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

w(x) = w G at G. At A, where the distributed load w is zero, the slope of the shear-force
diagram is also zero. Moving to the right along the beam span, the distributed load increases
to a small positive value at B and, accordingly, the slope of the shear-force diagram at B is
a small positive value (i.e., the shear-force diagram slopes slightly upward). At points C
through G, the magnitude of the distributed load gets larger and larger (i.e., more and more
positive). Similarly, the slope of the shear-force diagram at these points becomes increasingly
more positive. In other words, the V curve gets increasingly steeper as the distributed
load w gets larger.
In a similar manner, Equation (7.2) states that the slope of the bending-moment
diagram at any point is equal to the shear force V at that same point. At point A in Figure
7.9, the shear force V A is a relatively large negative value; therefore, the slope of
the bending-moment diagram is a relatively large negative value. In other words, the
M diagram slopes sharply downward. At points B and C, the shear forces V B and V C are
still negative, but not as negative as V A . Consequently, the M diagram still slopes downward,
but not as steeply as at A. At point D, the shear force V D is zero, meaning that the
slope of the M diagram is zero. (This fact represents an important detail because the
maximum and minimum values of a function are at those locations where the slope of
the V diagram is zero.) At point E, the shear force V E becomes a small positive number
and, accordingly, the M diagram begins to slope upward slightly. At points F and G, the
shear forces V F and V G are relatively large positive numbers, meaning that the M diagram
slopes sharply upward.
Equations (7.1) and (7.2) may be rewritten in the form dV = w(x) dx and dM = V dx. The
terms w(x) dx and V dx represent differential areas under the distributed-load and shear-force
diagrams, respectively. Equation (7.1) can be integrated between any two locations x 1 and
x 2 on the beam:
∫
V
V
1
2
dV =
∫
x
x
1
2
wx ( ) dx
207
graphical Method for
Constructing Shear and
Moment diagrams
For brevity, the shear-force
diagram is also termed the V
diagram or the V curve. The
bending-moment diagram is
also termed the M diagram
or the M curve.
Integrating gives the following relationship:
2
∆ V = V − V = w( x)
dx
2 1
Similarly, Equation (7.2) can be expressed in integral form as
which gives the relationship
M2
∫ dM =
M
1
∫
∫
x
x
1
x
x
2
1
Vdx
(7.3)
The terms load diagram and
distributed-load curve are
synonyms. For brevity, the
distributed-load curve is
referred to as the w diagram
or the w curve.
2
∆ M = M − M = Vdx
(7.4)
2 1
Equation (7.3) reveals that the change in shear force, ∆V, between any two points on the
beam is equal to the area under the distributed-load curve between those same two points.
Similarly, Equation (7.4) states that the change in bending moment, ∆M, between any two
points is equal to the corresponding area under the shear-force curve.
To illustrate the significance of Equations (7.3) and (7.4), consider the beam shown in
Figure 7.10. The change in shear force between points E and F can be found from the area
under the distributed-load curve between those same two points. Similarly, the change in
bending moment between points B and C is given by the area under the V curve between
those same two points.
∫
x
x
1
207