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

y
w a
w b
y
w 0
x
x
A
B
C
A
B
a
b
L
FIGURE p7.4
FIGURE p7.6
P
y
w 0
M B
A
a
B
b
C
x
A B C
a
b
x
FIGURE p7.5
FIGURE p7.7
7.3 Graphical method for constructing Shear
and moment Diagrams
As shown in Section 7.2, we can construct shear and moment diagrams by developing
functions that express the variation of the internal shear force V(x) and the internal bending
moment M(x) along the beam and then plotting these functions. When a beam has several
loads, however, this approach can be quite time consuming and a simpler method is desired.
The process of constructing shear and moment diagrams is much easier if specific relationships
among load, shear, and moment are taken into consideration.
Consider a beam subjected to several loads, as shown in Figure 7.8a. All loads are
shown in their respective positive directions. We will investigate a small portion of the
beam where there are no external concentrated loads or concentrated moments. This small
beam element has length ∆x (Figure 7.8b). An internal shear force V and an internal bending
moment M act on the left side of the beam element. Because the distributed load is acting
on this element, the shear force and bending moment on the right side must be slightly
different in order to satisfy equilibrium. Specifically, they must have values of V + ∆V and
M + ∆M, respectively. All shear forces and bending moments are assumed to act in their
positive directions, as defined by the sign convention shown in Figure 7.6. The distributed
load can be replaced by its resultant force w(x) ∆x, which acts at a fractional distance k ∆x
w(x) ∆x
w(x)
k ∆x
P 0
P 1
M+∆M
y
w(x)
V
O
FIGURE 7.8a
A
M 0
M 1
x
∆x
Generalized beam subjected to positive external loads.
B
x
M
∆x
V+∆V
FIGURE 7.8b Beam element
showing internal shear forces
and bending moments.
205