tortion of circula shafts

7/29/2019 TORTION OF CIRCULA SHAFTS

1/13

V

TO RS IO NOF CI RC UL ARSH AF TS

VOB2-3


2/13

V-1

TO RS IO NOF CI RC UL ARSH AF TS

Fr om the kinematics of deformation of an elastic circular shaft

shown in Figure 1, the geometry of small deformation gives the following

relationship:

x tan y = r 4 >( x)

where y and 1j I ( x) are small angles (less than 6 degrees).

observed that y is the same angle for all x, and therefore,

y t y( x)

Taking a derivative of this relation with respect to x, gives

tany = r[dljl(x)/dx]

It can be

as r is the same at all x being the radius of the cylindrical surface of

the free-body of the shaft. Since for small angles y.

tany " y

then

y : r[d${x)/dx]

--r--"~vmed O:nf igumm~----~~------~

KintJ m 3 '/ /c .sof G I o I M / Pe/Orm ll lionof' C/iru/~,. .Shaffs

Figure 1


3/13

If a finite slice of length zx is isolated from the shaft, and an

element of the si ze ~x by r~4I is exc:rnined before and after deformation,

as shown in Figure 2, then it can be observed that the torsional

deformation produces a small angle change y in the original right angle

of the element. 'This small angle is called the shear deformation. Hal f

of the shear deformation y is called the shear strain

Exs

= y/2

where s is the circllYlferential coordinate 1n the form of the arc-length.

~~~~~~~~t---X~ s = r d t #

--._ JrHbd I I==_ ; ( J ( H 1 l t i

Figure 2

The stress-strain relation between the shear strain and the shear

stress (J acting in the cross-section is also linear for smallxs

deform ation :

(J

xs =Gy = 2GExs = Gr [d4l(x)/dx]

where the Shear Modulus (also called Modulus of Rigidity) is dependent

on the t-bdulus of Elasticity E and the Poisson's Ratio \I for the

material,

G = E/2(1+v)

It is customary to use T for the torsional shear stress (Jxs

T = G r [d4l(x)/dx)


4/13

where < $(x) designates the small angle of rotation of the cross-section

at x brought about by torsion. rbwever. the cross-section i tsel f

remains undeformed: it rotates like a thin rigid disk. In torsion. T

is not an average shear stress but the actual shear stress acting at the

point.

Torsion of any shaft is produced by applying to the shaft a

torsional couple measured by a manent M (T). The torsional couple-x

manent M (T) acts along the axis of the shaft. The x-axis of the framex

of reference is chosen to coincide with the axis of the shaft. In order

to establish the fundamental Equation of Torsion of Shafts. the same

idea is followed as for the bending of beans. The equation of

torsion

of shafts is establ ished by expressing the torsional stress couple Mx

(-r)

in terms of the tor sional strain by means of the stress-strain relation.

The torsional couple-manent is given by

M (T) = ! r dF (T)= ! r[TdA]x A s A

where the torsional shear force acting on the cross-sectional element d A

is

dFs

(-r) = TdA

as shown in Figure 3.y

Cross-Sed/on41Geometryof Shaff

Figure 3


5/13

M (T) = J r{Gr[dt/dx l dA = G[d$(x)/dx] J r2 dAx A A

Substi tuting for T from the stress-strain relation gives

]

as G and dlj)(x)/dx are constant over the entire cross-section. This

expression. again, can be naturall y factored into a product of three

factors: G gives the material response to torsion, d~/dx gives the

kinematics of torsion, and

J = f r2dA

A

called the Polar Second Moment of the Cross-Section, is entirel y a

geometric property of the cross-section of the shaft. This equation can

be solved for the Rate of Torsion:

d~(x)/dx = Mx(T)/GJ

which represents the Equation of Torsion of Circular Shafts. It can be

integrated to obtain the angle of torsion at x:

Ij)(x)=J [d$(x)/dx] dx = J [MX(T)/GJJ dx + CA A

where the integration constant C is found from the boundary cond ition

of compatible deformation of the shaft. 4 > (x) represents the small ang l e

of rotation of the cross-section at x caused by torsion.

Torsional Shear stress

It can be seen from the stress-strain relations that the torsional

shear stress in the cross-section at any x solely depends upon the

radial distance r:

T = r G[dlj)(x)/dx]

It is evident that the torsional shear stress T is zero at the center of

the shaft, where r = 0, and maxtmun at the external surface of the shaft

where r = a. The torsional shear stress varies linearly with r from a


6/13

zero val ue at the ax is 0 f the shaft to a maxImun at the ex ternal sur fac e

of the shaft where r :: a.

Substi tuting for d .


7/13

tacit assumption was made that any cross-sectional radius in torsion

deforms into a radius. That this is a reasonable asaunpt i.on can be

observed from a simple thought experiment. Assume that a cross-

sectional radius deforms into a curved line in a circular shaft under

torsion as shown in Figure 5. If the shaft is turned 180 degrees about

this undeformed radius it is found that the radius is now deformed

symmetricall y to the opposite side of the undeformed rad ius despite the

fact that everything about the shaft is still the s ane : the material,

the geometry of the shaft, and the torsional couples applied at the ends

of the shaft.

Figure 5

The shear deformation at point r in the cross-section of the shaft

for the two cases of deformation are obviousl y distinct. Since a

linearly elastic deformation of the shaft can only have ;;j unique

response to load ing , so Lel y the case when the rad ius deforms into a

straight radial line is admissible because then the two rotated

configurations of the shaft give the same strain. Therefore. the

assunption that a plane cross-section of a circular shaft deforms in


8/13

V-7

torsion into a plane cross-section without distorsion is a valid

conclusion.

The above arg unent s do not apply if the shaft is not circular in

cross-section. as can be easily proven by similar argunents if shafts

with other than circular cross-sections are used in the rotation

experiment.

Shear Strain in a Plane:

Consider a rectangular element in (x-y) plane as shown in Figure 6.

y

l J y

r

J

L %

A)t

~ (J:+4lt):# lI)'(lt)f-.d.~ (1()

. . I "p-NJ~)={,Ix(X)+~U.jX)

x

SheiirD e f o rm 'd f io nofP/~ni rB e m e n t

Figure 6

~all shear strain by definition is hal f of the angle change yxy

right angle, where the average angle change

in a

_Yxy=tan a1

+ tan a2

= {[uy

(x + 6x;y) - uy

(x;y)/~x}

+ {[ u (x ; y +6Y) - U (x; y) ] / t : .y} = [ t :.u / t : .x +llu.

x x y x

IllY. ]


9/13

V-7Imposing the 1 imi t ll.x + 0 and ll.y +0 gives the shear deformation at the


10/13

x

=

x

V - 8

point (x ,y) :

y = (au lax) + (au lay)xy y

The shear strain

E := Y 12 = (1/2) (au lax) + (au lay)xy xy y x

For the torsion of the shaft, the small element of the shaft is almost

planar, for which dx = rde = ds, and thus,

a( )/ax = (1/r) (a( )/a~]

and

For the torsion of the shaft

du rdq,( x)y

u = 0

Therefore,

Yxs = r[aq,(x)/ax]

as r r( x), which is prec isel y the same resul t obtained earl ier by

considering the global deformation of the circular shaft.


11/13


12/13


13/13

v, -

tortion of circula shafts

Documents