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

Beam
Segment
x Coordinate
origin
limits
(mm)
I
m
(N ⋅ mm)
M
(N ⋅ mm)
BC C 0–165 20,000 -1x -900x
AB C 165–
465
160,000 -1x -2,300x +
1,400(165)
∫
⎛
m M ⎞
⎜ ⎟
⎝ EI ⎠
dx
2
67,381.875 N /mm
E
2
323,817.188 N /mm
E
2
391,199.063 N /mm
E
Virtual-Work Equation: From Equation (17.35), the beam deflection at C can now be
determined:
2 2
391,199.063 N /mm 391,199.063 N /mm
(1 N) ⋅D C = =
2
E
70,000 N/mm
∴D = 5.59 mm ↓ Ans.
C
ExAmpLE 17.15
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
The real loadings are discontinuous at points B and D,
while the virtual loading for the beam is discontinuous at
C. 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 x coordinate
origin 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 pertinent equations in
a tabular format.
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. A free-body diagram of the beam is
shown.
A
A
180 kN
B
3 m 1.5 m 1.5 m 3 m
B
45 kN/m
C
1 kN
3 m 1.5 m 1.5 m 3 m
1
2 kN 1
2 kN
C
D
D
x 3
x 1
x 2
x 4
E
E
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