TorsionTorsional Deformation of a Circular Shaft

Torque is a type of moment that tends to twist a member about its longitudinal axis.

Engineers need to understand the effects of torque to properly design axles or drive shafts used in vehicles and machinery.

Torque causes shear stresses to develop in shafts.

Before Deformation

After Deformation

T

T

Torque applied to a shaft fixed at one end

Distortion and Shear Strainof an element located at a radial distance ρ from the axis of the shaft.

Shear Strain increases linearly with ρ

The Shear Stress – Strain Diagram

Linear Variation in Shear Stress

For linear-elastic shafts subjected to shear strains, Hooke’s Law applies:

The linear variation in shear strain corresponds with a linear variation in shear stress:

Polar Moment of Inertia for a Tubular Shaft

Due to the complementary property of shear, not only does the internal torqueT develop a linear distribution of shear stress along each radial line in the planeof the cross-sectional area, but also an associated shear-stress distribution isdeveloped along an axial plane.

Power Transmission

Angle of Twist

Angle of Twist – Constant Torque and Cross-Sectional Area

Positive Sign Convention for T and Ф

Statically Indeterminate Torque-Loaded Members

Torsional Stress Concentration

5.68 Known: The A-36 bolt is tightened within a hole so that the reactive torque on the shank AB can be expressed by the equation t = (kx2) N∙m/m where x is in meters.

Find: If a torque of T = 50 N∙m is applied to the bolt head, determine the following:

(a) the constant k, and

(b) the amount of twist in the 50 mm length of the shank.

Assumption: the shank has a constant radius of 4 mm.

0.00 0.01 0.02 0.03 0.04 0.05x, m

0

500

1000

1500

2000

2500

3000D

istr

ibu

ted

To

rqu

e, t

, (Nm

)/m

Problem 5.68 - distributed torque on the shank of an A-36 bolt

t = (kx2) Nm/m

0.00 0.01 0.02 0.03 0.04 0.05x, m

0

10

20

30

40

50

To

rqu

e, T

, Nm

Problem 5.68 - torque on the shank of an A-36 bolt

T = 0.4(106)x3 Nm

5.82 Known: Two shafts are connected using gears fixed to their ends. Their other ends are attached to fixed supports at A and B. They are also supported by journal bearings at C and D, which allow free rotation of the shafts along their axes.

Find: If a 500 N∙m torque is applied to the gear at E as shown, determine the rotation of the gear at B.

Data: shafts are made from A-36 steel, and they each have a diameter of 25 mm.

Schematic:

● Small gear driving a big gear - increases torque and decreases speed of driven gear

● Large gear driving a small gear - decreases torque and increases speed of driven gear

● Car going up a hill.......more torque is needed.......thus a high gear ratio is used (3:1).......but speed is reduced.

● Car traveling on a highway......more speed is desired......thus a low gear ratio is used (1:1)......but torque is reduced.

Gear Basics

2

1

2

1

1

2

r

r