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

y
6 kips/ft
4 kips/ft
y
25 kN
30 kN/m
A B
C
D
8 ft 7 ft 10 ft 5 ft
FIGURE p7.30
E
x
A B C D
1 m 3 m
2 m
FIGURE p7.31
x
7.4 Discontinuity Functions to Represent
Load, Shear, and moment
In Section 7.2, we constructed shear and moment diagrams by developing functions that
express the variation of internal shear force V(x) and internal bending moment M(x) along
the beam and then plotting these functions. The method of integration used in Section 7.2
is convenient if the loads can be expressed as continuous functions acting over the entire
length of the beam. However, if several loadings act on the beam, this approach can become
extremely tedious and time consuming because a new set of functions must be developed
for each interval of the beam.
In this section, a method will be presented in which a single function is formulated
that incorporates all loads acting on the beam. This single load function w(x) will be constructed
in such a way that it will be continuous for the entire length of the beam even
though the loads may not be. The load function w(x) can then be integrated twice—first to
derive V(x) and a second time to obtain M(x). To express the load on the beam in a single
function, two types of mathematical operators will be employed. Macaulay functions will
be used to describe distributed loads, and singularity functions will be used to represent
concentrated forces and concentrated moments. Together, these functions are termed discontinuity
functions. Their usage has restrictions that distinguish them from ordinary
functions. To provide a clear indication of these restrictions, the traditional parentheses
used with functions are replaced by angle brackets, called Macaulay brackets, that take the
form 〈x − a〉 n .
macaulay Functions
Distributed loadings can be represented by Macaulay functions, which are defined in
general terms as follows:
n
〈 x − a〉 =
0 when x < a
n
( x − a) when x ≥ a
forn
≥ 0 ( n = 0,1,2,...) (7.7)
Whenever the term inside the brackets is less than zero, the Macaulay function equals zero
and it is as if the function does not exist. However, when the term inside the brackets is
greater than or equal to zero, the Macaulay function behaves like an ordinary function,
which would be written with parentheses. In other words, the Macaulay function acts like
a switch in which the function turns on for values of x greater than or equal to a.
Three Macaulay functions corresponding, respectively, to n = 0, n = 1, and n = 2 are
plotted in Figure 7.13. In Figure 7.13a, the function 〈x − a〉 0 is discontinuous at x = a,
producing a plot in the shape of a step. Accordingly, this function is termed a step function.
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