57:019:AAA Intro Mech. Def. Bodies

Period #9: Torsion

A. Context

A torque is a moment applied to mechanical system that causes it to twist or rotate.

In the present circumstances, we’ll be concerned with torsional moments (torques) applied to long slender shafts as shown in the figure below. We’ll especially want to determine what is the internal stress distribution in a shaft subjected to torques, and what is the resulting deformation (or twisting) of the shaft itself. Initial assumptions: • shaft has a circular cross-section; • material behaves as a linear, elastic solid.

9.1 The University of Iowa

The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.2

B. Kinematics of Torsion Kinematics of torsion: how the shaft deforms when subjected to a torque. Observe that when the shaft is loaded, the rectangular patch undergoes essentially pure shear deformation.

Fig. 9.1 Torsional deformation in a rubber shaft. (a) unloaded, and (b) deformed under torque loading.

(a) (b)

D’ D B’

max

max

'tan

'

:ionconfigurat undeformed

in the AB and AC fibers heConsider t

c

xc

xx

BB

BB

57:019:AAA Intro Mech. Def. Bodies 9.3 The University of Iowa

C. Shear Stresses

Kinematics of torsion established that =(/c)max. Since by Hooke’s Law, t=G it follows that on the cross-section of the shaft: t=(/c)tmax

max

max

c

cG

t

t

G

57:019:AAA Intro Mech. Def. Bodies 9.4 The University of Iowa

2

42

2

2

4

max

4

max

0

3max

0

2

c

c

c

c

dc

d

dAT

c

c

A

t

t

t

t

t

Jc

dc

T

dc

dc

dAdT

ddA

c

max

0

3max

3max

max

2

2

2

2

t

t

t

t

t

d

The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.5

circular cross-section (solid shaft)

annular cross-section (tubular shaft)

2

4cJ

2

44

io ccJ

The polar moment of inertia: dAJ 2

D. The Polar Moment of Inertia

E. The Torsion Formula

F. Torque/Twist Relation:

co ci

J

T

J

Tc tt ; max

dx

JG

T

dx

d

JG

T

G

GJ

T

t

The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.6

G. Power Transmission

• It is often the case that a source of power such as an engine exists at one location, but work must be done at a different location.

• Power can be transmitted from the engine to the location where work needs to be done using drive shafts. The rate of power transmission by shaft is:

(cycles) sshaft turn heat which t rate theis f

(radians) sshaft turn heat which t rate theis

shaft in the torque theis T

:where

f2*TTωdt

dθTP

The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.7

H. Example Problems

Example 9.1: The assembly consists of two sections of galvanized steel pipe connected together using a reducing coupling at B. The smaller pipe has an outer diameter of 0.75 in. and an inner diameter of 0.68 in., whereas the larger pipe has an outer diameter of 1 in. and an inner wall diameter of 0.86 in. If the pipe is tightly secured to the wall at C, determine the maximum shear stress developed in each section of the pipe when the couple shown is applied to the handles of the wrench.

The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.8

Example 9.2: If the applied torque on the shaft CD is T’=75N·m, determine the absolute maximum shear stress in each shaft. The bearings B, C, and D allow free rotations of the shafts, and the motor holds the shafts fixed from rotating