torsion of a circular cylinder_f06_revised

7/23/2019 Torsion of a Circular Cylinder_f06_revised

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Torsion of a Circular Cylinder

A Course Project

By

George Kodikulam

Mukesh Nagapuri

Bharath Kumar Vudhamari

Pradeep Kumar Bitlari!akshmi Ganoothula

Continuum Mechanics " M##N $%%&

'all (&&)


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*+er+ie,

-ntroduction

Basic Concepts

Theory

train Tensor

tress Tensor

#.ample

Conclusion


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-ntroduction

Not all deformation is elongational or compressi+e/ The concept of strain can

0e e.tended to inclined shearing or distortional effects/

The deformation of a circular cylinder rod 0y surface tractions applied at the

end forces can 0e studied using 0asic concepts of continuum mechanics/

The formulas for deformation and stresses in circular cylinder of linear

elastic material su0jected to torsion are de+eloped /The state of stress in pure

shear and state of strain are analy1ed/


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Continuum Mechanics Approach

olid Mechanics is the study of load carrying mem0ers in terms of forces2deformations and sta0ility/

The Continuum Mechanics Approach is applied2 0ecause the materialproperties are taken to 0e the same e+en ,hen ,e consider infinitesimalareas and +olumes/

Alternati+e Approach

There are many alternati+e approaches to study solid mechanics/ uch as from 0asic e3uations relating atomic forces and interactions are usedto 0uild up material properties and then e.tending it to large sets of suchentities/

'or e.ample Molecular 4ynamics


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Tor3ue

T,isting moments or tor3ues are forces acting through distance so as

to promote rotation/

#.ample

5sing a ,rench to tighten a nut in a 0olt/

-f the 0olt2 ,rench and force are all perpendicular to one another2 the moment is the force ' times the length of the ,rench/

imple tor3ue " T 6 ' 7 l


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Torsion

Torsion is thet,isting of a straight 0ar ,hen it is loaded 0y t,isting moments

or tor3ues that tend to produce rotation a0out the longitudinal a.es of the 0ar/

'or instance2 ,hen ,e turn a scre, dri+er to produce torsion our hand applies

tor3ue 8T9 to the handle and t,ists the shank of the scre, dri+er/


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Bars su0jected to Torsion

!et us no, consider a straight 0ar

supported at one end and acted upon

0y t,o pairs of e3ual and opposite

forces/

Then each pair of forces and

form a couple that tend to t,ist the

0ar a0out its longitudinal a.is2 thus

producing surface tractions andmoments/

Then ,e can ,rite the moments as

:P (P

: : :T P d= ( ( (T P d=


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Torsion

The moments that produce t,isting of 0ar2 such as and are called tor3ues

or t,isting moments/

The moments are indicated 0y the right hand rule for moment +ectors2

,hich states that if the fingers curl in the direction of the moments then the

direction of the thum0 represents the direction of +ector/

:T (T


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Torsional !oads on Circular hafts

-nterested in stresses and strains of

circular shafts su0jected to t,isting

couples or tor3ues/

Tur0ine e.erts tor3ue Ton the

shaft/

haft transmits the tor3ue to the

generator/

Generator creates an e3ual and

opposite tor3ue T/


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Net Tor3ue 4ue to -nternal tresses

Net of the internal shearing stresses is an

internal tor3ue2 e3ual and opposite to the

applied tor3ue2

Although the net tor3ue due to the shearing

stresses is kno,n2 the distri0ution of the

stresses is not/

4istri0ution of shearing stresses is statically

indeterminate ; must consider shaftdeformations/

5nlike the normal stress due to a.ial loads2

the distri0ution of shearing stresses due to

torsional loads can not 0e assumed uniform/

( )T dF dA = =


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A.ial hear Components

Tor3ue applied to shaft produces shearing stresses

on the faces perpendicular to the a.is/

Conditions of e3uili0rium re3uire the e.istence of

e3ual stresses on the faces of the t,o planescontaining the a.is of the shaft/

The e.istence of the a.ial shear components is

demonstrated 0y considering a shaft made up of a.ial

slats/

The slats slide ,ith respect to each other ,hen e3ual

and opposite tor3ues are applied to the ends of the

shaft/


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haft 4eformations

'rom o0ser+ation2 the angle of t,ist of the shaft is

proportional to the applied tor3ue and to the shaft

length/

non=a.isymmetric?shafts are distorted ,hen su0jected to torsion/

T

L


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Torsional deformation of circular shaft

Consider a cylindrical 0ar of

circular cross section t,isted 0y

the tor3ues T at 0oth the ends/

ince e+ery cross section of the

0ar -s symmetrical and is appliedto the same internal tor3ue T ,e

say that the 0ar is in pure torsion/

5nder action of tor3ue T the right

end of the 0ar ,ill rotate through

small angle kno,n as angle of

t,ist/ The angle of t,ist +aries along the

a.is of the 0ar at intermediate cross

section denoted 0y /

> ?x


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Torsional 4eformations of a Circular Bar

@ate of t,ist

hear train at the outersurface of the 0ar

'or pure torsion the rate of

t,ist is constant and e3ual tothe total angle of t,istdi+ided 0y the length ! of the0ar

d

dx

=

A

ma.

bb rd r

ab dx

= = =

ma.

rr

L

= =


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'or !inear #lastic Materials

'rom ooke9s !a,

G is shear modulus of elasticity

is shear strain

'rom hear train e3uation "

hear tress at the outer surface ofthe 0ar "

Torsion 'ormula " To determine therelation 0et,een shear stresses andtor3ue2 torsional formula is to 0eaccomplished/

hear tresses in cylindrical 0ar ,ith

circular cross section/

ma.

G r =

G =

ma.

rr

L

= =


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Torsional 'ormula

ince the stresses act continously they ha+e aresultant in the form of moment/

The Moment of a small element dA located at radialdistance and is gi+en 0y

The resultant moment > tor3ue T ? is the summationo+er the entire cross sectional area of all suchelemental moments/

Polar moment of inertia of circle ,ith radius r anddiameter d

Ma.imum hear tress

4istri0ution of stresses acting on across section/

(ma.dM dAr

=

(ma. ma.p

A A

T dM dA I r r

= = =

( %(p

r dI

= =

ma. %

:)

p

Tr T

I d

= =


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#.ample from DMechanics of MaterialsE 0y Gere F Timoshenko

A solid steel 0ar of circular cross section has diameter d6:/$in2 l6$in2

psi/The 0ar is su0jected to tor3ues T acting at the ends if the tor3ues ha+e

magnitude T6($& l0ft

a?,hat is the ma.imum shear stress in the 0ar 0?,hat is the angle of t,ist 0et,een the ends

olution

a?'rom torsional formula

0? Angle of t,ist

)

($&7:(7$

::/$7:&p

TL

GI= =

7:/$ /HI&

%( %(p

dI in

= = =

ma. % %

:) :)7($&7:($%&

7:/$

Tpsi

d

= = =

)G6::/$7:&


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tress and train Analysis

A shaft of circular cross section su0jected to tor3ues at opposite ends is considered/

A point in the cross section at the coordinate ,ill 0e rotated at an angle

The displacements are gi+en 0y 2

and /ince is 1ero in direction/ Components of strain tensor 0y using

,e ha+e

2

" Torsion does not result in the change in +olume it implies pure sheardeformation/

:> J J ?

(ik i k k iu u u x = +

&ii

=

: ( %u x x=

( : %u x x=

% &u =

%x

%x

%x

(

:

( :

& &(

& &(

&( (

ij

x

x

x x

=


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tress and train Analysis

'rom ooke9s !a, ,e ha+e

,here F are !ame9s Constant/

And 6G ,here G is modulus of rigidity/

Components of stress tensor

(ij ij kk ij = +

(

:

( :

& &

& &

&

ij

G x

G x

G x G x

=


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#.ample from DContinuum MechanicsE 0y Batra

Gi+en the displacement components

'ind the strain tensor and stress tensor at the material point >&2:2:? ,here

olution

&2:2:?

(

: ( ( %(u x x x= + %

( %u x= % &u =

:> J J ?

(ik i k k iu u u x = +

( ( (

( (%

(%

& ( (

&

(

&(

ij

x x x

x xx

xx

+

+ =

& (/$ /$

/$ & :

/$ : &

ij

=

)G6::/$7:& psi


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#.ample from DContinuum MechanicsE 0y Batra

'rom ooke9s !a, ,e ha+e (ij ij kk ij

= +

ij ijG =

)

& (/$ /$

/$ & : ::/$7:&

/$ : &

ij

=


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Conclusion

The e3uations deri+ed in this section are limited to 0ars of circular cross

section solid that 0eha+e in linear elastic manner or loads must 0e such that

the stresses do not e.ceed the proportional limit of the material/

-t can 0e emphasi1ed that the e3uations for the torsion of circular cylinder

cannot 0e used for non circular 0ars/


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ome,ork Pro0lem

Gi+en the displacement components

'ind the strain tensor and stress tensor at the material point >&2&2:? ,here

%

: ( %(u x x= +

( (

( : %u x x=

% &u =

)G6::/$7:& psi


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@eferences

Gere2 Timoshenko/2 DMechanics of Materials2E 'ourth #dition2 P

torsion of a circular cylinder_f06_revised

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