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

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ENERgy METHOdS
The total energy of a system in static equilibrium and subjected to any combination of
loads is the sum of the strain energies stored in the system as a result of each type of load.
Consequently, energy methods make it possible to readily determine the total deformation
of a solid body subjected to multiple loads—a situation frequently encountered in engineering
applications.
17.2 Work and Strain Energy
Work of a Force
Work W is defined as the product of a force that acts on a particle (often, in a body) and the
distance the particle (or body) moves in the direction of the force. For example, Figure 17.1
shows two forces acting on a body. As the body moves from initial position (a) to displaced
position (b), a particle in it moves from location A to location A′, a distance d 1 , and another
particle in it moves from location B to location B′, a distance d 2 .
Even though the particle acted on by force F 1 has moved a total distance d 1 , the work
done by this force is simply W 1 = F 1 s 1 because the work done by a force is defined as the
product of the force and the distance that the particle it acts on has moved in the direction
that the force acts. Similarly, the work done by force F 2 is W 2 = F 2 s 2 . Work can be either a
positive or a negative quantity. Positive work occurs when the particle moves in the same
direction that the force acts. Negative work occurs when the particle moves in a direction
opposite that in which the force acts. In Figure 17.1, the work done by forces F 1 and F 2 is
positive if the body moves from position (a) to position (b). The work done by forces F 1
and F 2 is negative if the body moves from position (b) to position (a).
Next, consider a prismatic bar of length L that is subjected to a constant external load
P as shown in Figure 17.2. The load will be applied to the bar very slowly—increasing
from zero to its maximum value P—so that any dynamic or inertial effects due to motion
are precluded. As the load is applied, the bar gradually elongates. The bar attains its maximum
deformation δ when the full magnitude of P is reached. Thereafter, both the load and
the deformation remain unchanged.
The work done by the load is the product of the magnitude of the force and the distance
that the particle (or body) the force acts on moves; however, in this instance the force
changes its magnitude from zero to its final value P. As a result, the work done by the load
as the bar elongates is dependent on the manner in which the force and the corresponding
y
L
(a) Initial
Position
A
B
F 2
d F 2
2
B′
F 1
s 2
d 1
F 1
A′
δ
s 1
(b) Displaced
Position
x
P
FIGURE 17.1 Forces acting on a body that
changes position.
FIGURE 17.2 Prismatic bar
with static load P.