Ch 3 Load and Stress Analysis Shigley Ed 9

8/18/2019 Ch 3 Load and Stress Analysis Shigley Ed 9

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Chapter 3

Load and Stress Analysis

Lecture Slides

The McGraw-Hill Companies © 2012


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Chapter Outline

Shigley’s Mechanical Engineering Design


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Free-Body Diagram Example 3-1



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Fig. 3-1


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Shear Force and Bending oments in Beams

Cut beam at any location x1

Internal shear force V and bending moment M must ensureequilibrium


Fig. 3−2


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Sign Con!entions "or Bending and Shear


Fig. 3−3


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Distri#uted Load on Beam

Distributed load q( x called load intensity

!nits of force "er unit length


Fig. 3−#


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$elationships #et%een Load& Shear& and Bending

$he change in shear force from A to B is equal to the area of theloading diagram bet%een x A and x B.

$he change in moment from A to B is equal to the area of the

shear-force diagram bet%een x A and x B.



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Shear-oment Diagrams

Shigley’s Mechanical Engineering DesignFig. 3−&


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oment Diagrams ' (%o )lanes


Fig. 3−2#


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Com#ining oments "rom (%o )lanes

'dd moments from t%o "lanes as "er"endicularectors

Shigley’s Mechanical Engineering DesignFig. 3−2#


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Singularity Functions

' notation useful

for integrating

across

discontinuities

'ngle brac)ets

indicate s"ecial

function to

determine %hether

forces and moments

are actie

Shigley’s Mechanical Engineering Design$able 3−1


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Example 3-*


Fig. 3-&


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Example 3-*



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Example 3-*



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Example 3-3


Fig. 3-*


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Example 3-3



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Example 3-3


Fig. 3-*


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Stress

Normal stress is normal to a surface+ designated by σ

Tangential shear stress is tangent to a surface+ designated by τ ,ormal stress acting out%ard on surface is tensile stress

,ormal stress acting in%ard on surface is compressive stress

!.. Customary units of stress are "ounds "er square inch ("si

I units of stress are ne%tons "er square meter (,m21 ,m2 / 1 "ascal (0a



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Stress element

e"resents stress at a point

Coordinate directions are arbitraryChoosing coordinates %hich result in ero shear stress %ill

"roduce "rinci"al stresses



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Cartesian Stress Components

Defined by three mutually orthogonal surfaces at a "oint %ithin

a body

ach surface can hae normal and shear stress

hear stress is often resoled into "er"endicular com"onents

First subscri"t indicates direction of surface normal

econd subscri"t indicates direction of shear stress

Shigley’s Mechanical Engineering DesignFig. 3−4Fig. 3−5 (a


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Defined by three mutually orthogonal surfaces at a "oint %ithin

a body

ach surface can hae normal and shear stress

hear stress is often resoled into "er"endicular com"onents

First subscri"t indicates direction of surface normal

econd subscri"t indicates direction of shear stress



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In most cases+ 6cross shears7 are equal

Plane stress occurs %hen stresses on one surface are ero


Fig. 3−5


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)lane-Stress (rans"ormation E+uations

Cutting "lane stress element at an arbitrary angle and balancing

stresses gies plane-stress transformation equations

Shigley’s Mechanical Engineering DesignFig. 3−8


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)rincipal Stresses "or )lane Stress

Differentiating q. (3-5 %ith res"ect to φ and setting equal toero ma9imies σ and gies

$he t%o alues of 2φ p are the principal directions

$he stresses in the "rinci"al directions are the principal stresses

$he "rinci"al direction surfaces hae ero shear stresses.

ubstituting q. (3-1: into q. (3-5 gies e9"ression for thenon-ero "rinci"al stresses.

,ote that there is a third "rinci"al stress+ equal to ero for "lanestress.



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Extreme-!alue Shear Stresses "or )lane Stress

0erforming similar "rocedure %ith shear stress in q. (3-8+ the

ma9imum shear stresses are found to be on surfaces that are

;#&< from the "rinci"al directions.

$he t%o e9treme-alue shear stresses are



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aximum Shear Stress

$here are al%ays three "rinci"al stresses. =ne is ero for "lane

stress.

$here are al%ays three e9treme-alue shear stresses.

$he maximum shear stress is al%ays the greatest of these three.q. (3-1# %ill not gie the maximum shear stress in cases

%here there are t%o non-ero "rinci"al stresses that are both

"ositie or both negatie.

If "rinci"al stresses are ordered so that σ 1 > σ 2 > σ 3+then τ ma9 / τ 1.3



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ohr,s Circle Diagram

' gra"hical method for isualiing the stress state at a "oint

e"resents relation bet%een 9-y stresses and "rinci"al stresses0arametric relationshi" bet%een σ and τ (%ith 2φ as "arameter

elationshi" is a circle %ith center at

! / (σ + τ / ?(σ x @ σ y2+ : A

and radius of


2

2

2

x y

xy "

σ σ

τ

−

= +


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ohr,s Circle Diagram

Shigley’s Mechanical Engineering DesignFig. 3−1:


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Example 3-



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Example 3-



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Example 3-



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Example 3-



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Example 3-

Shigley’s Mechanical Engineering DesignFig. 3−11(d


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Example 3-



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Example 3-



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Example 3- Summary

x-y

orientation

0rinci"al stress

orientation

Ba9 shear

orientation

. i i S


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.eneral (hree-Dimensional Stress

'll stress elements are actually 3-D.

0lane stress elements sim"ly hae one surface %ith ero stresses.

For cases %here there is no stress-free surface+ the "rinci"al

stresses are found from the roots of the cubic equation


. l (h Di i l S


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.eneral (hree-Dimensional Stress

'l%ays three e9treme shear alues

Maximum #hear #tress is the largest

0rinci"al stresses are usually ordered such that σ 1 > σ 2 > σ 3+

in %hich case τ ma9 / τ 1.3


El ti St i


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Elastic Strain

$oo%e&s la'

( is oungs modulus+ or modulus of elasticity

$ension in on direction "roduces negatie strain (contraction

in a "er"endicular direction.For a9ial stress in x direction+

$he constant of "ro"ortionality n is Poisson&s ratio

ee $able '-& for alues for common materials.


El ti St i


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Elastic Strain

For a stress element undergoing σ x+ σ y+ and σ ) + simultaneously+


El ti St i


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Elastic Strain

Eoo)es la% for shear

#hear strain γ is the change in a right angle of a stress element%hen subGected to "ure shear stress.

* is the shear modulus of elasticity or modulus of rigidity

For a linear+ isotro"ic+ homogeneous material+


/ i" l Di t i# t d St


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/ni"ormly Distri#uted Stresses

!niformly distributed stress distribution is often assumed for

"ure tension+ "ure com"ression+ or "ure shear.

For tension and com"ression+

For direct shear (no bending "resent+


0 l St " B i B di


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0ormal Stresses "or Beams in Bending

traight beam in "ositie bending

x a9is is neutral axis x) "lane is neutral plane

Neutral axis is coincident %ith the

centroidal axis of the cross section


Fig. 3−13

0 l St " B i B di


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Hending stress aries linearly %ith distance from neutral a9is+ y

+ is the second-area moment about the ) a9is




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Ba9imum bending stress is %here y is greatest.

c is the magnitude of the greatest y

, +.c is the section modulus


Assumptions "or 0ormal Bending Stress


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Assumptions "or 0ormal Bending Stress

0ure bending (though effects of a9ial+ torsional+ and shear

loads are often assumed to hae minimal effect on bending

stress

Baterial is isotro"ic and homogeneous

Baterial obeys Eoo)es la%

Heam is initially straight %ith constant cross sectionHeam has a9is of symmetry in the "lane of bending

0ro"ortions are such that failure is by bending rather than

crushing+ %rin)ling+ or side%ise buc)ling

0lane cross sections remain "lane during bending


Example 3


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Example 3-

Shigley’s Mechanical Engineering DesignDimensions in mmFig. 3−1&

Example 3


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Example 3-


Example 3


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Example 3-


Example 3


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Example 3-


Example 3-


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Example 3-


(%o-)lane Bending


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(%o-)lane Bending

Consider bending in both xy and x) "lanes

Cross sections %ith one or t%o "lanes of symmetry only

For solid circular cross section+ the ma9imum bending stress is


Example 3-2


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Example 3-2

Shigley’s Mechanical Engineering DesignFig. 3−1*

Example 3-2


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Example 3-2


Example 3-2


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Example 3 2


Example 3-2


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Example 3 2


Shear Stresses "or Beams in Bending


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Shear Stresses "or Beams in Bending


Fig. 3−14

(rans!erse Shear Stress


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(rans!erse Shear Stress

$ranserse shear stress is al%ays accom"anied %ith bending

stress.


Fig. 3−15

(rans!erse Shear Stress in a $ectangular Beam


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(rans!erse Shear Stress in a $ectangular Beam


aximum alues o" (rans!erse Shear Stress


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aximum alues o" (rans!erse Shear Stress


$able 3−2

Signi"icance o" (rans!erse Shear Compared to Bending


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S g c ce o s!e se S e Co p ed o e d g


9am"le Cantileer beam+ rectangular cross section

Ba9imum shear stress+ including bending stress ( My.+ and

transerse shear stress (V/ +0+

Signi"icance o" (rans!erse Shear Compared to Bending


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g p g


Critical stress element (largest τ ma9 %ill al%ays be either

◦ Due to bending+ on the outer surface ( y.c1+ %here the transerse

shear is ero◦ =r due to transerse shear at the neutral a9is ( y.c:+ %here the

bending is ero $ransition ha""ens at some critical alue of 1.h alid for any cross section that does not increase in %idth farther a%ay

from the neutral a9is.

◦ Includes round and rectangular solids+ but not I beams and channels

Example 3-4


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p

Shigley’s Mechanical Engineering DesignFig. 3−2:

Example 3-4


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p

Shigley’s Mechanical Engineering DesignFig. 3−2:(b

Example 3-4


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p


Fig. 3−2:(c

Example 3-4


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p


Example 3-4


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p


Example 3-4


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p


Example 3-4


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(orsion


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Torque vector J a moment ector collinear %ith a9is of a

mechanical element

' bar subGected to a torque ector is said to be in torsion

Angle of t'ist + in radians+ for a solid round bar


(orsional Shear Stress


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For round bar in torsion+ torsional shear stress is "ro"ortional to

the radius ρ

Ba9imum torsional shear stress is at the outer surface


Assumptions "or (orsion E+uations


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quations (3-3& to (3-34 are only a""licable for the follo%ing

conditions

◦ 0ure torque

◦ emote from any discontinuities or "oint of a""lication of

torque

◦ Baterial obeys Eoo)es la%

◦ 'dGacent cross sections originally "lane and "arallel remain

"lane and "arallel

◦ adial lines remain straight

De"ends on a9isymmetry+ so does not hold true fornoncircular cross sections

Consequently+ only a""licable for round cross sections


(orsional Shear in $ectangular Section


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hear stress does not ary linearly %ith radial distance for

rectangular cross section

hear stress is ero at the corners

Ba9imum shear stress is at the middle of the longest side

For rectangular 0 9 c bar+ %here 0 is longest side


)o%er& Speed& and (or+ue


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0o%er equals torque times s"eed

' conenient conersion %ith s"eed in r"m


%here $ / "o%er+ K n / angular elocity+ reolutions "er minute

)o%er& Speed& and (or+ue


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In !.. Customary units+ %ith unit conersion built in


Example 3-5


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Example 3-5


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Example 3-5


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Example 3-5


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Example 3-5


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Example 3-5


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Example 3-5


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Example 3-6


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Fig. 3−2#

Example 3-6


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Fig. 3−2#

Example 3-6


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Example 3-6


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Example 3-6


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Shigley’s Mechanical Engineering DesignFi . 3−2#

Example 3-6


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Closed (hin-7alled (u#es


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Kall thic)ness t 22 tube

radius r

0roduct of shear stress

times %all thic)ness is

constant

hear stress is inersely

"ro"ortional to %allthic)ness

$otal torque T is

Am is the area enclosed by

the section median line


Fig. 3−2&

Closed (hin-7alled (u#es


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oling for shear stress

'ngular t%ist (radians "er unit length

1m is the length of the section median line


Example 3-18


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Example 3-18


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Example 3-11


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Open (hin-7alled Sections


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Khen the median %all line is not closed+ the section is said to be

an open section

ome common o"en thin-%alled sections

$orsional shear stress

%here T / $orque+ 1 / length of median line+ c / %all thic)ness+* / shear modulus+ and θ 1 / angle of t%ist "er unit length


Fig. 3−24

Open (hin-7alled Sections


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hear stress is inersely "ro"ortional to c3

'ngle of t%ist is inersely "ro"ortional to c3

For small %all thic)ness+ stress and t%ist can become quite large

9am"le

◦ Com"are thin round tube %ith and %ithout slit

◦

atio of %all thic)ness to outside diameter of :.1◦ tress %ith slit is 12.3 times greater

◦ $%ist %ith slit is *1.& times greater


Example 3-1*


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Example 3-1*


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Example 3-1*


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Stress Concentration


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Localied increase of stress near discontinuities

4 t is $heoretical (Meometric tress Concentration Factor


(heoretical Stress Concentration Factor


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Mra"hs aailable for

standard configurations

ee '""endi9 '-1& and'-1* for common

e9am"les

Bany more in Peterson&s

#tress-!oncentration 5actors

,ote the trend for higher

4 t at shar"er discontinuity

radius+ and at greaterdisru"tion


Stress Concentration "or Static and Ductile Conditions


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Kith static loads and ductile materials

◦ Eighest stressed fibers yield (cold %or)

◦ Load is shared %ith ne9t fibers

◦ Cold %or)ing is localied

◦ =erall "art does not see damage unless ultimate strength is

e9ceeded

◦ tress concentration effect is commonly ignored for static

loads on ductile materials


(echni+ues to $educe Stress Concentration


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Increase radius

educe disru"tion

'llo% 6dead ones7 to sha"e flo%lines more gradually


Example 3-13


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Fig. 3−3:

Example 3-13


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Fig. '−1& −1

Example 3-13


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Example 3-13


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Fig. '−1&−&

Stresses in )ressuri9ed Cylinders


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Cylinder %ith inside radius r i+ outside radius r o+ internal

"ressure pi+ and e9ternal "ressure po

$angential and radial stresses+


Fig. 3−31



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"ecial case of ero outside "ressure+ po / :




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If ends are closed+ then longitudinal stresses also e9ist


(hin-7alled essels


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Cylindrical "ressure essel %ith %all thic)ness 11: or less of

the radius

adial stress is quite small com"ared to tangential stress

'erage tangential stress

Ba9imum tangential stress

Longitudinal stress (if ends are closed


Example 3-1


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Example 3-1


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Stresses in $otating $ings

i i h fl h l bl di )


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otating rings+ such as fly%heels+ blo%ers+ dis)s+ etc.

$angential and radial stresses are similar to thic)-%alled

"ressure cylinders+ e9ce"t caused by inertial forces

Conditions

◦ =utside radius is large com"ared %ith thic)ness (>1:1

◦ $hic)ness is constant

◦ tresses are constant oer the thic)ness

tresses are


)ress and Shrin: Fits

$ li d i l bl d i h di l i f δ


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$%o cylindrical "arts are assembled %ith radial interference δ

0ressure at interface

If both cylinders are of the same material


)ress and Shrin: Fits

(3 #8 f li d li


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q. (3-#8 for "ressure cylinders a""lies

For the inner member+ po p and pi / :

For the outer member+ po / : and pi / p


(emperature E""ects

, l t i d t i f t t h


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,ormal strain due to e9"ansion from tem"erature change

%here α is the coefficient of thermal expansion

Thermal stresses occur %hen members are constrained to

"reent strain during tem"erature change

For a straight bar constrained at ends+ tem"erature increase %ill

create a com"ressie stress

Flat "late constrained at edges


Coe""icients o" (hermal Expansion


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Cur!ed Beams in Bending

I thi ) d b


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In thic) cured beams

◦ ,eutral a9is and centroidal a9is are not coincident

◦ Hending stress does not ary linearly %ith distance from the

neutral a9is


Fig. 3−3#



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r o / radius of outer fiber

r i radius of inner fiber

r n radius of neutral a9is

r c / radius of centroidal a9is

h / de"th of sectionco/ distance from neutral a9is to outer fiber

ci distance from neutral a9is to inner fiber

e / distance from centroidal a9is to neutral a9is

M bending momentN "ositie M decreases

Fig. 3−3#


Location of ne tral a is


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Location of neutral a9is

tress distribution

tress at inner and outer surfaces


Example 3-1


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Fig. 3−3&

Example 3-1


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Fig. 3−3&(0

Example 3-1


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Shigley’s Mechanical Engineering DesignFig. 3−3&

Formulas "or Sections o" Cur!ed Beams ;(a#le 3-


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Formulas "or Sections o" Cur!ed Beams ;(a#le 3-


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Alternati!e Calculations "or e

'""ro9imation for e alid for large curature %here e is small


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'""ro9imation for e6 alid for large curature %here e is small

%ith r n r c

ubstituting q. (3-** into q. (3-*#+ %ith r n J y / r + gies


Example 3-12


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Contact Stresses

$%o bodies %ith cured surfaces "ressed together


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$%o bodies %ith cured surfaces "ressed together

0oint or line contact changes to area contact

tresses deelo"ed are three-dimensional

Called contact stresses or $ert)ian stresses

Common e9am"les

◦ Kheel rolling on rail

◦ Bating gear teeth

◦ olling bearings


Spherical Contact Stress

$%o solid s"heres of diameters d1 and d2 are "ressed together


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$%o solid s"heres of diameters d 1 and d 2 are "ressed together

%ith force 5

Circular area of contact of radius a



0ressure distribution is hemis"herical


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0ressure distribution is hemis"herical

Ba9imum "ressure at the center of

contact area


Fig. 3−3*


Ba9imum stresses on the ) a9is


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Ba9imum stresses on the ) a9is

0rinci"al stresses

From Bohrs circle+ ma9imum shear stress is



0lot of three "rinci"al


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0lot of three "rinci"al

stress and ma9imum

shear stress as a functionof distance belo% the

contact surface

,ote that τ ma9 "ea)s

belo% the contact surfaceFatigue failure belo% the

surface leads to "itting

and s"alling

For "oisson ratio of :.3:+τ ma9 / :.3 pma9

at de"th of

) / :.#5a


Fig. 3−34

Cylindrical Contact Stress

$%o right circular cylinders %ith length l and


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$%o right circular cylinders %ith length l and

diameters d 1 and d 2

'rea of contact is a narro% rectangle of %idth20 and length l

0ressure distribution is elli"tical

Ealf-%idth 0

Ba9imum "ressure


Fig. 3−35


Ba9imum stresses on ) a9is


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Ba9imum stresses on ) a9is



0lot of stress


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0lot of stress

com"onents and

ma9imum shearstress as a function

of distance belo%

the contact surface

For "oisson ratioof :.3:+

τ ma9 / :.3 pma9

at de"th of

) / :.45*0

Documents

Ch 3 Load and Stress Analysis Shigley Ed 9