Page 357 - Marks Calculation for Machine Design

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January 4, 2005
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MACHINE ASSEMBLY
There are several things to notice in Fig. 8.5. First, the load line is at a 45 angle;
◦
however, it starts at the constant value of the bolt preload divided by the tensile-stress area
(F preload /A T ). Second, the distance (d) represents the fatigue factor-of-safety (n fatigue ),
where the vertical distance (S a − σ a ) is equal to the horizontal distance (S m − σ m ). Third,
the endurance limit (S e ) and ultimate tensile strength (S ut ) are found as usual.
Again from Chap. 7, the Goodman theory was presented mathematically in Eq. (7.22),
and repeated here as
S a S m
+ = 1 (7.22)
S e S ut
As an expression is needed for the alternating strength of the bolt (S a ) to use in Eq. (8.57)
to determine the fatigue factor-of-safety (n fatigue ), substitute the mean strength of the bolt
(S m ) from Eq. (8.58) into the Goodman theory formula in Eq. (7.22) to give
F preload
S a +
S a A T
+ = 1 (8.59)
S e S ut
Leaving out the numerous algebra steps, Eq. (8.59) can be rearranged to give the alter-
nating strength of the bolt (S a ) as
F preload
S ut −
A T
S a = (8.60)
S ut
1 +
S e
Remember, the stress-concentration factor (K f ) for the bolt, which is due to the short-
radius fillet under the head of the bolt and the imperfections at the start of the threads from
the nominal diameter of the bolt, must be incorporated in the endurance limit (S e ) through
the miscellaneous effects factor (k e ) of the Marin equation. Otherwise, the load line will
◦
not be at 45 . Stress-concentration factors (K f ) can be found in references such as Marks’
Standard Handbook for Mechanical Engineers.
Before declaring the bolted assembly design safe, it is a good practice to determine the
factor-of-safety against yielding (n yield ), where
S y S y
n yield = = (8.61)
σ max σ m + σ a
Examples 1 through 4 considered a bolt and nut assembly, but without washers. Example 5
will consider a cap screw assembly with a single washer. The geometry and notation for a
cap screw assembly is shown in Fig. 8.6.
The length of the grip (L grip ) is the sum of three terms as given in Eq.
L grip = t washer + t 1 + h (8.62)
where (h) is the effective depth of the threads of the cap screw into the thickness (t 2 ) of the
threaded member and determined by the guidelines given in Eq. (8.63) as
 t
 2
 t 2 < d
2

h = (8.63)
 d
 t 2 ≥ d

2