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

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EQuILIbRIuM OF bEAMS
w(x)
w A
w B
w C
w D
w E
w F
w G
A B C D E F G
x
The change in shear force, ∆V, between
any two locations is equal to the area
under the distributed-load curve.
V E
V F
∆V
V
M
The positive direction for a
concentrated load is upward.
M B
∆M
An upward load P causes the
shear diagram to jump upward.
Similarly, a downward load P
causes the shear diagram to
jump downward.
M
V
∆x
P 0
O
FIGURE 7.11a Free-body
diagram of beam element
subjected to concentrated
load P 0 .
V O
M
M 0
∆x
M +∆M
V+∆V
M +∆M
V+∆V
FIGURE 7.11b Free-body
diagram of beam element
subjected to concentrated
moment M 0 .
The change in bending moment, ∆M, between any
two locations is equal to the area under the V curve.
FIGURE 7.10 Relationships between areas for the load, shear, and moment diagrams.
Regions of concentrated Loads and moments
Equations (7.1) through (7.4) were derived for a portion of the beam subjected to a distributed
load only. Next, consider the free-body diagram of a very thin portion of the beam (see
Figure 7.8a) directly beneath one of the concentrated loads (Figure 7.11a). Force equilibrium
for this free body can be stated as
Σ F = V + P − ( V +D V) = 0 ∴ D V = P
y 0 0
(7.5)
This equation shows that the change in shear force, DV, between the left and right sides of
a thin beam element is equal to the intensity of the external concentrated load P 0 acting on
the beam element. At the location of a positive external load, the shear-force diagram is
discontinuous: It “jumps” upward by an amount equal to the intensity of an upward concentrated
load. A downward external concentrated load causes the shear-force diagram to
jump downward. (See Example 7.1.)
Next, consider a thin beam element located at a concentrated moment (Figure 7.11b).
Moment equilibrium for this element can be expressed as
As Dx approaches zero,
M C
Σ M = −M −VD x + M + ( M + D M) = 0
O 0
D M = − M 0
(7.6)
The moment diagram is discontinuous at locations where external concentrated moments
are applied. Equation (7.6) reveals that the change in internal bending moment, DM, between