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

m10.5 Superposition Warm-up. A series of examples
and exercises that illustrate basic skills required for successful
application of the super position method to beam deflection
problems.
ExAmpLE 10.9
v
v
v
80 kN/m
x
A
B
C
150 kN.m
3 m
2 m
80 kN/m
x
A
B
C
3 m
2 m
Case 1—Cantilever with uniform load.
x
A
B
C
150 kN.m
3 m
2 m
Case 2—Cantilever with concentrated moment.
The cantilever beam shown consists of a structural steel wide-flange
shape [E = 200 GPa; I = 650 × 10 6 mm 4 ]. For the loading shown,
determine
(a) the beam deflection at point B.
(b) the beam deflection at point C.
Plan the Solution
To solve this problem, the given loading will be separated into two
cases: (1) a cantilever beam with a uniformly distributed load and
(2) a cantilever beam with a concentrated moment acting at the free
end. Pertinent equations for these two cases are given in the beam
table found in Appendix C. For case 1, we will use equations for the
deflection and rotation angle at the free end of the cantilever to determine
the beam deflections at B and C. For case 2, the elastic curve
equation will be used to compute beam deflections at both locations.
SolutioN
For this beam, the elastic modulus is E = 200 GPa and the moment
of inertia is I = 650 × 10 6 mm 4 . Since the term EI will appear in all
of the equations, it may be helpful to start by computing that value:
EI = (200 GPa)(650 × 10 mm ) = 130 × 10 Nmm ⋅
6 4 12 2
= 130 × 10 kN⋅m
3 2
As in all calculations, it is essential to use consistent units throughout the computations.
This rule is particularly important in the superposition method. When we substitute numbers
into the various equations obtained from the beam table, it is easy to lose track of the
units. If this happens, you may find that you have calculated a beam deflection that seems
absurd, such as a deflection of 1,000,000 mm for a beam that spans only 3 m. To avoid
this situation, always be aware of the units associated with each variable and make sure
that all units are consistent.
Case 1—Cantilever with uniform load
From the beam table in Appendix C, the deflection at the free end of a cantilever beam
that is subjected to a uniformly distributed load over its entire span is given as
v
max
4
wL
=− (a)
8EI
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