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

ExAmpLE 17.19
Compute the deflection at point C for the simply supported
beam shown. Assume that EI = 3.4 × 10 5 kN ⋅ m 2 .
Plan the Solution
Since the deflection is desired at C and no external load acts at
that location, a dummy load P will be required at C. With
dummy load P placed at C, the bending-moment equation will
be discontinuous at points B, C, and D. Therefore, this beam
must be considered in four segments: AB, BC, CD, and
DE. To facilitate the derivation of moment equations, it
will be convenient to locate the origin of the x coordinate
system at A for segments AB and BC, and at E for segments
CD and DE. To organize the calculation, it will
also be convenient to summarize the relevant equations
in a tabular format.
SolutioN
Place a dummy load P at point C, which is located at the
center of the 9 m beam span. Determine the beam reactions,
taking care to include both the actual loads and the
dummy load P. The reaction forces at A and E are shown
on the free-body diagram of the beam.
Draw a free-body diagram around support A of the beam, cutting through segment
AB. The origin of the x coordinate system will be placed at A. From the diagram, derive
the following equation for the internal moment M in segment AB of the beam:
M
45 kN/m ⎛ P ⎞
=- x + ⎜ + 300 kN⎟x
2 ⎝ 2 ⎠
0 ≤ x ≤ 3 m
1 2 1
1
Repeat the process with a free-body diagram cut through segment BC of the beam.
From that diagram, derive the following equation for the internal moment M in segment
BC of the beam:
45 kN/m
⎛ P ⎞
M =- x - (180 kN)( x - 3m) + ⎜ + 300 kN⎟x
2
⎝ 2 ⎠
3m≤
x ≤ 4.5 m
2 2 2 2
2
+
For segment CD, draw a free-body diagram around support E, cutting through
segment CD of the beam. From the diagram, derive the following equation for the
internal moment M in segment CD of the beam:
M
45 kN/m
2
⎛ P ⎞
=- ( x3
- 3 m) + ⎜ + 150 kN⎟x
2
⎝ 2 ⎠
3m≤
x ≤ 4.5 m
3
3
+
785