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

Maximum Bending Moment
The maximum bending moment for a 10 ft long cantilever beam with a uniformly distributed
load w = 100 lb/ft is
M
max
2 2
wL (100 lb/ft)(10ft)
=− =− =−5,000 lb⋅ ft = −60,000 lb⋅in.
2
2
Flexure Formula
The flexure formula [Equation (8.7)] gives the bending stress at any coordinate location y;
however, the flexure formula is valid only if the beam consists of a homogeneous material.
The transformation process used to replace the aluminum plate with an equivalent amount
of wood was necessary to obtain a homogeneous cross section that satisfies the limitations
of the flexure formula.
The transformed section consisting entirely of wood is equivalent to the actual cross
section. The transformed section is equivalent because the bending strains produced in
the transformed section are identical to the strains produced in the actual cross section.
The bending stresses in the transformed section, however, require an additional adjustment.
The bending stresses computed for the original wood portion of the cross section
[i.e., area (1)] are correctly computed from the flexure formula. The bending stresses
computed for the aluminum plate must be multiplied by the modular ratio n to account for
the difference in elastic moduli of the two materials.
(2)
3 in.
y
0.25 in.
100 lb/ft
y
+660 με
+614 με
y
6,730 psi
1,044 psi 6,260 psi
z
8 in.
x
ε x
σ x
60,000 lb∙in.
(1)
–853 με
–1,450 psi
3 in.
(contraction) (elongation)
(compression) (tension)
Beam cross section. Profile view of beam. Bending strains. Bending stresses.
Maximum Bending Stresses in the Wood
The maximum bending stress in the wood portion (1) of the cross section occurs at the
lower surface of the beam. Since the wood was not transformed, Equation (8.17) is used
to compute the maximum bending stress:
My ( 60,000 lb in.)( 4.6513 in.)
σ x1 =− =− − ⋅ − =− 1, 450 psi = 1,450 psi (C) Ans.
4
I
192.5 in.
t
Maximum Bending Stresses in the Aluminum
The aluminum portion of the cross section was transformed in the analysis to an equivalent
width of wood. While the bending strains for the transformed section are correct, the
bending stress for the transformed material must be multiplied by the modular ratio n to
account for the differing elastic moduli of the two materials. The maximum bending
stress in the aluminum portion (2) of the cross section (a stress that occurs at the upper
surface of the beam) is computed from Equation (8.18):
278
n My ( 60,000 lb in.)(3.5987 in.)
σ x2 =− =−6 − ⋅ = 6,730 psi = 6,730 psi (T) Ans.
4
I
192.5 in.
t