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

ExAmpLE 17.3
A cantilever beam AB of length L and flexural rigidity EI supports
the linearly distributed loading shown. Determine the elastic
strain energy due to bending stored in this beam.
Plan the Solution
Consider a free-body diagram that cuts through the beam at a
distance x from the free end of the cantilever. Derive the bendingmoment
equation M(x), and then use it in Equation (17.20) to
determine the elastic strain energy.
A
L
w 0
B
x
SolutioN
Draw a free-body diagram that cuts through the beam at an arbitrary distance x from
the origin. The equilibrium equation for the sum of moments about section a–a is
wx 0 ⎛ x⎞
x
∑ Ma a=
⎜ ⎟ ⎛ M 0
L ⎝ 2⎠
⎝ ⎜ ⎞
−
⎟ + =
3⎠
Therefore, the bending-moment equation for this beam is
A
x
a
a
w 0 x
L
V
M
x
wx
Mx ( ) =−
6L
0 3
The elastic strain energy in a beam is given by Equation (17.20) as
U
=
L
∫ 0
2
M
2EI dx
For the cantilevered beam considered here,
U
1 wx
w
= ⎛ ⎞
⎜-
⎟ dx =
2EI
⎝ 6L
⎠ 72EI L
0
∫ 2 0 2 ∫ 0
L 0 3 2
L
6
x dx
or
U
wL
0 2 7 wL
0 2 5
= = Ans.
2
504EI L 504EI
ExAmpLE 17.4
A simply supported beam ABC of length L and flexural
rigidity EI supports the concentrated load shown. What is
the elastic strain energy due to bending that is stored in this
beam?
Plan the Solution
Determine the beam reactions from a free-body diagram of
the entire beam. Then, consider two free-body diagrams that
cut through the beam. The first free-body diagram cuts
through the beam at a distance x from pin support A. From
this diagram, derive the bending-moment equation M(x) for
A
a
L
B
P
b
C
x
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