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

order to complete this FBD, however, the vertical reaction force A y
and the moment reaction M A must be determined. Perhaps it
might be simpler to consider an FBD of the right portion of the
cantilever, since the reactions at fixed support A do not appear on
that FBD.
An FBD of the right portion of the cantilever beam is shown.
A common mistake at this stage of the analysis is to define the
beam length between section a–a and B as x. Note, however, that
the origin of the x–v coordinate system is located at support A,
with positive x extending to the right. Therefore, to be consistent
with the defined coordinate system, the length of the beam segment
must be denoted L − x. This simple coordinate transformation
is the key to success with this type of problem.
Accordingly, cut through the beam at section a–a, and consider
the beam and its loads between a–a and the free end of the
cantilever at B. Note that a clockwise internal moment M is
shown acting on the beam segment at a–a. Clockwise is the positive
direction for an internal moment acting on the left face of a
bending element, and this direction is consistent with the sign
convention shown in Figure 10.5.
The equilibrium equation for the sum of moments about a–a is
−
Σ = − −
⎛ L x
M
⎞
a−a
w( L x)
⎝ ⎠ − M
2
= 0
Therefore, the bending-moment equation for this beam is
v
M A
w
a
M
A
a
V
x
L – x
B
L
Free-body diagram of the left portion
of the cantilever beam.
v
w (L – x)
w
M
a
x
A y
A
a
V
L – x
B
Free-body diagram of the right portion
of the cantilever beam.
x
M
w
=− L − x
2 ( )2 (a)
Notice that this equation is valid for the interval 0 ≤ x ≤ L. Substitute the expression for
M into Equation (10.1) to obtain
EI d 2
v
2
dx
w
=− L − x
2 ( ) 2
(b)
integration
The first integration of Equation (b) gives
EI dv
dx
w
=+ L − x + C
6 ( )3 1 (c)
where C 1 is a constant of integration. Note the sign change on the first term. Integrating
again gives
w
EIv =− L − x + Cx + C
24 ( )4 1 2 (d)
where C 2 is a second constant of integration.
403