Page 278 - Marks Calculation for Machine Design

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Brown.cls
Brown˙C06
STRENGTH OF MACHINES
260
Notice that the stress at the hole (σ o ) is 50 percent greater than the axial stress (σ axial ),
and that the design normal stress (σ xx ) is almost three and a half times greater than the axial
stress. It should be clear that stress concentrations cannot be ignored.
Notch Sensitivity. As mentioned earlier, some brittle materials are not as sensitive to stress
concentrations as others, so a reduced value of the stress-concentration factor (K t ), denoted
(K f ),isdefined in Eq. (6.23),
K f = 1 + q(K t − 1) (6.23)
where (q) is the notch sensitivity. The subscript f on this reduced value of the stress-
concentration factor stands for fatigue, which will be discussed shortly in Chap. 7. However,
notch sensitivity is important to static loading conditions, just as it is to dynamic or fatigue
loading conditions.
Notch sensitivity (q), which ranges from 0 to 1, is a function not only of the material but
the notch radius as well. The smaller the notch radius, the smaller the value of the notch
sensitivity, and therefore, the smaller the reduced value of the stress-concentration factor
(K f ). Based on Eq. (6.23), a notch sensitivity of zero gives a reduced stress-concentration
factor(K f )equalto1,meaningthematerialisnotsensitivetonotches.Foranotchsensitivity
of 1, the reduced stress-concentration factor (K f ) equals the geometric stress-concentration
factor (K t ), meaning the material is fully sensitive to notches. Values of the notch sensitivity
(q) are available in various references; however, if a value of the notch sensitivity is not
known, use a value of 1 to be safe.
6.2 COLUMN BUCKLING

Column buckling occurs when a compressive axial load acting on a machine element being
modeled as a column exceeds a predetermined value. This machine element typically does
not fail exactly at this value; however, the design is unsafe if this value is exceeded. The
discussion on column buckling will be divided into four areas.
1. Euler formula for long slender columns
2. Parabolic formula for intermediate length columns
3. Secant formula for eccentric loading
4. Compression of short columns
These four areas are primarily differentiated relative to a slenderness ratio (L/k), where
(L) is the length of the column and (k) is the radius of gyration of the cross-sectional area
of the column. If the cross-sectional area has a weak and a strong axis, then the radius of
gyration used in the slenderness ratio should be for the weak axis. The radius of gyration
(k) is found from the relationship in Eq. (6.24),

I
I = Ak 2 or k = (6.24)
A
where (I) is the area moment of inertia and (A) is the cross-sectional area of the column.
For example, suppose the cross section of a column is rectangular as shown in Fig. 6.19.
As the height (h) is larger than the width (b), the x-axis is the strong axis and the y-axis is the
weak axis. Therefore, the area moment of inertia for the weak axis is given in Eq. (6.25) as
1 3
I weak = hb (6.25)
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