Page 344 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
Brown˙C08
Brown.cls
326
U.S. Customary 15:14 APPLICATION TO MACHINES SI/Metric
Step6. Usingthethreadedstiffness(k T )found Step6. Usingthethreadedstiffness(k T )found
in step 4 and the unthreaded stiffness (k UT ) in step 4 and the unthreaded stiffness (k UT )
found in step 5, calculate the bolt stiffness (k bolt ) found in step 5, calculate the bolt stiffness (k bolt )
using Eq. (8.9) as using Eq. (8.9) as
k T k UT k T k UT
k bolt = k bolt =
k T + k UT k T + k UT
2
8
9
2
6
6
(8.52 × 10 )(4.70 × 10 ) lb /in 2 (1.16 × 10 )(7.80 × 10 ) N /m 2
= =
6
9
8
6
(8.52 × 10 ) + (4.70 × 10 ) lb/in (1.16 × 10 ) + (7.80 × 10 ) N/m
2
2
4.00 × 10 13 lb /in 2 9.05 × 10 17 N /m 2
= = 9
1.32 × 10 lb/in 1.94 × 10 N/m
7
8
6
= 3.03 × 10 lb/in = 4.66 × 10 N/m
= 3,030 kip/in = 466 MN/m
8.2.2 The Members
As stated earlier, the members in a bolted connection, including any washers, are under
compression, and they too act as linear springs. Therefore, the expression for the series
combination of the threaded and unthreaded stiffnesses given in Eq. (8.8) can be extended
to allow for more than two member stiffnessses as given in Eq. (8.18) as
1 1 1 1 1
= + + + ··· + (8.18)
k members k 1 k 2 k 3 k N
where (N) is the number of members in the joint.
If one of the members within the joint is a gasket, which usually has a significantly lower
stiffness than the other members, then it will dominate the overall stiffness of the joint in a
negative way. For this reason gaskets should never be used in a structural joint, one that is
primarily carrying high loads. This is in contrast to joints that are merely sealing off a gas
or a liquid in a tank or pipeline. However, if both strength and sealing are important, such
as in pressure vessel applications, then special designs are needed to provide the necessary
sealing without compromising the stiffness of the joint.
The stiffness of a member in compression is not as straightforward as was the case for
a cap screw or bolt in tension. Much research has been conducted and is continuing. The
most accepted of the theories currently in practice proposes that the distribution of pressure
in the members spreads out from under the washer face of the bolt and the washer face of
the nut in a volume resembling the frustum of a hollow cone. The geometry and notation
for this theory is shown in Fig. 8.2.
Note that the hollow frustum from the washer face of the bolt and the hollow frustum
from the washer face of the nut meet at the midpoint of the grip. As the thicknesses of the
two members (t 1 ) and (t 2 ) shown in Fig. 8.2 are not equal, there is a third thickness in the
middle, denoted by (t middle ), and given by Eq. (8.19) as

L grip
t middle = t 2 − − t washer (8.19)
2
Therefore, there are actually five hollow frusta that act as linear springs; one for each
washer, two in the member with the greater thickness, and one in the other member.