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3.5 Angle of twist in elastic range

In this section, a relation will be derived between the angle of twist f of a circular shaft and the

torque T exerted on the shaft. The entire shaft will be assumed to remain elastic. Consideringfirst the case of a shaft of lengthL and of uniform cross section of radius c subjected to a torque

T at its free end we recall from sections before that the angle of twist f and the maximum

shearing strain gmax are related as follows:

Angle of twist

max=

But, in the elastic range, the yield stress is not exceeded anywhere in the shaft, Hookes law

applies, and we have gmax 5 tmaxyG.

max=

=

and solving for f, we write

=


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Whereis expressed in radians. The relation obtained shows that, within the elastic range, the

angle of twistis proportional to thetorque T applied to the shaft.

Example

Problem 3.38


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Solution


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Design of Transmission Shafts

What is the power and torque ?

Power is How much work can be done in specified time

Torque is the load carry capacity

Principal transmission shaft performance specifications are:

power

speed

Designer must select shaft material and cross-section to meet performance specifications

without exceeding allowable shearing stress.

Determine torque applied to shaft at specified power and speed,

f

PPT

fTTP

2

2


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Find shaft cross-section which will not exceed the maximum allowable shearing stress,

Example of transmission shaft

shaftshollow2

shaftssolid2

max

41

42

22

max

3

max

Tcc

cc

J

Tc

c

J

J

Tc


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Statically Indeterminate Shafts

Given the shaft dimensions and the applied torque, we would like to find the torque

reactions atAandB.

From a free-body analysis of the shaft

which is not sufficient to find the end torques. The problem is statically indeterminate.

Divide the shaft into two components which must have compatible deformations,

Substitute into the original equilibrium equation

mN120 BA TT

ABBA T

JL

JLT

GJ

LT

GJ

LT

12

21

2

2

1

1

21 0

mN120

12

21

AA TJL

JL

T


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Sample Problem 3.4

Two solid steel shafts are connected by gears. Knowing that for each shaft G= 77 GPa and that

the allowable shearing stress is 55 MPa, determine (a) the largest torque T0that may be applied

to the end of shaftAB, (b) the corresponding angle through which endAof shaftABrotates.

SOLUTION:

Apply a static equilibrium analysis on the two shafts to find a relationship between TCD

and T0 .

Apply a kinematic analysis to relate the angular rotations of the gears.

Find the maximum allowable torque on each shaftchoose the smallest.

Find the corresponding angle of twist for each shaft and the net angular rotation of endA.

Apply a static equilibrium analysis on the two shafts to find a relationship between TCD

and T0.

Apply a kinematic analysis to relate the angular rotations of the gears.


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0

0

73.2

mm600

mm220

TT

TFM

TFM

CD

CDC

B

CB

CC

B

CB

CCBB

r

r

rr

73.2

mm20

mm60


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Stress Concentrations


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