CIVL222 STRENGTH OF MATERIALS

Chapter 6a

Torsion

Definition

Torque is a moment that tends totwist a member about itslongitudinal axis. Slender memberssubjected to a twisting load are saidto be in torsion.

Example of TorsionWhen 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 TorsionShafts 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 TorsionFor a non-circular section member or an opensection member subjected to torsion:

Plane cross sections of the member do notremain plane and the cross sections distort in amanner which is called warping. In otherwords, the fibers in the longitudinal directiondeform unequally.

Example of Torsion

Example of TorsionFor a circular shaft or a closed circular sectionmember subjected to torsion:

Plane circular cross sections remain plane andthe cross sections at the ends of the memberremain flat.

The length and the radius of the memberremain unchanged.

Plane circular cross sections remainperpendicular to the longitudinal axis.

Analogy Between Axial Deformation and Torsion

• Axial Force (P)• Elongation ()• Normal Stress ()• Extensional Strain ()• Modulus of Elasticity (E)

• Torque (T)• Twist Angle ()• Shear Stress ()• Shear Strain ()• 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 of200 mm (see figure). If the allowable stress in shear is 60 MPa, what is themaximum permissible torque Tmax that may be exerted with the wrench?Through what angle f (in degrees) will the shaft twist under the action of themaximum torque? (Assume G = 78 GPa and disregard any bending of theshaft.)

Example #4

A hollow steel shaft used in a constructionauger has outer diameter d2 =150 mm. andinner diameter d1 = 115 mm. (see figure).The steel has shear modulus of elasticity G= 80 GPaFor an applied torque of 17 kN.m,determine the following quantities:

(a) shear stress t2 at the outer surface ofthe shaft,

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

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

Also, draw a diagram showing how theshear stresses vary in magnitude along aradial line in the cross section.

Example #5A hollow aluminum tube used in a roof structure has an outside diameter d2 =100 mmand an inside diameter d1 =80 mm (see figure). The tube is 2.5 m long, and thealuminum has shear modulus G= 28 GPa.

(a) If the tube is twisted in pure torsion by torques acting at the ends, what is the angleof 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 torquewith 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 inthe 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 insidediameter of 25 mm ?

Work Done and Power TransmittedWhen a force moves in a straight line with constantvelocity the work done is given by the product ofthe magnitude of the force and the distance throughwhich it has moved.

Work done = force × distance

The power transmitted by this action is defined asthe rate at which this work is done, i.e. the workdone 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 TransmittedUsually 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 revolutionsper minute and subject to torque of T (N.m) will be given by:

WattsNTPower602

w: the shaft’s angular velocity (rad/s) : the frequency of  shaft’s rotation (Hz = 1 revolution/s)

rpm : revolutions per minutehp: horsepower , 1 hp = 746 W

fPPT

fTTP

2

2

f 2

Example #7

A 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 stresstmax in the shaft?

(b) If the maximum allowable shear stress is 40 MPa, what is the minimumpermissible diameter dminof the shaft?

Example #8

The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) isrunning at 2500 rpm (see figure).

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

(b) If the allowable shear stress is 30 MPa, what is the maximum power thatcan be transmitted?