Section 5 – WORK AND ENERGY PRINCIPLES
The concepts of work and energy are fundamental to the analysis of structures.
They provide our most helpful tools and allow us to simplify the processes of
satisfying the three basic requirements of equilibrium, stressstrain and
compatibility.
Without these principles it would be very difficult to solve many
problems with any degree of assurance; they form the basis of modern analysis
including the finite element techniques implemented in a variety of software.
Engineers, mathematicians and physicists try to find grand unifying principles to
explain what they do.
For mechanics, including dynamics, materials and
structures, work and energy relationships tie together forces, displacements,
velocities and accelerations.
For structures we normally focus on the
relationships between forces and displacements; this makes work the most
obvious place to look to find a fundamental principle.
Consider a particle that can translate in threedimensional space.
Let us
suppose that several forces act on it while it undergoes a small displacement du.
The situation is illustrated by the following figure.
F
1
F
2
F
3
x
y
O
z
Figure 5.1 – Particle Translating in Space
The work done by any of these forces during the displacement is
i
dW
F du
=
r
r
g
.
Over a finite displacement from
to
a
b
u
u
r
r
, the work that this force does is given by
the integral.
i
where
is the component of F in the direction of the displacement
b
b
a
a
u
u
i
i
iu
u
u
iu
W
F du
F du
F
=
=
∫
∫
r
r
r
r
g
r
© 1999, 2000 Robert O. Meitz
41
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Lecture Notes
Aerospace Structures
Page 42
Note that the nature of the force is not defined in any particular way.
All that has
been said is that the force exists and that its point of application has moved by
some amount.
The work done by the force depends only on the history of the
force as the displacement takes place as implied by the integral expressions.
With several forces acting as shown in Figure 1, the total work is the sum of the
work done by the several forces.
1
1
b
a
n
n
u
i
i
u
i
i
W
W
F du
=
=
=
=
∑
∑
∫
r
r
r
r
g
The force system acting on a particle is statically equivalent to a single force
called the resultant and may be replaced by it in any expression.
If the forces on
the particle are in equilibrium, the resultant force is zero and the net work done
by the force system is also zero.
This will be true no matter what direction the
displacement may take.
This observation makes it clear that we can use the concept of work to establish
whether a force system is in equilibrium.
Specifically, if we discover that the net
work is zero for any possible displacement, then the forces on the object are in
equilibrium. The precise formulation of any work principle will depend on the
nature of the problem being considered.
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 Fall '06
 Meitz
 Energy, Force, Aerospace Structures, Robert O. Meitz

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