Page 251 - Marks Calculation for Machine Design

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CHAPTER 6
STATIC DESIGN AND
COLUMN BUCKLING

6.1 STATIC DESIGN

The question now arises as to whether the values of the principal stresses (σ 1 ) and (σ 2 ) and
the maximum and minimum shear stresses (τ max ) and (τ min ) found for a machine element
in Chap. 5, either mathematically or using the Mohr’s circle graphical process, represent a
safe operating condition. Depending on whether the material used for the machine element
can be considered ductile or brittle, the most commonly accepted criteria, or theories,
predicting that a design is safe under static conditions will be presented. The most common
ways to define a factor-of-safety (n) for a machine element will also be presented, again
based on whether the material being used is ductile or brittle.

Static Design Coordinate System. For the static design theories that follow, all the theories
can be represented by mathematical expressions; however, as was the case with Mohr’s
circle, a graphical picture of these expressions provides a significant insight into what the
theory really means in terms of predicting that a design is safe under static conditions.
Figure 6.1 shows the coordinate system that will be used, where the horizontal axis is the
maximum principal stress (σ 1 ) and the vertical axis is the minimum principal stress (σ 2 ).
For ductile materials, the yield strength (S y ) in tension and in compression are relatively
equal in magnitude, whereas for brittle materials the ultimate compressive strength (S uc )
is significantly greater in magnitude than the ultimate tensile strength (S ut ). Figure 6.1
reflects the difference between the yield and ultimate strengths, and the difference between
the magnitudes of the ultimate tensile and compressive strengths.
(Note that capital S is used for the term strength of a material, whereas the Greek letter σ
is used for the calculated normal stresses and the principal stresses and τ for the calculated
shear stresses and the maximum and minimum shear stresses.)
The four quadrants of this coordinate system, labeled I, II, III, and IV as shown, represent
the possible combinations of the principal stresses (σ 1 , σ 2 ). As it is usually assumed that
the maximum principal stress (σ 1 ) is always greater than or at least equal to the minimum
principal stress (σ 2 ), combinations in the second (II) quadrant where (σ 1 ) would be negative
and (σ 2 ) would be positive, are not possible. However, the graphical representations of the
analytical expressions will include the second quadrant just from a mathematical standpoint.
Primarily, the most common combinations are in the first (I) quadrant where (σ 1 ) and (σ 2 )
are both positive and in the fourth (IV) quadrant where (σ 1 ) is positive and (σ 2 ) is negative.
Combinations can occur in the third (III) quadrant where (σ 1 ) is negative, however (σ 2 )
must be equally or more negative.

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