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If
p
i
and
p
Vi
are constant during deformation, Eq. (2.32a) gives
=−
−
U
e
p
i
u
i
dS
p
Vi
u
i
dV
(2.32b)
S
p
V
By comparison with Eq. (2.27), it follows that for linear response and conservative loads
U
e
=−
2
W
e
(2.33)
The complementary potential functions are obtained in a similar fashion. For prescribed
displacements,
∂
T
U
0
e
∂
u
T
=−
(2.34)
p
and
U
e
=−
S
u
u
i
p
i
dS
=−
S
u
p
T
u
dS
, where
U
0
e
is the complementary energy density of
the prescribed displacements.
2.1.4 Virtual Work
It is useful to define the work done by the loads on a body during a small, admissible change
in the displacements. An admissible or possible change is a displacement which varies con-
tinuously as a function of the coordinates and does not violate displacement boundary
conditions. Although the actual displacements may be large, the
change
in the displace-
ments must be small. Traditionally, these infinitesimal, possible changes in displacements
have been named
virtual displacements
. Virtual displacements are designated by
u
i
, which
indicates that they correspond to a variation of a function as defined in Appendix I.
The use of virtual displacements is usually traced to Johann Bernoulli, although the
concept of work as the basis for investigating mechanical problems dates back to Leonardo
da Vinci,
2
and some believe it can be traced to Aristotle.
3
A brief historical survey is given
in Lanczos (1970). A more detailed discussion of the historical developments of variational
formulations is provided in Oravas and McLean (1966).
The definition of virtual work follows directly from the definitions of work considered
earlier in this chapter. Thus, from Eq. (2.6),
internal virtual work
is given by
δ
δ
W
i
=−
V
σ
δ
ij
dV
(2.35)
ij
and
external virtual work
would be (Eq. 2.26)
δ
W
e
=
p
i
δ
u
i
dS
+
p
V
i
δ
u
i
dV
(2.36)
S
p
V
2
Leonardo da Vinci (1452-1519) was an Italian artist and scientist. An illegitimate son who was raised by his father,
he served an apprenticeship in art and studied anatomy, astronomy, botany, mathematics, engineering, and music.
His accomplishments are legendary. He is famous not only for his paintings, but also for his engineering, including
the design of a glider, a parachute, an elevator, a canal, an irrigation system, and a cathedral. He studied stresses
in columns, arches, and walls. His dissection of cadavers led to significant improvements in the understanding of
anatomy, including the heart and valves.
3
Aristotle (384-322 B.C.), the Greek philosopher, contributed to the fields of physics, astronomy, meteorology,
mathematics, biology, and psychology. At 17 he joined Plato's academy in Athens, where he remained until
Plato's death in 347. In 342 he began to tutor the future Alexander the Great. On the death of Alexander in 323,
Aristotle was charged with impiety. This led to voluntary exile. He died a few months later. At the Academy,
Aristotle is credited with initiating a fundamental change in philosophy when he steered away from Plato's
dialectic, Socratic examination in mathematics in favor of an axiomatic approach in which theorems are based on
basic principles (hypotheses, definitions, and axioms). This proof-theory approach of Greek mathematics is still
followed today.
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