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

ExAmpLE 17.14
1,400 N
900 N
Calculate the deflection at end C of the cantilever beam shown.
Assume that E = 70 GPa for the entire beam.
A
I = 160,000 mm 4
B
I = 20,000 mm 4 C
300 mm 165 mm
C
1 N
Plan the Solution
For the cantilever beam considered here, the virtual-moment equation
will be continuous over the entire span, but the real-moment
equation is discontinuous at B. Therefore, the beam must be considered
in two segments: AB and BC. The moment equations will be
simpler to derive if the origin of the x coordinate system is placed
at the free end C.
A
m
B
x
300 mm 165 mm
1 N
SolutioN
Virtual Moment m: Remove the real loads from the beam and
apply a virtual unit load downward at C, where the deflection is
desired. At this stage of the calculation, it makes no difference that
the beam depth changes along the span, since only the virtual
moment m is needed here.
Now draw a free-body diagram around end C of the beam.
Place the origin of the x coordinate system at C. From the free-body
diagram, derive the following equation for the virtual internal
moment m:
v
x
C
m =−(1 N) x 0 ≤ x ≤ 465 mm
M
900 N
Real Moment M: Remove the virtual load, and reapply the real loads at B and
C. Draw a free-body diagram around end C that cuts through segment BC of the
beam. The same x coordinate used to derive the virtual moment must be used to
derive the real moment; therefore, the origin of the x coordinate system must be
placed at C. From the free-body diagram, derive the following equations for the
real internal moment M:
V
x
C
M =−(900 N) x 0 ≤ x ≤ 165 mm
Now draw a second free-body diagram that cuts through segment AB of the
beam and includes the free end of the cantilever. From the free-body diagram,
derive the following equations for the real internal moment M:
M
1,400 N
900 N
M =−(1,400 N)( x −165 mm) − (900 N) x
=− (2,300 N) x + (1,400 N)(165 mm)
165 mm < x ≤ 465 mm
V
B
x
165 mm
C
The moment of inertia differs for segments AB and BC. This difference will
be incorporated into the calculation in the term M/EI. The equations developed
so far, along with the limits of integration, are conveniently summarized in the
following table:
768