Page 258 - Marks Calculation for Machine Design

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January 4, 2005
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Brown˙C06
STRENGTH OF MACHINES
240
Biaxial where
2
s 1 = s > 0
s 2
S y
Boundary of allowable
combinations
Biaxial where
s = 2s > 0
2
1
s 1
–S y S y
Uniaxial where
s > 0, s = 0
1
2
–S y
Pure shear where
s > 0, s = –s 1
1
2
FIGURE 6.7 Load lines for uniaxial, biaxial, and pure shear combinations.
Consider the following example in both the U.S. Customary and SI/metric systems.

U.S. Customary
Example 1. Plot the combinations given in the table below of the principal stresses (σ 1 , σ 2 )
from several selected examples presented earlier in Chap. 5, on a static design coordinate
system for ductile materials. Show the boundaries of the recommended theories for deter-
mining if the combinations are safe, along with the four special load lines shown in Fig.
6.7. Also, determine the factor-of-safety for each combination. Use a yield strength (S y ) of
12 kpsi that is at the low end for magnesium alloys.

Summary of the principal stresses from selected examples (in kpsi)
Example Principal stress (σ 1 ) Principal stress (σ 2 )
5(§5.2) 11 −4
2(§5.3) 12 0
3(§5.3) 16 8
4(§5.3) 10 −10

solution
Step 1. Plot the combinations of principal stresses from the given table.
This is shown in Fig. 6.8. Notice that the combination of principal stresses for Example 5
(Sec. 5.2) falls outside the boundary in the fourth (IV) quadrant, the combination for
Example 2 (Sec. 5.3) falls on the uniaxial load line directly on the boundary, the combination