chapter 1_stress and strain

8/19/2019 Chapter 1_stress and Strain

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STRESS ANDSTRAIN

http://www.youtube.com/watch?v=S6gu7bzvMqQDAM 21! " M#$A%&$ '#'#(A)*

http://../VISUAL/Stress%20and%20Strain.flvhttp://../VISUAL/Stress%20and%20Strain.flv


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Topics:

• Introduction• Main Principles of Static's

Stress

• Normal Stress• Shear Stress• Bearing Stress• Thermal Stress

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1.1 Introduction

Mechanics : The study of how bodies react to forces acting on them

RIGID BODIES

Things that do not change shape!

Statics : The study of bodies

in an e"uilibrium

DEFORMABLE BODIES

Things that do change shape!FLUIDS

Mechanics of Materials :The study of the relationships

between the external loads

applied to a deformable body and

the intensity of internal forces

acting within the body#

Incompressible $ompressible

%ynamics :

Kinematics concerned

with the geometric aspects

of the motion

(# Kinetics concerned

with the forces causing the

motion#

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1.2 Main Principles of Statics

+ A ,oa- may be -e0e a the comb0e- eect o ete30a, o3ce

act0g o0 the bo-y.

Point Load:A po0t ,oa- o3 co0ce0t3ate- ,oa- o0e whch co0-e3e- to actat a po0t. &0 actua, p3actce4 the ,oa- ha to be -t3bute- ove3ma,, a3ea4 becaue uch ma,, 50e+e-ge co0tact a3e ge0e3a,,y

0ethe3 pob,e 0o3 -e3ab,e.

Distributed Load:A -t3bute- ,oa- o0e whch -t3bute- o3 p3ea- 0 omema00e3 ove3 the ,e0gth o the beam. & the p3ea- u0o3m ".e.at the u0o3m 3ate4 ay w 5% o3 %/mete3 3u0 * t a- to beu0o3m,y -t3bute- ,oa- a0- abb3evate- a D). & thep3ea- 0ot at u0o3m 3ate4 t a- to be 0o0+u0o3m,y-t3bute- ,oa-. 3a0gu,a3 a0- t3apezo-a,,y -t3bute- ,oa-a,, u0-e3 th catego3y.

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Point Load Uniform Load Non-uniformlyload

Figure

Free bodydiagram

(FBD)

w 5%w 5%/m w 5%/m

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Axial Load Normal Stress Shear Stress

Bearing Stress Alloable Stress De!ormation o! Stru"tural under Axial Load Stati"ally indeterminate #roblem

Thermal Stress

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$e"hani"s o! material is a study o! the

relationshi# beteen the external loads a##liedto a de!ormable body and the intensity o!internal !or"es a"ting ithin the body%

Stress & the intensity o! the internal !or"e on as#e"i'" #lane (area) #assing through a #oint%

Strain & des"ribe the de!ormation by "hangesin length o! line segments and the "hanges inthe angles beteen them

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9/741.1 Introduction

• Normal Stress : stress which acts perpendicular) or normal to) the

*! cross section of the load+carrying member#

: can be either compressi,e or tensile#• Shear Stress : stress which acts tangent to the cross section of

-! the load+carrying member# : refers to a cutting+li.e action#

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Normal Stress σ the intensity o! !or"e or !or"e #er unit area a"tingnormal to ∆A

A positive sign ill be used to indi"ate a tensile stress (member in tension)

A negative sign ill be used to indi"ate acompressive stress (member in compression)

= P / A

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"a*

"b*

Stress ( ) = Force (P)

Cross Section (A)

Unit : Nm-² N/mm( or MPa

N/m( or Pa

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13/741.4 Axial Loading – Normal Stress

Assumptions :

0niform deformation: Bar

remains straight before and

after load is applied) and

cross section remains flat or plane during deformation

(# In order for uniform

deformation) force be

applied along centroidal a1isof cross section $

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A P

A P

A F F F A

!

=

=

=Σ=↑+ ∫ ∫

σ

σ

σ dd2

" 3 a,erage normal stress at any point

on cross sectional area

P 3 internal resultant normal force

A 3 cross+sectional area of the bar

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• 0se e"uation of " 3 P / A for cross+sectional area of a member when

section sub4ected to internal resultant force

Internal Loading

• Section member perpendicular to its longitudinal a1is at ptwhere normal stress is to be determined

• %raw free+body diagram• 0se e"uation of force e"uilibrium to obtain internal a1ial

force at the section

• %etermine member5s 1+sectional area at the section• $ompute a,erage normal stress " 3 P / A

Average Normal Stress

1.4 Axial Loading – Normal StressDAM 21! " M#$A%&$ '#'#(A)*


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8ee3e0ce: http://e0.w5pe-a.o3g/w5/De,ta9",ette3*

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Exam#le %*

To solid "ylindri"al rods AB and B+ are elded together at Band loaded as shon% ,noing that d)&-.mm and

d/&/.mm 'nd a0erage normal stress at the midse"tion o!*(a) rod AB1 and

(b) rod B+%

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Exam#le %/

To solid "ylindri"al roads AB and B+ are elded

together at B and loaded as shon% ,noing thatd) & -. mm and d/ & 2. mm 'nd the a0erage

normal stress in the mid se"tion o!*

(a) rod AB1 and

(b) rod B+%

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#amp,e 1.!

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Normal strain ε is the elongation or"ontra"tion o! a line segment #er unito! length

∆L & elongation

Lo & length

ε = ") )o* / )o ∆) / )o

strainnormal

6

=δ

=ε

; ∆)= δ

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Exam#le %3*Determine the "orres#onding strain !or a bar o!

length L&.%4..m and uni!orm "ross se"tionhi"h undergoes a de!ormation δ&2.×.54m%

7

7

7

&89 &9 m(89 &9 m m

6 9 799 m

(89 &9 (89

/.

@

−

−

−

δ ×ε = = = ×

= × µ

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#amp,e 1.< :A tee, 3o- 2 mm -amete3 a0- m,o0g ub>ecte- to a0 aa, pu,, o 5%. &t ete0-e- by 2.


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So,uto0 :Damete3 o tee, 3o-4 - = 2 mm @ .2 m

)e0gth4 , = m'3eu3e4 ' = 5%#te0o04 -, = 2.


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Tensile test is an ex#eriment to determinethe load5de!ormation beha0ior o! thematerial%

Data !rom tensile test "an be #lot into stressand strain diagram%

Exam#le o! test s#e"imen

5 note the dog5bone geometry

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6ni0ersal Testing $a"hine 5 e7ui#mentused to sub8e"t a s#e"imen to tension

"om#ression bending et"% loads andmeasure its res#onse

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Stress5Strain Diagrams

A number o! im#ortant me"hani"al #ro#erties o! materials that "an be dedu"ed !rom the stress5strain

diagram are illustrated in 'gure abo0e%

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!

'o0t C+A = ,0ea3 3e,ato0hp betwee0 t3e a0- t3a0

'o0t A = p3opo3to0a, ,mt "σ')*he 3ato o t3e to t3a0 0 th ,0ea3 3ego0 o t3e+t3a0 -ag3am ca,,e- ou0g Mo-u,u o3 the Mo-u,u o #,atcty gve0

ε

σ

∆

∆Ε

σ E σ')0t: M'aDAM 21! " M#$A%&$ '#'#(A)*


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!

'o0t A+F = pecme0 beg0 ye,-0g.

'o0t F = ye,- po0t'o0t F+G = pecme0 co0t0ue to e,o0gate wthout a0y 0c3eae 0 t3e. &t3ee3 a pe3ect,y p,atc zo0e'o0t G = t3e beg0 to 0c3eae'o0t G+D = 3ee3 a the zo0e o t3a0 ha3-e00g'o0t D = u,tmate t3e/t3e0gth pecme0 beg0 to 0ec5+-ow0

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9oint : to A

9oint + to D

9oint D to E

At #oint E

Normal or engineering stress "an be determined by di0iding thea##lied load by the s#e"imen original "ross se"tional area%

True stress is "al"ulated using the a"tual "ross se"tional area atthe

instant the load is measured%

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Some o! the materials li;e aluminum (du"tile) does not ha0e"lear

yield #oint li;es stru"tural steel% There!ore stress 0alue "alled theo


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Brittle material su"h as "erami" and glass ha0e lotensile

stress 0alue but high in "om#ressi0e stress% Stress5straindiagram !or brittle material%

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#amp,e 1 6 :


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#amp,e 1.6 :

he mm -amete3 cab,e FG ma-e o a tee, wth #=2H'a.$0ow0g that the mamum t3e 0 the cab,e mut 0ot ecee-1BM'a a0- that the e,o0gato0 o the cab,e mut 0ot ecee-6mm4 0- the mamum ,oa- ' that ca0 be app,e- a how0.

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Elasticity re!ers to the #ro#erty o! a material su"hthat it returns to its original dimensions a!ter

unloading% Any material hi"h de!orms hen sub8e"ted to load

and returns to its original dimensions hen unloaded

is said to be elasti"%

I! the stress is #ro#ortional to the strain the materialis said to be linear elastic otherise it is non-linear

elastic% Beyond the elasti" limit some residual strain or permanent strains ill remain in the material u#on

unloading% The residual elongation "orres#onding to the

#ermanent strain is "alled the permanent set %

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I he amou0t o t3a0 whch 3ecove3e- upo0 u0,oa-0g ca,,e- the e,a1t/c 3ecove3y.

!


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@hen an elasti" homogenous and isotro#i" materialis sub8e"ted to uni!orm tension it stret"hes axially

but "ontra"ts laterally along its entire length% Similarly i! the material is sub8e"ted to axial

"om#ression it shortens axially but bulges outlaterally (sideays)%

The ratio o! lateral strain to axial strain is a "onstant

;non as the 9oissons ratio

here the strains are "aused by uniaxial stress

only

axial

lateral

$ε

ε −=

!6

pa.si 1

sisi y

6

6

b d

b d

@

@

δε ε =

δ δε ε = − = −

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E l


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Exam#le %

A . "m diameter steel rod is loaded ith C4/ ;Nby tensile !or"es% ,noing that the E&/. 9a and

ν& .%/ determine the de!ormation o! rod

diameter a!ter being loaded%Solutionσ in rod σ &

Lateral strain

∴

!

MPa

m

N x

A

p;#&9=

!<

&

&9>7(

((

:

==

π

9998:#9&9(9;

;#&9=

:

===

MPa x

MPa

% a

σ ε

#"$"#$ 9998:9(=9al

ε ν ε

999&8


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Exer"ises % A steel #i#e o! length L&%/ m outside diameter d/&2.mm and

inside diameter d)&.mm is "om#ressed by an axial !or"e 9&

4/.;N%The material has modulus o! elasti"ity E& /..9a and

9oissons Ratio v & .%-.%Determine *

a) the shortening G (ans* 5.%322 mm)

b) the lateral strainH lateral (ans* -%x.54)

") the in"rease d/ in the outer diameter and the in"rease d)

in

the inner diameter

(ans* .%. mm and .%./2mm)

d) the in"rease t in the all thi";ness

(ans* .%..//C mm)

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/% A hollo "ir"ular #ost AB+ as shon in Figure / su##orts a load9)&%2 ;N a"ting at the to#% A se"ond load 9/ is uni!ormly

distributed around the "a# #late at B% The diameters andthi";nesses o! the u##er and loer #arts o! the #ost are dAB&-/

mm tAB& /mm dB+ 2 mm and tB+&mm res#e"ti0ely% a) +al"ulate the normal stress JAB in the u##er #art

o! the #ost% (ans* %2 $9a)

b) I! it is desired that the loer #art o! the #ost

ha0e the same "om#ressi0e stress as the u##er

#art hat should be the magnitude o! the load 9/K

(ans * 9/&4;N)

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- A standard tension test is used to determine the


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-% A standard tension test is used to determine the#ro#erties o! an ex#erimental #lasti"% The tests#e"imen is a 2 mm diameter rod and it issub8e"ted to a -%2 ;N tensile !or"e% ,noing that an

elongation o! mm and a de"rease in diameter o!.%4/ mm are obser0ed in a /. mm gage length%Determine the modulus o! elasti"s the modulus o!rigidity and 9oissons ratio o! the material%

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A !or"e a"ting parallel or tangential to a se"tion ta;enthrough a material (i%e% in the plane o! the material) is

"alled a shear force The shear !or"e intensity i%e% shear !or"e di0ided by thearea o0er hi"h it a"ts is "alled the average shear stress, τ

τ & shear stress

V & shear !or"e

A & "ross5se"tional area

Shear stress arises as a result o! the dire"t a"tion o! !or"estrying to "ut through a material it is ;non as dire"t shear

!or"e

Shear stresses "an also arise indire"tly as a result o!

tension torsion or bending o! a member%

A

' =τ

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De#ending on the ty#e o! "onne"tion a "onne"tingelement (bolt ri0et #in) may be sub8e"ted tosingle shear or double shear as shon%

Ri0et in Single Shear

<

(d

P

A

'

π

τ ==

2

Ri0et in Double Shear

((

(

!


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A

F

A

P ==a,eτ

Single Shear

A

F

A

P

(

a,e ==τ

%ouble Shear

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The e


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Bearing stress is also ;non as a "onta"t stress

Bearing stress in sha!t ;ey1

Bearing stress in ri0et and #lat1

r(L

M

L(

r M

A

P

)

)

(

!(===σ

td

P

)

=σ

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6

So,uto0 :

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It also ;non as Shear $odulus o! Elasti"ity or the$odulus o! Rigidity%

alue o! shear modulus "an be obtained !rom thelinear region o! shear stress5strain diagram%

The modulus young (E) #oissons ratio( ν) and themodulus o! rigidity () "an be related as

γ τ * = 0t : 'aca, o3 'a o3%/m2

!&( ν += % *

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Be"ause o! the "hange in the dimensions o! a bodyas a result o! tension or "om#ression the 0olume

o! the body also "hanges ithin the elasti" limit% +onsider a re"tangular #arallel #i#ed ha0ing sidesa b and " in the x y and M dire"tions res#e"ti0ely%


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The tensile !or"e 9 "auses an axial elongation o! aε and lateral "ontra"tions o! b νε and " νε in the x y andM dire"tions res#e"ti0ely% en"e

Initial 0olume o! body o & ab"

Final 0olume ! & (a P aε)(b 5 b νε)(" 5 " νε)& ab"( P ε)( 5 νε)/

&0ta,bo-y


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Ex#anding and negle"ting higher orders o! ε (sin"e ε is0ery small)

Final 0olume ! & ab"( P ε 5 / νε)

+hange in 0olume

∆ & Final olume 5 Initial olume& ab"( P ε 5 / νε) 5 ab"& ab"( P ε 5 / νε 5 )& ab"(ε 5 / νε)& o ε( 5 / ν)

en"e

6

!(&

!(&

ν σ

ε

ν ε

−=

−=∆

%

'

'

o

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Isotro#i" material is sub8e"ted to general triaxialstress σx σy and σM%

Sin"e all strain satis!y ε QQ so ε0 & εx P εy P εM

εx &

εy &

εM &

[ ]!& + x %

σ σ ν σ +−

[ ]!& x + %

σ σ ν σ +−

[ ]!& + x %

σ σ ν σ +−

!(&

+ x$ %

σ σ σ ν

ε ++−

=

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#amp,e 2


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#amp,e 2.

A tta0um a,,oy ba3 ha the o,,ow0g o3g0a, -me0o0: =1cm y = cm a0- z = 2cm. he ba3 ub>ecte- to t3ee σ

= 1 % a0- σ y = + 6 %4 a 0-cate- 0 gu3e be,ow. he3ema00g t3ee "σz4 τy4 τz a0- τ yz* a3e a,, ze3o. )et # = 16 5%a0- ν = .!! o3 the tta0um a,,oy.

"a*Dete3m0e the cha0ge 0 the ,e0gth o3

∆4 ∆ y a0- ∆z.

"b* Dete3m0e the -,atato04 εv.

z

y

1 %1 %

6 %

6 %

62-/,atat/o0 = pe0gemba0ga0DAM 21! " M#$A%&$ '#'#(A)*


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A##lied load that is less than the load the member "an !ullysu##ort% (maximum load)

:ne method o! s#e"i!ying the alloable load !or the design oranalysis o! a member is use a number "alled the Fa"tor o! Sa!ety(FS)%

Alloable5Stress Design

allo,

fail

F

F FS =

KS L 1

FS

or

FS

+ield

allo,

+ield

allo,

τ

τ

σ

σ ==

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I! a bar is 'xed at both ends, as shon in

'g% (a) to un;non axial rea"tionso""urs and the !or"e e7uilibriume7uation be"omes1

9P??

29?

@B

y

=−+

=Σ↑+

9B/@ =δ

I &0 th cae4 the ba3 ca,,e- tatca,,y0-ete3m0ate4 0ce the equ,b3umequato0 a3e 0ot uce0t to -ete3m0ethe 3eacto0.

I the 3e,ato0hp betwee0 the o3ce act0g o0the ba3 a0- t cha0ge 0 ,e0gth a3e 50ow0 ao3ce+-p,aceme0t 3e,ato0

I the 3e,atve -p,aceme0t o o0e e0- o the ba3wth 3epect to the othe3 e0- equa, to ze3o

0ce the e0- uppo3t a3e e-. e0ce

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@A

P6)9B/@ =δ=δ 9B@ =δ+δ

@$

$BB@

@$

$BB@

$BB@$@

$BB@$@

6

6??

6

@A

@A

6??

@A

6?

@A

6?

9

@A

6?

@A

6?

=

×=

=

=−

+=

+=

=−

&6

6?P

?6

6?P

6

6??P

@$

$BB

B@$

$BB

@$

$BB

B

I 8ea,z0g that the 0te30a, o3ce 0 egme0t AG NKA4 a0- 0 egme0t GF4

the 0te30a, o3ce KF. he3eo3e4 the equato0 ca0 be w3tte0 a

=

=

+=

+=

66P?

6

6?P

6

66?P

6

6

6

6?P

@$B

@$B

@$

@$$BB

@$

@$

@$

$BB

B@@B ?P?)9P?? −==−+

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#amp,e 2 1:


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#amp,e 2.1:

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#amp,e 2 2:


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#amp,e 2.2:

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#amp,e 2.!:

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@ B

B @

? 9 ? ? (9 &9 N 9 &

? (9 &9 ?

, ( ) ................( )

( )

+ → Σ = − − + =

= −

( ) ( )

B @

@ B

@ @$ B $B

@

( (= ( = (

@ B

9 99&m

9 99&m

? 6 ? 6 9 99&m@A @A

? 9 m9 99&m

9 99(8m (99 &9 Nm 9 99(8m (99 &9 Nm

or

? 9 m =(; 9N (

Substitute e" & o e" (

?

/ .

.

.

( . ) ( . ).

. .

( . ) ( . ) . ................( )

( )int ( )

− −

δ =

δ − δ =

− =

− = π × π ×

− =

@ @

@

9 m =(; 9N

? &7 7.N

?B =.N

( . ) ( , )( . ) .

.

.

− − =

=

=

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A "hange in tem#erature "an "ause material to "hange itsdimensions%

I! the tem#erature in"reases generally a material ex#andshereas i! the tem#erature de"reases the material ill"ontra"t%

I! this is the "ase and the material is homogenous andisotro#i" it has been !ound !rom ex#eriment that thede!ormation o! a member ha0ing a length L "an be"al"ulated using the !ormula1

δ T&α(∆ T)L here

α&linear "oe"ient o! thermal ex#ansion (unit* >+°) ∆ T&"hange in tem#erature L&original length o! the member (m or mm) δ T&"hange in length o! the member

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E " %


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Hve0: α=121+6/G°

Eam!"e #$%:Eam!"e #$%:

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"" i


67/74

???

9?

B@

C

===Σ↑+

So"ution:So"ution:

@B 9=δ he cha0ge 0 ,e0gth o the ba3 ze3o "becaue the uppo3t -o0ot move*

@B T ?( )+ ↑ δ = δ − δ

o -ete3m0e the cha0ge 0,e0gth4 3emove the uppe3 uppo3to the ba3 a0- obta0 a ba3 e- at the bae a0- 3ee to-p,ace at the uppe3 e0-.So the ba3 w,, e,o0gate by a0

amou0t O whe0 o0,ytempe3atu3e cha0ge act0gA0- the ba3 ho3te0 by a0amou0t OK whe0 o0,y the 3eacto0

act0g

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@B T ?

T ?

7

( =

<

( =

< ( =

9

?6T6 9@A

? &&( &9 79 9 & 9

9 9& (99 &9

? & 7 &9

9 9& (99 &9

? 7 &9 9 9& (99 &9

; (.N

( )

( )( )( )

. ( )

( ).

. ( ). . ( )

.

−

−

−

+ ↑ δ = δ − δ

δ − δ =

α∆ − =

× ° − ° − =×

× =

×= × × ×=

(

? ; (.N;(MPa

@ 9 9&

.;

.σ = =

Ave3age 0o3ma, the3ma, t3e:

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y st al? 9 (? ? =9 &9 N 9 e" &, ( ) ......... ( )+ ↑ Σ = + − =

st al

st st T st ?

al al T al ?

st T st ? al T al ?

st al

st al

7 st( =

7

e" (

? 6 ? 6T6 T6

@ A @ A

? 9 (8&( &9 >9 (9 9 (89 9( (99 &9

(: &9 >9 (9

............................... ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

( . )( )( . )( . ) ( )

(

−

−

δ = δ

+ ↑ δ = δ − δδ = δ − δ

δ − δ = δ − δ

α∆ − =α∆ −

× ° − ° − =π ×

× ° − ° al( =

< &9 : 8 &9& > &9 = =


71/74

st al

al al

al al

al

al

al

st

Substitute e" o e" &

(? ? =9 &9 N 9

( &78 >> &9 & (&7? ? =9 &9 N 9& ;7 &9 (


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66

-1 = 22.6 mm a0- -2 = .2 mm

DAM 21! " M#$A%&$ '#'#(A)*

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6 :

To "ylindri"al rods +D made o! steel (E&/.. 9a) and

A+ made o! aluminum (E&/ 9a) are 8oined at + and

restrained by rigid su##orts at A and D% Determine

(a) the rea"tions at A and D1 and (RA&2/%;N RD& C%

;N)

(b) the dee"tion o! #oint +% (.%.C4 mm)

67DAM 21! " M#$A%&$ '#'#(A)*

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At room tem#erature (/o+) a .%2 mm ga# existsbeteen the ends o! the rods shon% At a later timehen the tem#erature has rea"hed 4..+ determine

(a) the normal stress in the aluminum rod1 and

(Ja &52.%4 $9a)

(b) the "hange in length o! the aluminum rod

(Ga& .%-4 mm)

chapter 1_stress and strain

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