Page 310 - Marks Calculation for Machine Design

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STRENGTH OF MACHINES
By similar triangles, the mean stress (σ m | c ) can be found from Eq. (7.32) as
S e
σ m | c = (7.32)
S e σ a
+
S ut σ m
The mean stress (σ m | c ), or the alternating stress (σ a | c ), can also be found graphically
if all the information is plotted to scale in a diagram similar to Fig. 7.15, as will be done
shortly in an example.
Remember, the factor-of-safety (n) associated with the perpendicular distance (d) from
the point (σ m ,σ a ) to the Goodman line is given by Eq. (7.25), or it too can be found
graphically by plotting all the information in a diagram similar to Fig. 7.12.
The following examples, in both the U.S. Customary and SI/metric system of units, will
use both the mathematical expressions presented on the previous pages, as well as a graph-
ical approach, to determine the various factors-of-safety for a particular design.
U.S. Customary
Example 1. For the cantilevered beam shown in Fig. 7.16, which is acted upon by a
fluctuating tip force (F) of between 2.4 lb and 5.6 lb, determine
a. The factor-of-safety (n) using the Goodman theory
b. The factor-of-safety (n m ) where the mean stress (σ m ) is held constant
c. The factor-of-safety (n a ) where the alternating stress (σ a ) is held constant
d. The factor-of-safety (n c ) where the ratio of the alternating stress (σ a ) to the mean stress
(σ m ) is held constant

F 1 "
16

1 "
" 2
2 1 2
FIGURE 7.16 Cantilevered beam for Example 1 (U.S. Customary).

The beam is made of cold-drawn steel, ground to the dimensions shown, then welded
to the vertical support at its left end. The beam operates at room temperature. Also, S ut is
85 kpsi and K f is 1.2 (due to welds at left end of beam).

solution
Step 1. Using Eq. (7.8) and values for the coefficient (a) and exponent (b) from Table 7.1,
calculate the surface finish factor (k a ) as
b −0.085
k a = aS ut = (1.34 kpsi)(85 kpsi) = (1.34)(0.6855) = 0.92
Step 2. Using Eq. (7.12) calculate the effective diameter (d e ) as

1/2 1 1 2
d e = 0.808 (bh) = 0.808 in in = (0.808) (0.03125 in )
16 2
= (0.808)(0.1768 in) = 0.143 in