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

SolutioN
Section Properties
The cross-sectional area of the post is
b
120 mm
a
A = (80 mm)(120 mm) = 9,600 mm 2
The moment of inertia of the cross-sectional area about the x centroidal
axis is
I
x
3
(120 mm)(80 mm)
= = 5.120 × 10 mm
12
6 4
80 mm
c
55 mm
Cross-sectional dimensions and
location of application of load.
z
x
P
d
30 mm
and the moment of inertia about the z centroidal axis is
I
z
3
(80 mm)(120 mm)
= = 11.52 × 10 mm
12
6 4
Since I z > I x for the coordinate axes shown, the x axis is termed the
weak axis and the z axis is termed the strong axis.
y
Equivalent Forces in the Post
The vertical load P = 70 kN applied 30 mm from the x axis and 55 mm
from the z axis is statically equivalent to an internal axial force F =
70 kN, an internal bending moment M x = 2.10 kN · m, and an internal
bending moment M z = 3.85 kN · m. The stresses created by each of
these will be considered in turn.
z
c
F = 70 kN
b
d
M z = 3.85 kN·m
a
M x = 2.10 kN·m
x
Axial Stress Due to F
The internal axial force F = 70 kN creates compressive normal stress
that is uniformly distributed over the entire cross section. The magnitude
of the stress is computed as
y
F = 70 kN
b
7.29 MPa
F (70 kN)(1,000 N/kN)
s axial = = = 7.29 MPa(C)
2
A 9,600 mm
c
d
a
x
Bending Stress Due to M x
The bending moment acting as shown about the x axis creates
compressive normal stress on side cd, and tensile normal stress on side
ab, of the post. The maximum bending stress occurs at a distance
z = ± 40 mm from the neutral axis (which is the x centroidal axis for
M x ). The maximum bending stress magnitude is
z
16.41 MPa
b
y
16.41 MPa
Mz x
(2.10 kN⋅ m)( ± 40mm)(1,000 N/kN)(1, 000 mm/m)
s bend = =
6 4
Ix
5.120 × 10 mm
=± 16.41 MPa
z
c
d
a
M x = 2.10 kN·m
x
Bending Stress Due to M z
The bending moment acting as shown about the z centroidal axis
creates compressive normal stress on side ad, and tensile normal
stress on side bc, of the post. The maximum bending stress occurs at a
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