Civil Engineering Reference
In-Depth Information
k
1
Deflection,
Figure 2.8
Force-deflection
relation for a linear elastic
spring.
free length is
0
0
1
2
k
2
0
=
W
=
F
d
=
k
d
=
U
e
(2.40)
0
0
and we observe that the work and resulting elastic potential energy are quadratic
functions of displacement and have the units of force-length. This is a general
result for linearly elastic systems, as will be seen in many examples throughout
this text.
Utilizing Equation 2.28, the strain energy for an axially loaded elastic bar
fixed at one end can immediately be written as
1
2
k
1
2
AE
L
2
2
U
e
=
=
(2.41)
However, for a more general purpose, this result is converted to a different form
(applicable to a bar element only) as follows:
PL
AE
2
P
A
P
AE
AL
1
2
k
1
2
AE
L
1
2
1
2
ε
2
U
e
=
=
=
=
V
(2.42)
where
V
is the total volume of deformed material and the quantity
2
ε
is
strain
energy per unit volume,
also known as
strain energy density
. In Equation 2.42,
stress and strain values are those corresponding to the
final
value of applied
force. The factor
2
arises from the linear relation between stress and strain as the
load is applied from zero to the final value
P
. In general, for uniaxial loading, the
strain energy per unit volume
u
e
is defined by
ε
u
e
=
d
ε
(2.43)
0
which is extended to more general states of stress in subsequent chapters. We note
that Equation 2.43 represents the area under the elastic stress-strain diagram.
