP O R T S O R S U R R O U N D I N G
S T R U C T U RE S , I NT E R N A L FO R C E S D E V E L O P T O R E S I S T T H E
D E FO R M A T I O N .
I T C A N B E D E FI NE D A S :
WHERE:
𝜎T = E∝ ∆T
𝜎T
= THERMAL STRESS
E = YOUNG’S MODULUS OF
ELASTICITY
= COEFFICIENT OF THERMAL
∝
∆T
T H E R M A L D E FO R M A T I O N
T H E R M A L D E FO R M A T I O N R E FE RS T O T H E C H A N G E IN T H E
S H A P E , S IZE , O R D IM E N S IO N S O F A M A T E R I A L D U E T O
T E M PE R A T U R E VA R I A T I O NS . T H I S O C C U R S B E C A U S E T H E
A T O M S I N T H E MA T E R I A L M O V E M O R E O R L E S S
D E PE N D I N G O N T E M PE R A T U R E , AF FE C T I N G T H E O V E R A L L
S T R U C T U RE .
I T C A N B E D E FI NE D A S :
WHERE:
𝛿 T = ∝ L∆T
𝛿T
= CHANGE IN LENGTH
L
∝ = ORIGINAL LENGTH
∆T= C O E F F I C I E N T O F T H E R M A L
EXPANSION
= CHANGE IN TEMPERAT URE
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