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Chapter 4

Load and Stress AnalysisLoad and Stress Analysis

A. Bazoune

TorsionTorsion

LecLec. 7. 7

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https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=me&chap_sec=02.1&page=theory

TORSIONTORSION

TORSIONTORSION

Any moment vector that is collinear with an axisof a mechanical element is called a torquevector, because the moment causes the element

to be twisted about that axis.

A bar subjected to such a moment is also said tobe in torsion.

Designated by drawing arrows on the surface ofthe bar to indicate direction, or by drawingtorque-vector arrows along the axes of twist ofthe bar23-Sep-07 3

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PRACTICAL EXAMPLESPRACTICAL EXAMPLES

Transmit power from one device to another

Turbine to a generator

Engine to the wheels

Motor to the pulley

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AssumptionsAssumptions

The bar is acted upon by a pure torque.

The section under consideration are remote fromthe point of application of the load and from achange in diameter.

Adjacent cross sections originally plane andparallel remain plane and parallel after twisting,and any radial line remains straight.

The material obeys Hookes law.23-Sep-07 5

Fig. 4.2323-Sep-07 6

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Shear StressShear Stress

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The angle of twist for a solid round bar is

where

T= torque

l = length

G = modulus of rigidity

J = polar second moment of area

T l

G J = (4-35)

Angle of TwistAngle of Twist

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For a solid round bar, the shear stress is zero atthe centre and maximum at the surface.

The distribution is proportional to radius and isgiven by

Designating ras the radius to the outer surface,

we have

Eq. (4-37) applies onlyto circular sections.

T

J

=

max

Tr

J =

(4-36)

(4-37)

Shear StressShear Stress

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For a solid round section,

where d= diameter of the bar.

For a hollow round section,

4

32

d

J

=

( )4 432

o iJ d d

=

(4-38)

(4-39)

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To obtain the Torque T from the power andspeed of a rotating shaft, using USC units, use thefollowing equation

where

H= power, hp

T = torque, lb-in

n = shaft speed, rev./min

F= force, lb

V= velocity, ft/min.

2

33,000 33,000(12) 63,025

F V Tn T nH

= = = (4-40)

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When SI units are used, the equation is

where

H = power, W

T= torque, N-m

= angular velocity, rad./s

The torque Tcorresponding to the power in watts

is given approximately by

where n : rev./min.

9.55H

Tn

=

H T= (4-41)

(4-42)

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Torsional Stresses in NonTorsional Stresses in Non--CircularCircularCrossCross--section Memberssection Members

There are some applications in machinery for non-circular cross-section members and shafts where a

regular polygonal cross-section is useful intransmitting torque to a gear or pulley that canhave an axial change in position. Because no keyor keyway is needed, the possibility of a lost keyis avoided.

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Saint Venant (1855) showed that in a

rectangular bc section bar occurs in the middle of

the longest side b and is of magnitudeformula

where b is the longer side, and a factor that isfunction of the ratio b/c as shown in Table of Page 139.

The angle of twist is given by

where is a function of the ratio b/c as shown in

Table of Page 139.

max 2 2

1.83

/

T T

bc b c b c

= = +

(4-43)

3

Tl

bc G

= (4-44)

max

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ExampleExample 44--88 TextbookTextbook

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ExampleExample 44--88 TextbookTextbookContdContd

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ExampleExample 44--99TextbookTextbook

Fig. 4.26a

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ExampleExample 44--99 TextbookTextbook

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Closed ThinClosed Thin--Walled TubesWalled Tubes (t

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ExampleExample 44--1111 TextbookTextbook

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Open ThinOpen Thin--Walled SectionsWalled Sections

When the median wall line is not closed, it is to be open.Fig. 4-29 shows some examples.

Fig. 4-29Some open thin-walled sections

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Open ThinOpen Thin--Walled SectionsWalled Sections

Open sections in torsion, where the wall is thin, haverelations derived from the membrane analogy theory asfollows:

where

: shear stress

G : shear modulus

1: the angle of twist per unit length

L : length of the median line

c : wall thikness

1 2

3TG c

Lc = =

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ExampleExample 44--1212 TextbookTextbook

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