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

11.5 in.
6 in.
Ref. axis
0.5 in.
1 in.
4 in.
(1)
(2)
8 in.
(3)
1 in.
10 in.
1 in.
12 in.
A i
(in. 2 )
y i
(in.)
y i A i
(in. 3 )
(1) 4.0 11.5 46.0
(2) 10.0 6.0 60.0
(3) 8.0 0.5 4.0
yA i i
y = Σ Σ A
i
22.0 110.0
3
110.0 in.
= = 5.0 in.
2
22.0 in.
Therefore, the z centroidal axis is located 5.0 in. above the reference axis for this cross
section.
z
7 in.
5 in.
1 in.
4 in.
(1)
y
(2)
(3)
Moment of inertia
Since the centroids of areas (1), (2), and (3) do not coincide with the z centroidal axis for
the entire cross section, the parallel-axis theorem must be used to calculate the moment
of inertia of the cross section about this axis. The complete calculation is
summarized in the following table:
1 in.
12 in.
10 in.
I c
(in. 4 )
| d i |
(in.)
d i2 A i
(in. 4 )
I z
(in. 4 )
(1) 0.333 6.5 169.000 169.333
(2) 83.333 1.0 10.000 93.333
(3) 0.667 4.5 162.000 162.667
425.333
8 in.
1 in.
Thus, the moment of inertia of the cross section about its z centroidal axis
is I z = 425.333 in. 4 .
+M
–M
258
+M
–M
Flexure Formula
A positive bending moment produces compressive normal stress at the top of the
beam and tensile normal stress at the bottom. Since the cross section of the beam is
not symmetric about the axis of bending (i.e., the z axis), the magnitude of the bending
stress at the top of the beam will be greater than that at the bottom of the beam.
The maximum positive internal bending moment is M = 5,625 lb · ft. For
this positive moment, the compressive bending stress produced on the top of
the flanged shape (at y = +7 in.) is calculated as
My (5,625 lb⋅ft)(7in.)(12 in./ft)
σ x =− =−
=− 1,111 psi = 1,111 psi(C)
4
I
425.333 in.
z
and the tensile bending stress produced on the bottom of the flanged shape (at y =
–5 in.) is calculated as
My (5,625 lb⋅ft)( −5in.)(12 in./ft)
σ x =− =−
=+ 793 psi = 793 psi(T)
4
I
425.333 in.
z
A negative bending moment produces tensile stress at the top of the beam and
compressive stress at the bottom. The maximum negative internal bending moment is