Page 322 - Marks Calculation for Machine Design

P1: Shashi
January 4, 2005
15:4
Brown.cls
Brown˙C07
STRENGTH OF MACHINES
304
Step 16. Subtract the constant axial force (F 2 ) from the limiting values in step 15 to give
the limiting range of the fluctuating axial force (F 1 ) forcing the factor-of-safety to be 1.
max lim
F = F − F 2 = (7,943 N) − (2,250 N) = 5,693 N
1 max
min lim
F = F − F 2 = (−743 N) − (2,250 N) =−2,993 N
1 min
This means the limiting range on the fluctuating force (F 1 ) is −2,993 to 5,693 N.
Alternative Method to Account for Stress Concentrations. The factor-of-safety (n)
according to the Goodman theory was by Eq. (7.25), repeated here
σ a σ m 1
+ = (7.25)
S e S ut n
In the determination of the endurance limit (S e ) in the denominator of the first term, one
of the modifying factors in the Marin equation was the miscellaneous effect factor (k e ),
where if there were stress concentrations, this factor was given by Eq. (7.16), also repeated
here
1
k e = (7.16)
K f
where the reduced stress concentration factor (K f ) was found from Eq. (6.23) as
K f = 1 + q(K t − 1) (6.23)
with (K t ) being the geometric stress concentration factor and (q) being the notch sensitivity.
If the miscellaneous effect factor (k e ) is separated from the endurance limit (S e ) in the
Goodman theory equation, then Eq. (7.25) can be rearranged as follows:

σ a σ m σ a σ m K f σ a σ m 1
+ = + = + =
S e (k e ) S ut 1 S ut S e S ut n
S e

K f move K f to numerator
separate out k e

substitute for K f
where now the reduced stress concentration factor (K f ) is multiplied by the alternating
stress (σ a ). This is a very important point, that any stress concentrations affect only the
alternating stress (σ a ), not the mean stress (σ m ). However, extreme care must be taken
to make sure the reduced stress concentration factor (K f ) is not left out, or included
twice.
Fluctuations in Torsional Loading. If the fluctuating loading on a machine element is
torsional, then there will be a mean shear stress (τ m ) and an alternating shear stress (τ a ).

The test specimen endurance limit (S ) is still determined from the guidelines in Eq. (7.1);
e
however, there will be an ultimate shear strength (S us ) defined as
S us = (0.67)S ut (7.33)
where the factor 0.67 is due to the work by Robert E. Joerres [Chap. 6, Springs, in Shigley,
Mischke, & Brown, 2004] at Associated Spring—Barnes Group.
Also, when calculating the endurance limit (S e ) from the Marin equation, Eq. (7.7), use
a loading factor (k c ) of 0.577. The other modifying factors are the same.