Page 270 - Marks Calculation for Machine Design

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STRENGTH OF MACHINES
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Biaxial where
s = s > 0
2
1
s 2
Boundary of allowable
combinations
S ut
Biaxial where
s = 2s > 0
S ut 1 2
s 1
–S uc
–S ut Uniaxial where
s > 0, s = 0
2
1
Pure shear where
s > 0, s = –s 1
2
1
–S uc
FIGURE 6.15 Load lines for uniaxial, biaxial, and pure shear combinations.
the horizontal line in the third (III) quadrant, represents the maximum-normal-stress theory.
The solid line that connects the point (0,−S uc ) to the point (S ut ,0) represents the Coulomb-
Mohr theory. The dotted line from point (0,−S uc ) to the point (S ut ,−S ut ) represents the
modified Coulomb-Mohr theory.
To conclude the discussion for brittle materials, Fig. 6.15 shows the load lines for uniaxial,
biaxial, and pure shear combinations of the principal stresses (σ 1 , σ 2 ).
Consider the following example in both the U.S. Customary and SI/metric systems.
U.S. Customary
Example 2. Plot the combinations given in the table below of the principal stresses (σ 1 , σ 2 )
on a static design coordinate system for brittle materials. Show the boundaries of the recom-
mended theories for determining if the combinations are safe, along with the four special
load lines shown in Fig. 6.15. Also, determine the factor-of-safety for each combination.
Use an ultimate strength in tension (S ut ) of 30 kpsi and an ultimate strength in compression
(S uc ) of 90 kpsi that are the typical values for cast iron.

Principal stresses (in kpsi)
Point Principal stress (σ 1 ) Principal stress (σ 2 )
1 40 −15
2 30 0
3 20 20
4 25 −25
5 15 −55