Page 261 - Marks Calculation for Machine Design

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January 4, 2005
Brown.cls
Brown˙C06
STATIC DESIGN AND COLUMN BUCKLING
Example 4 (Sec. 5.3):
σ 1 − σ 2 1 10 − (−10) 20 243
= = = = 1.67
S y n 12 12
1
n = = 0.60 (unsafe)
1.67
whereagainthefactors-of-safetyaresmallerthanobtainedwiththedistortion-energytheory.
This is what is meant by being more conservative, or more restrictive.
SI/Metric
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
84 MPa that is at the low end for magnesium alloys.

Summary of the principal stresses from selected examples (in MPa)
Example Principal stress (σ 1 ) Principal stress (σ 2 )
5(§5.2) 83 −33
2(§5.3) 84 0
3(§5.3) 112 56
4(§5.3) 70 −70

solution
Step 1. Plot the combinations of principal stresses from the given table.
This is shown in Fig. 6.9. 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
for Example 3 (Sec. 5.3) falls on the (σ 1 = 2σ 2 > 0) biaxial load line outside the boundary,
and the combination for Example 4 (Sec. 5.3) falls on the pure shear load line outside the
boundary.
Step 2. Identify which theory is appropriate for each combination.
For all the examples the distortion-energy theory gives the most accurate information.
However, for Examples 5 and 4 the maximum-shear-stress theory would be okay, but would
be more conservative. For Example 3, the maximum-normal-stress theory would be okay,
but would be more conservative. For Example 2, all three theories are appropriate as they
intersect at a point on the (σ 1 ) axis.
Step 3. Calculate the factor-of-safety for each combination, using the appropriate static
design theory.
As stated in step 2, the distortion-energy theory gives the most accurate information, so
use Eq. (6.9) to make the following calculations for each combination.