Page 257 - Marks Calculation for Machine Design

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best predicts whether a design is safe or not, specifically in the first (I) and fourth (IV)
quadrants. STATIC DESIGN AND COLUMN BUCKLING 239
First (I): Distortion-energy theory is the most accurate. Maximum-normal-stress theory
is okay, but conservative. Maximum-shear-stress theory does not apply.
Fourth (IV): Distortion-energy theory is the most accurate. Maximum-shear-stress
theory is okay, but conservative. Maximum-normal-stress theory does not apply.
Graphically, these recommendations are shown in Fig. 6.6.

s 2
Maximum-normal-stress
S y theory
Boundary of allowable
combinations

II I
s
–S y S y 1
IV
Distortion-energy
theory
III
Maximum-shear-stress
–S y theory

Maximum-normal-stress
theory

FIGURE 6.6 Summary of recommendations (ductile).

The line at 45 passing through the origin of the coordinate system in Fig. 6.6 establishes
◦
the left boundary of the possible combinations of the principal stresses (σ 1 , σ 2 ). The vertical
line in the first (I) quadrant represents the maximum-normal-stress theory, the line at 45 ◦
in the fourth (IV) quadrant represents the maximum-shear-stress theory, and the ellipse in
both the first (I) and fourth (IV) quadrants represents the distortion-energy theory. Notice
that the ellipse passes through the three corner points (S y ,S y ),(S y ,0), and (0,−S y ). Observe
that the distortion-energy theory is the more accurate predictor for both quadrants, and is
less conservative than the other two theories.
To conclude the discussion for ductile materials, Fig. 6.7 shows the load lines for uniaxial,
biaxial, and pure shear combinations of the principal stresses (σ 1 , σ 2 ).