7/23/2019 Torsion of a Circular Cylinder_f06_revised
1/24
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 (&&)
7/23/2019 Torsion of a Circular Cylinder_f06_revised
2/24
*+er+ie,
-ntroduction
Basic Concepts
Theory
train Tensor
tress Tensor
#.ample
Conclusion
7/23/2019 Torsion of a Circular Cylinder_f06_revised
3/24
-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/
7/23/2019 Torsion of a Circular Cylinder_f06_revised
4/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
5/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
6/24
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/
7/23/2019 Torsion of a Circular Cylinder_f06_revised
7/24
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=
7/23/2019 Torsion of a Circular Cylinder_f06_revised
8/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
9/24
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/
7/23/2019 Torsion of a Circular Cylinder_f06_revised
10/24
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 = =
7/23/2019 Torsion of a Circular Cylinder_f06_revised
11/24
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/
7/23/2019 Torsion of a Circular Cylinder_f06_revised
12/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
13/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
14/24
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
= =
7/23/2019 Torsion of a Circular Cylinder_f06_revised
15/24
'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
= =
7/23/2019 Torsion of a Circular Cylinder_f06_revised
16/24
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
= =
7/23/2019 Torsion of a Circular Cylinder_f06_revised
17/24
#.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::/$7:&
7/23/2019 Torsion of a Circular Cylinder_f06_revised
18/24
tress and train Analysis
A shaft of circular cross section su0jected to tor3ues at opposite ends is considered/
A point in the cross section at the coordinate ,ill 0e rotated at an angle
The displacements are gi+en 0y 2
and /ince is 1ero in direction/ Components of strain tensor 0y using
,e ha+e
2
" Torsion does not result in the change in +olume it implies pure sheardeformation/
:> J J ?
(ik i k k iu u u x = +
&ii
=
: ( %u x x=
( : %u x x=
% &u =
%x
%x
%x
(
:
( :
& &(
& &(
&( (
ij
x
x
x x
=
7/23/2019 Torsion of a Circular Cylinder_f06_revised
19/24
tress and train Analysis
'rom ooke9s !a, ,e ha+e
,here F are !ame9s Constant/
And 6G ,here G is modulus of rigidity/
Components of stress tensor
(ij ij kk ij = +
(
:
( :
& &
& &
&
ij
G x
G x
G x G x
=
7/23/2019 Torsion of a Circular Cylinder_f06_revised
20/24
#.ample from DContinuum MechanicsE 0y Batra
Gi+en the displacement components
'ind the strain tensor and stress tensor at the material point >&2:2:? ,here
olution
&2:2:?
(
: ( ( %(u x x x= + %
( %u x= % &u =
:> J J ?
(ik i k k iu u u x = +
( ( (
( (%
(%
& ( (
&
(
&(
ij
x x x
x xx
xx
+
+ =
& (/$ /$
/$ & :
/$ : &
ij
=
)G6::/$7:& psi
7/23/2019 Torsion of a Circular Cylinder_f06_revised
21/24
#.ample from DContinuum MechanicsE 0y Batra
'rom ooke9s !a, ,e ha+e (ij ij kk ij
= +
ij ijG =
)
& (/$ /$
/$ & : ::/$7:&
/$ : &
ij
=
7/23/2019 Torsion of a Circular Cylinder_f06_revised
22/24
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/
7/23/2019 Torsion of a Circular Cylinder_f06_revised
23/24
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
7/23/2019 Torsion of a Circular Cylinder_f06_revised
24/24
@eferences
Gere2 Timoshenko/2 DMechanics of Materials2E 'ourth #dition2 P
Top Related