Page 142 - Shigley's Mechanical Engineering Design

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Load and Stress Analysis 117
and, for the outer member

2 2
r + R
o
= p (3–59)
2
r − R 2
(σ t ) o
r=R o
Assumptions
It is assumed that both members have the same length. In the case of a hub that has been
press-fitted onto a shaft, this assumption would not be true, and there would be an increased
pressure at each end of the hub. It is customary to allow for this condition by employing a
stress-concentration factor. The value of this factor depends upon the contact pressure and
the design of the female member, but its theoretical value is seldom greater than 2.
3–17 Temperature Effects
When the temperature of an unrestrained body is uniformly increased, the body expands,
and the normal strain is
x = y = z = α( T) (3–60)
where α is the coefficient of thermal expansion and T is the temperature change, in
degrees. In this action the body experiences a simple volume increase with the compo-
nents of shear strain all zero.
If a straight bar is restrained at the ends so as to prevent lengthwise expansion and
then is subjected to a uniform increase in temperature, a compressive stress will develop
because of the axial constraint. The stress is
σ =− E =−α( T)E (3–61)
In a similar manner, if a uniform flat plate is restrained at the edges and also sub-
jected to a uniform temperature rise, the compressive stress developed is given by the
equation
α( T)E
σ =− (3–62)
1 − ν
The stresses expressed by Eqs. (3–61) and (3–62) are called thermal stresses.
They arise because of a temperature change in a clamped or restrained member. Such
stresses, for example, occur during welding, since parts to be welded must be clamped
before welding. Table 3–3 lists approximate values of the coefficients of thermal
expansion.

Table 3–3
Material Celsius Scale (°C ) Fahrenheit Scale (°F )
1
−1
Coefficients of Thermal −6 −6
Aluminum 23.9(10) 13.3(10)
Expansion (Linear −6 −6
Brass, cast 18.7(10) 10.4(10)
Mean Coefficients −6 −6
Carbon steel 10.8(10) 6.0(10)
for the Temperature −6 −6
Cast iron 10.6(10) 5.9(10)
Range 0–100°C) −6 −6
Magnesium 25.2(10) 14.0(10)
Nickel steel 13.1(10) −6 7.3(10) −6
Stainless steel 17.3(10) −6 9.6(10) −6
Tungsten 4.3(10) −6 2.4(10) −6