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Classes #3 & #4 Classes #3 & #4 Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110

TorsionTorsion

Fall 2010Fall 2010

Dr. GuptaDr. Gupta

Dr. PickettDr. Pickett

Torque and Torsional Deformation

Torque is a MOMENT

applied about the LONG axis

of a circular shaft.

Shaft twists about the LONG axis.

Circles remain circles.

Each Longitudinal grid line

deforms into a HELIX,

that intersects the circles

at EQUAL angles.

Torque and Torsional Deformation

Torque is a MOMENT

applied about the LONG axis

of a circular shaft.

Shaft length & radius

do NOT change.

Ends of shaft remain flat,

perpendicular to long axis.

Radial lines remain STRAIGHT.

Torque and Torsional Deformation

If shaft is FIXED at one endandTorque applied at other end,

Radial lines - remain straight- rotate thru an angle of twist

- angle of twist varies along length of shaft

)(x

Torque and Torsional Deformation

An element located x

from the back end of the shaft,

will have different

angles of twist (rotations)

on its front and back faces.

The difference in angles of twist

causes

Torsional SHEAR STRAINS.

)()( xandx

Angles change from 90˚.

xxTanBD

thusx

BDTan

planehorizontalthein

BAalongAB

CAalongAC

__

__

'lim

2

Angles change from 90˚.

xthusxgives

BDBDsetting

TanBD

thus

BDTan

planeverticalthein

BAalongAB

CAalongAC

__

__

'lim

2

For all elements on the

cross section at x,

max

max

var

sectan

c

cedgeoutsideatimumto

centeratzerofrom

withlinearilyiesstrainshear

lyconsequent

tioncrosstheovertconsisdx

d

thus

samethearedanddx

For all elements on the

cross section at x,

max

max

max

var

c

cedgeoutsideatimumto

centeratzerofrom

withlinearilyiesstrainshear

Torsional Shear Stress

For circular shafts,

hollow or solid,

If material is

- Homogeneous

- Linear-Elastic

Hooke’s Law gives:

Thus, like Shear Strain,

Shear Stress,

varies linearly with

Radial distance from center of shaft.

G

max

c

Torsional Shear Stress

Each elemental area, dA

located at ρ from the center,

will have an internal Force, dF,

that will produce an internal

Resisting Torque, T.

)(dAdF

A

AA

cT

dAc

dAdFT

2max

max )(

max

c

Torsional Shear Stress

Polar Moment of Inertia, J:

for a Solid Shaft;

44

0

443

0

22

2

2)(

4

22)2(

)2(

2sin

insideoutside

A

cc

ccJ

shafthollowafor

ccdddAJ

ddAis

dthicknessofringaofareathe

ncecircumferece

max

c

Axial Shear Stress (solid shaft)In order for any element

to be in equilibrium

the internal torque, T,

must develop an

Axial Shear Stress,

equal to the

Radial Shear Stress,

also

varying Linearly with radial position.

max

cJ

Tradialaxial

Axial Shear Stress (hollow shaft)In order for any element

to be in equilibrium

the internal torque, T,

must develop an

Axial Shear Stress,

equal to the

Radial Shear Stress,

also

varying Linearly with radial position.

max

cJ

Tradialaxial

Axial Shear Stress (wood shaft )In order for any element

to be in equilibrium

the internal torque, T,

must develop an

Axial Shear Stress,

equal to the

Radial Shear Stress,

also

varying Linearly with radial position.

max

cJ

Tradialaxial

Torque and Torsional Deformation

If shaft is FIXED at one endandTorque applied at other end,

Radial lines - remain straight- rotate thru an angle of twist

- angle of twist varies along length of shaft

)(x

Torque and Torsional Deformation

An element located x

from the back end of the shaft,

will have different

angles of twist (rotations)

on its front and back faces.

The difference in angles of twist

causes

Torsional SHEAR STRAINS.

)()( xandx

Angle of TWIST

Angles change from 90˚.

xxTanBD

thusx

BDTan

planehorizontalthein

BAalongAB

CAalongAC

__

__

'lim

2

Angle of TWIST - Angles change from 90˚.

xthusxgives

BDBDsetting

TanBD

thus

BDTan

planeverticalthein

BAalongAB

CAalongAC

__

__

'lim

2

Angle of TWIST

GJ

TLtconsareTorqueandradiusifand

dxxJ

xT

Gthus

tconsGsoogeneousismaterialtheusually

xxJxG

xTgiveslengththeoveregrating

xxJxG

xTx

xJxG

xxTand

xJxG

xxTthen

lengthshaftalongymaterialorradiusorTorqueifor

TorqueappliedtconsandradiustconshasshaftifGJ

Tthen

J

Tand

Grecalling

xthus

L

L

tan

)(

)(1

tanhom

)()(

)(int

)()(

)(

)()(

)()(

)()(

)()(

var_

__tantan_

0

0

Angle of TWIST

GJ

TLthen

materialorradiusorTorqueinchangesarethereif

Material must be:- Homogeneous- Isotropic- Linear Elastic- Stress < (Yield Stress = Proportional Limit)

Angle of TWIST

Sign Convention: Positive direction is:- Point right hand thumb outward from shaft

- Right hand fingers curl in positive rotation