Torsion Mechanics of Material

7/21/2019 Torsion Mechanics of Material

1/17

MECHANICS OF

MATERIALS

Seventh Edition

Ferdinand P. Beer

E. Russell Johnston Jr.

John T. !e"ol#!avid F. Ma$ure%

Le&ture Notes'

Bro&% E. Barr(

).S. Militar( A&ade*(

CHAPTER

Copyright 2015 McGraw-Hill Education. Permiion re!uired "or reproducti

3Torsion


2/17

Copyright 2015 McGraw-Hill Education. Permiion re!uired "or reproduction or diplay.

MECHANICS OF MATERIALSSeventh

Beer + Johnston + !e"ol# + Ma$ure%

Torsional Loads on Circular Shafts

, -

Stresses and strains in members of

circular cross-section are subjected

to twisting couples or torques

Generator creates an equal and

opposite torque T

Shaft transmits the torque to thegenerator

Turbine exerts torque Ton the shaft

Fig. 3.2 (a) A generator provides power at aconstant revolution per minute to a turbinethrough shaft AB. (b) Free body diagram of shaftAB along with the driving and reaction torques onthe generator and turbine, respectively.


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Net Torque Due to Internal Stresses

, - ,

( ) == dAdFT

Net of the internal shearing stresses is aninternal torque equal and opposite to the

applied torque

!lthough the net torque due to the shearing

stresses is "nown the distribution of the

stresses is not#

$nli"e the normal stress due to axial loads the

distribution of shearing stresses due to torsional

loads cannot be assumed uniform#

%istribution of shearing stresses is staticall&

indeterminate ' must consider shaft

deformations#

Fig. 3.24 (a)Free body diagram ofsection BC with torque at Crepresented by the representablecontributions of small elements of areacarrying forces dF a radius from thesection center. (b) Freebody diagram

of section BC having all the small areaelements summed resulting in torque

Fig. 3.3 "haft sub#ect to torquesand a section plane at C.

S


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Axial Shear Components

, - /

Torque applied to shaft produces shearing

stresses on the faces perpendicular to the

axis#

(onditions of equilibrium require the

existence of equal stresses on the faces of the

two planes containing the axis of the shaft#

The slats slide with respect to each other

when equal and opposite torques are applied

to the ends of the shaft#

The existence of the axial shear components is

demonstrated b& considering a shaft made up

of slats pinned at both ends to dis"s#

Fig. 3.6 $odel of shearing in shaft(a) undeformed% (b) loaded and

deformed.

Fig. 3.5 "mall element in shaftshowing how shear stress componentsact.

MECHANICS OF MATERIA SS


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Shaft Deformations

, - 0

)rom obser*ation the angle of twist of theshaft is proportional to the applied torque and

to the shaft length#

L

T

+hen subjected to torsion e*er& cross-section

of a circular shaft remains plane and

undistorted#

(ross-sections for hollow and solid circular

shafts remain plain and undistorted because acircular shaft is axis&mmetric#

(ross-sections of noncircular ,non-

axis&mmetric shafts are distorted when

subjected to torsion#Fig. 3.8 Comparison of deformations

in circular (a) and square (b) shafts.

Fig. 3.7 "haft with &'ed support andline AB drawn showing deformationunder torsion loading (a) unloaded%(b) loaded.

MECHANICS OF MATERIALSS


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Shearin Strain

, - 1

(onsider an interior section of the shaft# !s a

torsional load is applied an element on the

interior c&linder deforms into a rhombus#

Shear strain is proportional to twist and radius

maxmax and

cL

c==

LL

== or

.t follows that

Since the ends of the element remain planar

the shear strain is equal to angle of twist#

Fig. 3.13 "hearing "train inematicde&nitions for torsion deformation. (a)

!he angle of twist (b) *ndeformedportion of shaft of radius with (c)+eformed portion of the shaft having

same angle of twist, and strain, anglesof twist per unit length, .



7/17Copyright 2015 McGraw-Hill Education. Permiion re!uired "or reproduction or diplay.



Stresses in Elastic Rane

, - 2

Jc

dAc

dAT max/max

= ==

0ecall that the sum of the moments of the

elementar& forces exerted on an& cross

section of the shaft must be equal to the

magnitude T of the torque1

andmax J

T

J

Tc

==

The results are "nown as the elastic torsion

formulas

2ultipl&ing the pre*ious equation b& the

shear modulus

max Gc

G =

max

c

=

)rom oo"es 4aw G= so

The shearing stress *aries linearl& with thedistance from the axis of the shaft#

Fig. 3.14 +istribution of shearingstresses in a torqued shaft% (a) "olidshaft, (b) hollow shaft.






Normal Stresses

,- 3

Note that all stresses for elements aand cha*e

the same magnitude#

5lement cis subjected to a tensile stress ontwo faces and compressi*e stress on the other

two#

5lements with faces parallel and perpendicular

to the shaft axis are subjected to shear stressesonl Normal stresses shearing stresses or a

combination of both ma& be found for other

orientations#

( )

max6

6max78

6max6max

/

/

/78cos/

o

===

==

A

A

A

F

AAF

(onsider an element at 78oto the shaft axis

5lement ais in pure shear#

Fig. 3.17 Circular shaft with stresselements at dierent orientations.

Fig. 3.18 Forces on faces at -/ to shafta'is.

Fig. 3.19 "haft elements with onlyshear stresses or normal stresses.






Torsional !ailure "odes

, - 4

%uctile materials generall& fail in shear# 9rittle materials are wea"er in

tension than shear#

+hen subjected to torsion a ductile specimen brea"s along a plane of

maximum shear i#e# a plane perpendicular to the shaft axis#

+hen subjected to torsion a brittle specimen brea"s along planes

perpendicular to the direction in which tension is a maximum i#e# along

surfaces at 78oto the shaft axis#

Photo 3.2 "hear failure of shaft sub#ect totorque.






Sample Pro#lem $%&

, - 56

ShaftBCis hollow with inner and outer

diameters of :6 mm and ;/6 mm

respecti*el ShaftsABand CDare solid

and of diameter d# )or the loading showndetermine ,a the minimum and maximum

shearing stress in shaftBC ,b the

required diameter dof shaftsABand CD

if the allowable shearing stress in these

shafts is 4$T.>N1

(ut sections through shaftsAB

andBCand perform static

equilibrium anal&ses to find

torque loadings#

Gi*en allowable shearing stress

and applied torque in*ert the

elastic torsion formula to find therequired diameter#

!ppl& elastic torsion formulas tofind minimum and maximum

stress on shaftBC.






, - 55

Sample Pro#lem $%&

S>4$T.>N1 (ut sections through shaftsABandBC

and perform static equilibrium anal&sisto find torque loadings#

( )

CDAB

ABx

TT

TM

==

==

m"N

Torsion Mechanics of Material

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