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Screws, Fasteners, and the Design of Nonpermanent Joints 427
threaded portion. Thus the stiffness constant of the bolt is equivalent to the stiffnesses
of two springs in series. Using the results of Prob. 4–1, we find
1 1 1 k 1 k 2
= + or k = (8–15)
k k 1 k 2 k 1 + k 2
for two springs in series. From Eq. (4–4), the spring rates of the threaded and
unthreaded portions of the bolt in the clamped zone are, respectively,
A t E A d E
k t = k d = (8–16)
l t l d
where A t = tensile-stress area (Tables 8–1, 8–2)
l t = length of threaded portion of grip
A d = major-diameter area of fastener
l d = length of unthreaded portion in grip

Substituting these stiffnesses in Eq. (8–15) gives

A d A t E
k b = (8–17)
A d l t + A t l d
where k b is the estimated effective stiffness of the bolt or cap screw in the clamped
zone. For short fasteners, the one in Fig. 8–14, for example, the unthreaded area is
small and so the first of the expressions in Eq. (8–16) can be used to find k b . For long
fasteners, the threaded area is relatively small, and so the second expression in
Eq. (8–16) can be used. Table 8–7 is useful.

8–5 Joints—Member Stiffness
In the previous section, we determined the stiffness of the fastener in the clamped zone.
In this section, we wish to study the stiffnesses of the members in the clamped zone.
Both of these stiffnesses must be known in order to learn what happens when the
assembled connection is subjected to an external tensile loading.
There may be more than two members included in the grip of the fastener. All
together these act like compressive springs in series, and hence the total spring rate
of the members is
1 1 1 1 1
= + + + ··· + (8–18)
k m k 1 k 2 k 3 k i
If one of the members is a soft gasket, its stiffness relative to the other members is
usually so small that for all practical purposes the others can be neglected and only
the gasket stiffness used.
If there is no gasket, the stiffness of the members is rather difficult to obtain,
except by experimentation, because the compression region spreads out between the
bolt head and the nut and hence the area is not uniform. There are, however, some
cases in which this area can be determined.
2
Ito has used ultrasonic techniques to determine the pressure distribution at the mem-
ber interface. The results show that the pressure stays high out to about 1.5 bolt radii.

2 Y. Ito, J. Toyoda, and S. Nagata, “Interface Pressure Distribution in a Bolt-Flange Assembly,” ASME paper
no. 77-WA/DE-11, 1977.