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

conditions. The reactions for statically determinate beams are easily computed in a fashion
that is familiar to all engineering students. Accordingly, beam reaction forces and moments
will be incorporated into the load function w(x) in the examples presented subsequently in
this section.
To summarize, constants of integration arise in the double integration of w(x) to obtain
V(x) and M(x). If w(x) is formulated solely in terms of the applied loads, then these constants
must be explicitly determined with the use of boundary conditions. However, if beam reaction
forces and moments are included in w(x) along with the applied loads, then constants of
integration are redundant and thus unnecessary for the V(x) and M(x) functions.
Application of Discontinuity Functions to Determine V and M. Table 7.2 summarizes
discontinuity expressions for w(x) that are required for various common loadings. It
is important to keep in mind that Macaulay functions continue indefinitely for x > a. In
other words, once a Macaulay function is switched on, it stays on for all increasing values
of x. In accordance with the concept of superposition, a Macaulay function is cancelled by
the addition of another Macaulay function to the beam’s w(x) function.
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dISCONTINuITy FuNCTIONS
TO REPRESENT LOAd,
SHEAR, ANd MOMENT
Macaulay functions continue
indefinitely for x > a. Therefore,
a new Macaulay function (or, in
some cases, several functions)
must be introduced to terminate
a previous function.
ExAmpLE 7.10
Use discontinuity functions to obtain expressions for the internal
shear force V(x) and internal bending moment M(x) in the
beam shown. Then, use these expressions to plot the shearforce
and bending-moment diagrams for the beam.
Plan the Solution
Determine the reactions at simple supports A and F. Using
Table 7.2, write expressions for w(x) for each of the three
A x
A
A
y
y
A y
120 kN.m
loads acting on the beam, as well as for the two support reactions. Integrate w(x) to
determine an equation for the shear force V(x), and then integrate V(x) to determine an
equation for the bending moment M(x). Plot these functions to complete the shearforce
and bending-moment diagrams.
SolutioN
Support Reactions
An FBD of the beam is shown. The equilibrium equations
are as follows:
Σ F = A = 0 (trivial)
x
x
Σ Fy = Ay + Fy
− 45 kN − (30 kN/m)(3 m) = 0
Σ M = 120 kN⋅m − (45 kN)(4 m)
A
− (30 kN/m)(3 m)(7.5 m) + F (12m) = 0
y
B
45 kN
C
2 m 2 m 2 m 3 m 3 m
120 kN.m
B
45 kN
C
D
D
30 kN/m
30 kN/m
2 m 2 m 2 m 3 m 3 m
E
E
x
F
x
F
F y
From these equations, the beam reactions are
A
y
= 73.75 kN and F = 61.25 kN
Discontinuity Expressions
Reaction force A y : The upward reaction force at A is expressed by
y
−
wx ( ) = A 〈 x − 0m〉 = 73.75 kN〈 x − 0m〉
y
1 −1
(a)