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

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EQuILIbRIuM OF bEAMS
w(x)
w(x)
ε
P 0 x – a –1
—ε
P 0
x
x
a
(a) Concentrated force as a special case of a distributed load
a
w(x)
w(x)
ε
—ε
P 0 M
= —ε 0
2
M x – a –2
0
x
x
FIGURE 7.14 Singularity
functions to represent (a)
concentrated forces and (b)
concentrated moments.
a
ε
(b) Concentrated moment as a special case of a distributed load
a
As before, the function has a value of M 0 only at x = a and is otherwise zero. In Equation
(7.10), notice that n = −2, which ensures that the result of the function has consistent units
of force per unit length.
Integrals of Discontinuity Functions
Integration of discontinuity functions is defined by the following rules:
n+
1
〈 x − a〉
∫ 〈 − 〉 n
x a dx = n + 1
forn
≥ 0
n+
1
〈 x − a〉 forn
< 0
(7.11)
Notice that, for negative values of the exponent n, the only effect of integration is that n
increases by 1.
Constants of integration. The integration of Macaulay functions produces constants
of integration. The constant of integration that results from the integration of w(x) to obtain
V(x) is simply the shear force at x = 0—that is, V(0). Similarly, the second constant of
integration that results when V(x) is integrated to obtain M(x) is the bending moment at
x = 0—that is, M(0). If the loading function w(x) is written solely in terms of the applied
loads, then constants of integration must be included in the integration process and evaluated
with the use of boundary conditions. As these constants of integration are introduced
into either the V(x) or M(x) functions, they are expressed by singularity functions in the
form C〈x〉 0 . After their introduction into either V(x) or M(x), the constants are integrated
in the usual manner in subsequent integrals.
However, the same result for both V(x) and M(x) can be obtained by including the reaction
forces and moments in the load function w(x). The inclusion of reaction forces and
moments in w(x) has considerable appeal, since the constants of integration for both V(x)
and M(x) are automatically determined without the need for explicit reference to boundary