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

ExAmpLE 8.7
A cantilever beam 10 ft long carries a uniformly
distributed load w = 100 lb/ft. The beam is
constructed from 3 in. wide by 8 in. deep wood
timber (1) that is reinforced on its upper surface
by a 3 in. wide by 0.25 in. thick aluminum
plate (2). The elastic modulus of the wood
is E = 1,700 ksi, and the elastic modulus of the
aluminum plate is E = 10,200 ksi. Determine
the maximum bending stresses produced in the
timber (1) and the aluminum plate (2).
100 lb/ft
Plan the Solution
The transformed-section method will be used
to transform the cross section consisting of two
materials into an equivalent cross section consisting
of a single material. This transformed
Cantilever beam with w = 100 lb/ft.
section will be used for calculation purposes.
The centroid location and the moment of inertia of the transformed section about its
centroid will be calculated. With these section properties, the flexure formula will be used
to compute the bending stresses in both the wood and the aluminum for the maximum
internal bending moment produced in the cantilever span.
10 ft
z
(2)
(1)
3 in.
y
3 in.
Cross-sectional
dimensions.
0.25 in.
8 in.
SolutioN
Modular Ratio
The transformation procedure is based on the ratio of the elastic moduli of the two
materials, termed the modular ratio and denoted by n. The modular ratio is defined as the
elastic modulus of the transformed material divided by the elastic modulus of the reference
material. In this example, the stiffer material (i.e., the aluminum) will be transformed
into an equivalent amount of the less stiff
material (i.e., the wood); therefore, the wood will
be used as the reference material. The modular ratio
y
for this transformation is
(2)
Etrans
E
n = =
E E
ref
10,200 ksi
= = 6
1, 700 ksi
The width of the aluminum portion of the cross section
is multiplied by the modular ratio n. The
resulting cross section, consisting solely of wood, is
equivalent to the actual cross section, which consists
of both wood and aluminum.
Section Properties
The centroid location for the transformed section
is shown in the figure on the left. The moment of
inertia of the transformed section about the z centroidal
axis is I t = 192.5 in. 4 .
2
1
3.5987 in.
4.6513 in.
z
6 × 3 in. = 18 in.
(1)
3 in.
transformed cross section.
(2)
z
6 × 3 in. = 18 in.
(1)
y
3 in.
0.25 in.
8 in.
0.25 in.
8 in.
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