Simple Stress 2013a

7/23/2019 Simple Stress 2013a

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LARRY E. ROCELA

CIVIL ENGINEER



1.1 LOADLoad is defned as the set o external orces acting

on a mechanism or engineering structure hicharise rom ser!ice conditions in hich the

com"onents or#Common loads in engineering a""lications are

tension and com"ression

$ension%& 'irect "ull. Eg%(orce "resent in liting hoist

Com"ression%& 'irect "ush. Eg%& (orce acting on the"illar o a )uilding

*ign con!ention olloed% $ensile orces are"ositi!e and com"ressi!e negati!e



1.1.1TYPES OF LOAD

$here are a num)er o di+erent a,s in hich loadcan )e a""lied to a mem)er. $,"ical loading t,"esare%

A- Dead/ Static load& Non uctuating orcesgenerall, caused ), gra!it,

/- Live load& Load due to d,namic e+ect. Loadexerted ), a lorr, on a )ridge

C- Impact load or shock load& 'ue to sudden)los

'- Fatige or !ctati"g or alter"ati"g loads%0agnitude and sign o the orces changing ith time



1.# ST$ESS1hen a material is su)2ected to an external

orce3 a resisting orce is set u" ithin thecom"onent3 this internal resistance orce "erunit area is called stress. *I unit is N4m567a-.

8#7a98:::7a3 807a98:;< 7a3 8G"a98:;=7a3 8 $erra 7ascal98:;8> 7a

In engineering a""lications3 e use thethe original cross section area o the s"ecimen

and it is #non as con!entional stress or

Engineering stress



1.% ST$AI&1hen a )od, is su)2ected to some external orce3 there is

some change o dimension o the )od,. $he ratio ochange o dimension o the )od, to its original dimensionis #non as strain

*train is a dimensionless ?uantit,*train ma, )e%& a- $ensile strain )- Com"ressi!e strain c-

Volumetric strain d- *hear strainTe"sile strai"' Ratio o increase in length to original

length o the )od, hen it is su)2ected to a "ull orce

(ompressive strai"& Ratio o decrease in length tooriginal length o the )od, hen it is su)2ected to a "ushorce

)olmetric strai"' Ratio o change o !olume o the)od, to the original !olume

Shear strai"'*train due to shear stress



1.* TYPE OF ST$ESSES



1.*.1TYPES OF DI$E(TST$ESS'irect stress ma, )e normal stress or shear

stress

&ormal stress +,- is the stress hich acts indirection "er"endicular to the area. Normalstress is urther classifed into tensile stress

Te"sile stress is the stress induced in a )od,3

hen it is su)2ected to to e?ual and o""osite"ulls 6tensile orces- as a result o hich thereis a tendenc, in increase in length

It acts normal to the area and "ulls on the area



1.*.1 TYPES OF DI$E(T

ST$ESS +Te"sile stress-Consider a )ar su)2ected to a tensile orce 7 at

its ends. Let

A9 Cross sectional area o the )od,

L9Original length o the )od,

dL9 Increase in length o the )od, due to its "ull7

@9 *tress induced in the )od,e9 $ensile strain

Consider a section & hich di!ides the )od, intoto hal!es



1.*.1 TYPES OF DI$E(TST$ESS +Te"silestress-

$he let "art o the section x&x3 ill )e ine?uili)rium i 79R 6Resisting orce-. *imilarl,the right "art o the section x&x ill )e ine?uili)rium i 79R 6Resisting orce-




ST$ESS +Te"silestress- $ensile stress 6@-9 Resisting orce4 Cross sectional

area9 A""lied orce4Cross sectional area974A

$ensile strain9 Increase in length4Original length9 dL4L

Com"ressi!e stress%& *tress induced in a )od,3 hensu)2ected to to e?ual and o""osite "ushes as a resulto hich there is a tendenc, o decrease in length othe )od,

It acts normal to the area and it "ushes on the area

In some cases the loading situation is such that thestress ill !ar, across an, gi!en section. In such casesthe stress at an, gi!en "oint is gi!en ),

@9 Lt BA : B74 BA9 d74dA9 deri!ati!e o orce .r.tarea




ST$ESS+(ompressive stress-

(ompressive stress9Resisting orce4 cross sectional area9A""lied orce4 cross sectional area

Com"ressi!e strain9 'ecrease in length4 Original length9 &dL4L*ign con!ention or direct stress and strain%& $ensile stresses

and strains are considered "ositi!e in sense "roducing anincrease in length. Com"ressi!e stresses and strains areconsidered negati!e in sense "roducing decrease in length




ST$ESS +Shearstress-Shear stress %& *tress Induced in a )od,3 hen

su)2ected to to e?ual and o""osite orces hich areacting tangentiall, across the resisting section as a

result o hich the )od, tends to shear o+ across thatsection

Consider a rectangular )loc# o height h3 length L andidth unit,. Let the )ottom ace A/ o the )loc# )e fxed

to the surace as shon. Let 7 )e the tangential orce

a""lied along to" ace C' o the )loc#. (or thee?uili)rium o the )loc#3 the surace A/ ill o+er atangential reaction orce R hich is e?ual in magnitude

and o""osite in direction to the a""lied tangential orce 7




ST$ESS +Shearstress-Consider a section & cut "arallel to the a""lied orce hich

s"lits rectangle into to "arts

(or the u""er "art to )e in e?uili)rium A""lied orce

79Resisting orce R(or the loer "art to )e in e?uili)rium A""lied orce79Resisting orce R

Dence3 shear stress 9 Resisting orce4Resisting area974L x8974L

*hear stress is tangential to the area on hich it acts




ST$ESS +Shearstress-As the ace A/ is fxed3 the rectangular section A/C'

ill )e distorted to A/C8'83 such that ne !erticalace A'8 ma#es an angle F ith the initial ace A'

Angle F is called shear strain. As F is !er, small3

F9tan F9''84A'9dl4h

Dence shear strain9dl4h



1. ELASTI(ITY

ELASTI( LI0IT $he "ro"ert, o a )od, ), !irtue o hich it undergoes

deormation hen su)2ected to an external orce andregains its original confguration 6sie and sha"e- u"on theremo!al o the deorming external orce is called elasticit,.

$he stress corres"onding to the limiting !alue o externalorce u" to and ithin hich the deormation disa""earscom"letel, u"on the remo!al o external orce is calledelastic limit

A material is said to )e elastic i it returns to its original3unloaded dimensions hen load is remo!ed.

I the external orce is so large that the stress exceeds theelastic limit3 the material loses to some extent its "ro"ert,o elasticit,. I no the orce is remo!ed3 the material illnot return to its original sha"e and sie and there ill )e aresidual deormation in the material



1. 2OO3E4S LA5

ELASTI( 0OD6LIDoo#eHs la states that% 1hen a )od, is

loaded ithin elastic limit3 the stress is"ro"ortional to strain de!elo"edJ or 1ithin the

elastic limit the ratio o stress a""lied to strainde!elo"ed is a constantJ

$he constant is #non as 0odulus o elasticit,or Elastic modulus or YoungHs modulus

0athematicall, ithin elastic limit*tress4*train9@4e9E

@9 74A e 9BL4L

E97L4A B L



1.7 2OO3E4S LA5

ELASTI( 0OD6LI YoungKs modulus 6E- is generall, assumed to )e the

same in tension or com"ression and or most oengineering a""lications has a high numerical !alue.

$,"icall,3 E9>8: x 8:;= N4m5 69>8: G7a- or steel0odulus o rigidit,3 G9 4F9 *hear stress4 shear

strain

(actor o saet,9 ltimate stress47ermissi)le stress

In most engineering a""lications strains donot otenexceed :.::M so that the assum"tion thatdeormations are small in relation to orinaldimensions is generall, !alid



1.8 ST$ESS'ST$AI&

(6$)E +TE&SILE TEST-*tandard tensile test in!ol!es su)2ecting a

circular )ar o uniorm cross section to agraduall, increasing tensile load until theailure occurs

$ensile test is carried out to com"are thestrengths o !arious materials

Change in length o a selected gauge lengtho )ar is recorded ), extensometers

A gra"h is "lotted ith load !s extension orstress !s strain



1.8 ST$ESS'ST$AI&

(6$)E +TE&SILE TEST-



1.8 ST$ESS'ST$AI&

(6$)E +TE&SILE TESTDIA9$A0-A Limit o "ro"ortionalit, It is the "oint here the

linear nature o the stress strain gra"h ceases/ Elastic limit It is the limiting "oint or the condition

that material )eha!es elasticall,3 )ut hoo#eKs la does

not a""l, . (or most "ractical "ur"oses it can )e otenassumed that limit o "ro"ortionalit, and elastic limits arethe same

/e,ond the elastic limits3 there ill )e some "ermanentdeormation or "ermanent set hen the load is remo!ed

C 6""er Yield "oint-3 ' 6Loer ,ield "oint- 7oints aterhich strain increases ithout corres"ondingl, highincrease in load or stress

E ltimate or maximum tensile stress 7oint here thenec#ing starts

( (racture "oint



1.: (O&STIT6TI)E

$ELATIO&S2IPS;ET5EE& ST$ESS

ST$AI&

A- 1'Dime"sio"al case 6due to "ull or "ush or shearorce-

@9Ee

;- #'Dime"sio"al caseConsider a )od, o length L3 idth / and height D. Let

the )od, )e su)2ected to an axial load. 'ue to thisaxial load3 there is a deormation along the length othe )od,. $his strain corres"onding to this deormation

is called longitudinal strain.*imilarl, there are deormations along directions

"er"endicular to line o a""lication o ore. $he strainscorres"onding to these deormations are called lateralstrains



1.: (O&STIT6TI)E


ST$AI&



1.: (O&STIT6TI)E


ST$AI&

Longitudinal strain is ala,s o o""osite sign othat o lateral strain. Ie i the longitudinal strain istensile3 lateral strains are com"ressi!e and !ice

!ersa E!er, longitudinal strain is accom"anied ), lateral

strains in orthogonal directions

Ratio o lateral strain to longitudinal strain is called

7oissonHs ratio 6- 0athematicall,39&Lateral strain4Longitudinal strain

Consider a rectangular fgure A/C' su)2ected a

stress in @x direction and in @ , direction



1.: (O&STIT6TI)E


ST$AI&

*train along x direction due to @x9 @ x4E*train along x direction due to @ ,9& x @,4E

$otal strain in x direction ex9 @ x4E & x @,4E

*imilarl, total strain in , direction3 e,9 @ ,4E &

x @x4EIn the a)o!e e?uation tensile stresses are

considered as "ositi!e and com"ressi!estresses as negati!e

(- % Dime"sio"al case<'Consider a M ' )od, su)2ected to M orthogonalnormal stresses in x3, and directionsres"ecti!el,



1.: (O&STIT6TI)E


ST$AI&

*train along x direction due to @x9 @ x4E

*train along x direction due to @ ,9& x @,4E

*train along x direction due to @ 9& x @4E $otal strain in x direction ex9 @ x4E & x 6@,4E @4E -

*imilarl, total strain in , direction3 e,9 @ ,4E &

x 6@xE @4E -

*imilarl, total strain in direction3 e9 @ 4E & x 6@xE @,4E -



1.1= A&ALYSIS OF ;A$S

OF )A$YI&9 ($OSSSE(TIO&Consider a )ar o di+erent lengths and o di+erent

diameters 6and hence o di+erent cross sectional areas- asshon )elo. Let this )ar )e su)2ected to an axial load 7.

$he total change in length ill )e o)tained ), adding thechanges in length o indi!idual sections

$otal stress in section 8% @89E8 x BL84L8@8 x L84E89BL8@8974A8 Dence BL897L84A8E8

*imilarl,3 BL>97L>4A>E> BLM97LM4AMEM



1.1= A&ALYSIS OF ;A$S

OF )A$YI&9 ($OSSSE(TIO&Dence total elongation BL97x 6L84A8E8L>4A>E>

LM4AMEM-I the YoungHs modulus o di+erent sections are the same3

E89E>9EM9E Dence BL974Ex 6L84A8L>4A> LM4AM-1hen a num)er o loads are acting on a )od,3 the resulting

strain3 according to "rinci"le o su"er"osition3 ill )e thealge)raic sum o strains caused ), indi!idual loads

1hile using this "rinci"le or an elastic )od, hich issu)2ected to a num)er o direct orces 6tensile or

com"ressi!e- at di+erent sections along the length o the)od,3 frst the ree )od, diagram o indi!idual section isdran. $hen the deormation o each section is calculatedand the total deormation is e?ual to the alge)raic sum odeormations o indi!idual sections



1.11 A&ALYSIS OF

6&IFO$0LY TAPE$I&9(I$(6LA$ $ODConsider a )ar uniorml, ta"ering rom a

diameter '8 at one end to a diameter '> atthe other end

Let

7 Axial load acting on the )ar

L Length o )ar

E YoungHs modulus o the material



1. 11 A&ALYSIS OF

6&IFO$0LY TAPE$I&9(I$(6LA$ $ODConsider an infnitesimal element o thic#ness dx3 diameter

'x at a distance x rom ace ith diameter '8.

'eormation o the element d6Bx-9 7 x dx4 6Ax E-

Ax9P4Q x 'x5 'x9 '8 & 6'8 '>-4L x xLet 6'8&'>-4L9# $hen 'x9 '8&#x

d6BLx-9 Q x 7 x dx46P x 6'8&#x-5 x E-

Integrating rom x9: to x9L Q7L46PE'8'>-

Let '8&#x9S then dx9 &6d S4#-

1hen x9:3 S9'8 1hen x9L3 S9'>



8.8> ANALY*I* O(

NI(OR0LY $A7ERINGREC$ANGLAR /AR



1.1% A&ALYSIS OF ;A$S

OF (O0POSITESE(TIO&SA )ar3 made u" o to or more )ars o e?ual lengths

)ut o di+erent materials rigidl, fxed ith eachother and )eha!ing as one unit or elongation and

shortening hen su)2ected to axial loads is calledcom"osite )ar.

Consider a com"osite )ar as shon )elo

Let

7 A""lied loadL Length o )ar

A8 Area o cross section o Inner mem)er

A> Cross sectional area o Outer mem)er



1.1% A&ALYSIS OF ;A$S

OF (O0POSITESE(TIO&S*train de!elo"ed in the outer mem)er9 *train

de!elo"ed in the inner mem)er

@84E8 9 @>4E>

$otal load 67-9 Load in the inner mem)er 678- Load in the outer mem)er 67>-

@8 x A8 @> x A>9 7

*ol!ing a)o!e to e?uations3 e get the!alues o @83 @> T e8 and e>



1.1* ST$ESS ELO&9&.

P$OD6(ED I& A ;A$ D6ETO ITS SELF 5EI92TConsider a )ar o length L3 area o cross section A rigidl,

fxed at one end. Let U )e the densit, o the material.Consider an infnitesimal element o thic#ness d, at adistance , rom the )ottom o the )ar.

$he orce acting on the element considered9 eight othe "ortion )elo it9UAg,



1.1* ST$ESS ELO&9&.

P$OD6(ED I& A ;A$ D6ETO ITS SELF 5EI92T $ensile stress de!elo"ed9 (orce acting on the

element4Area o cross section9 Ug,.

(rom the a)o!e e?uation3 it is clear that themaximum stress at the section here ,9L3 ieat the fxed end 6UgL- and minimum stress is

at the ree end69:-Elongation due to sel eight



8.8 *$RE** IN /AR 'E $O

RO$A$ION



8.8 *$RE** IN /AR 'E $O

RO$A$ION



1.1 T2E$0AL ST$ESS $hermal stresses are the stresses induced in a )od, dueto change in tem"erature. $hermal stresses are set u" ina )od,3 hen the tem"erature o the )od, is raised orloered and the )od, is restricted rom ex"anding or

contractingConsider a )od, hich is heated to a certain tem"erature

Let L9 Original length o the )od,

B $9Rise in tem"

E9YoungKs modulusW9CoeXcient o linear ex"ansion

dL9 Extension o rod due to rise o tem"I the rod is ree to ex"and3 $hermal strain de!elo"ed

et9 B L4L9W x B $



1.1 T2E$0AL ST$ESS $he extension o the rod3 B L9 L x W x B $

I the )od, is restricted rom ex"anding reel,3 $hermal stress de!elo"ed is @t4et9E

@t9 E x W x B $*tress and strain hen the su""ort ,ields%&

I the su""orts ,ield ), an amount e?ual to 3then the actual ex"ansion is gi!en ), the

di+erence )eteen the thermal strain and Actual strain3 e9 6L x W x B $ -4L

Actual stress9 Actual strain x E9 6L x W x B $ -4L x E



T2A&3 YO6

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Simple Stress 2013a

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