CIVL222 STRENGTH OF MATERIALS

Chapter 6

Torsion

Definition

Torque is a moment that tends to

twist a member about its

longitudinal axis. Slender members

subjected to a twisting load are said

to be in torsion.

Example of Torsion

When opening the lid of a common plastic drinks bottle, a

torque T applied to the cap is gradually increased until the

plastic connectors between the cap and the bottle experience

shear failure.

Example of Torsion

Shafts are structural members with length significantly greater

than the largest cross-sectional dimension used in transmitting

torque from one plane to another.

Example of Torsion

Example of Torsion

For a non-circular section member or an open

section member subjected to torsion:

Plane cross sections of the member do not

remain plane and the cross sections distort in a

manner which is called warping. In other

words, the fibers in the longitudinal direction

deform unequally.

Example of Torsion

Example of Torsion

For a circular shaft or a closed circular section

member subjected to torsion:

Plane circular cross sections remain plane and

the cross sections at the ends of the member

remain flat.

The length and the radius of the member

remain unchanged.

Plane circular cross sections remain

perpendicular to the longitudinal axis.

Analogy Between Axial Deformation and Torsion

• Axial Force (P)

• Elongation (d)

• Normal Stress (s)

• Extensional Strain (e)

• Modulus of Elasticity (E)

• Torque (T)

• Twist Angle (f)

• Shear Stress (t)

• Shear Strain (g)

• Shear Modulus (G)

Torsion Theory for circular sections

Absence of Warping

Investigate Deformation

Investigate Deformation

State of Pure Shear

Shear Strain Relate to Angel of Twist

Linear Variation of Shear Stress

Linear Variation of Shear Stress

Shear Stress Surfaces

Shear Stress Surfaces

Moment “dM” developed on “dA”

Setup the Integration “dM” over the Area “A”

Relating Torque and Stress

Torsion Formula

“J” for Solid Circular Shape

“J” for Hollow Circular Shape

Angle of Twist Formula

Summery of Key Equations

Sign Conventions

Sign Conventions

Sign Conventions

Sign Conventions

Example #1

Example #2

Example #3

The steel shaft of a socket wrench has a diameter of 8.0 mm. and a length of 200

mm (see figure). If the allowable stress in shear is 60 MPa, what is the

maximum permissible torque Tmax that may be exerted with the wrench?

Through what angle f (in degrees) will the shaft twist under the action of the

maximum torque? (Assume G = 78 GPa and disregard any bending of the

shaft.)

Example #4

A hollow steel shaft used in a construction

auger has outer diameter d2 =150 mm. and

inner diameter d1 = 115 mm. (see figure).

The steel has shear modulus of elasticity G

= 80 GPa

For an applied torque of 17 kN.m,

determine the following quantities:

(a) shear stress t2 at the outer surface of

the shaft,

(b) shear stress t1 at the inner surface, and

(c) rate of twist f (degrees per unit of

length).

Also, draw a diagram showing how the

shear stresses vary in magnitude along a

radial line in the cross section.

Example #5A hollow aluminum tube used in a roof structure has an outside diameter d2 =100

mm and an inside diameter d1 =80 mm (see figure). The tube is 2.5 m long, and

the aluminum has shear modulus G= 28 GPa.

(a) If the tube is twisted in pure torsion by torques acting at the ends, what is the

angle of twist f (in degrees) when the maximum shear stress is 50 MPa?

(b) What diameter d is required for a solid shaft (see figure) to resist the same

torque with the same maximum stress?

(c) What is the ratio of the weight of the hollow tube to the weight of the solid

shaft?

Example #6Four gears are attached to a circular shaft and transmit the torques shown in the

figure. The allowable shear stress in the shaft is 68 MPa.

(a) What is the required diameter d of the shaft if it has a solid cross section?

(a) What is the required outside diameter d if the shaft is hollow with an inside

diameter of 25 mm ?

Work Done and Power Transmitted

When a force moves in a straight line with constant

velocity the work done is given by the product the

magnitude of the force and the distance through

which it has moved.

Work done = force × distance

The power transmitted by this action is defined as

the rate at which this work is done, i.e. the work

done in unit time.

Power = work done

time

Work Done and Power TransmittedThe ‘distance’ travelled by a rotating body is

measured by the number of radians through which

it rotates. The work done by a torque acting on a

shaft is therefore given by the product of the

magnitude of the torque and the amount of rotation

in radians.

For one revolutions of the shaft:

The work done = T ×2p(since the shaft turns through 2p radians in one revolution)

If the shaft is rotating at N revolutions per minute,

then

work done = T ×2p N (units of work per minute)

Work Done and Power Transmitted

Usually the torque will measured in Newton meters

(N.m) and therefore the units of work will also be

N.m. However, it is more usual to give work in

joules (J) which are equal numerically to Newton

meters.

1 joule = 1 Newton meter

Power is measured in watts (W),

1 W = 1 J/s

= 1 N.m/s

=(1/60) N.m/min

Work Done and Power Transmitted

Hence, the power transmitted by a shaft rotating at

N revolutions per minute and subject to torque of T(N.m) will be given by:

WattsNT

Power60

2p

w: the shaft’s angular velocity (rad/s)

: the frequency of shaft’s rotation (Hz = 1 revolution/s)

rpm : revolutions per minute

hp: horsepower , 1 hp = 746 W

f

PPT

fTTP

pw

pw

2

2

fpw 2

Example #7A motor drives a shaft at 12 Hz and delivers 20 kW of power (see figure).

(a) If the shaft has a diameter of 30 mm, what is the maximum shear stress tmax

in the shaft?

(b) If the maximum allowable shear stress is 40 MPa, what is the minimum

permissible diameter dmin of the shaft?

Example #8The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is

running at 2500 rpm (see figure).

(a) If the shaft transmits 150 kW, what is the maximum shear stress in the

shaft?

(b) If the allowable shear stress is 30 MPa, what is the maximum power that

can be transmitted?