Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

(c) the principal moments of inertia and the angle from the +z
axis to the axis of the maximum moment of inertia.
(d) the maximum compressive bending stress in the cross section.
A
b 1
y
B
t
section are b 1 = 40 mm, b 2 = 70 mm, d = 90 mm, and t = 10 mm.
Determine
(a) the moments of inertia I y and I z .
(b) the product of inertia I yz .
(c) the principal moments of inertia and the angle from the +z
axis to the axis of the maximum moment of inertia.
(d) the maximum tensile bending stress in the cross section.
z
M z
FIGURE p8.67
D
t
C
b 2
d
z
A
b 1
y
B
M z
t (typ)
p8.68 The extruded beam cross section shown in Figure P8.68
is subjected to a bending moment of magnitude M z = 260 N ⋅ m,
acting in the indicated direction. The dimensions of the cross
D
FIGURE p8.68
b 2
C
8.9 Stress concentrations Under Flexural Loadings
In Section 5.7, it was shown that the introduction of a circular hole or other geometric discontinuity
into an axially loaded member could cause a significant increase in the stress
near the discontinuity. Similarly, increased stresses occur near any reduction in diameter of
a circular shaft subjected to torsion. This phenomenon, termed stress concentration, occurs
in flexural members as well.
In Section 8.3, it was shown that the normal stress magnitude in a beam of uniform
cross section in a region of pure bending is given by Equation (8.10):
σ max =
Mc
I z
For a rectangular beam, the
nominal bending stress used
in Equation (8.26) is the stress at
its minimum depth. For a
circular shaft, the nominal
bending stress is computed for
its minimum diameter.
The bending stress magnitude computed from Equation (8.10) is termed a nominal stress
because it does not account for the stress-concentration phenomenon. Near notches,
grooves, fillets, or any other abrupt change in cross section, the normal stress due to bending
can be significantly greater. The relationship between the maximum bending stress at
the discontinuity and the nominal stress computed from Equation (8.10) is expressed as the
stress-concentration factor
K
σ max
= (8.26)
σ
nom
The nominal stress used in Equation (8.26) is the bending stress computed for the minimum
depth or diameter of the flexural member at the location of the discontinuity. Since
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