Section 5.2
Solid Mechanics Part I
Kelly
180
5.2 Elastic Strain Energy
The strain energy stored in an elastic material upon deformation is calculated below for a
number of different geometries and loading conditions.
These expressions for stored
energy will then be used to solve some elasticity problems using the energy methods
mentioned in the previous section.
5.2.1
Strain energy in deformed Components
Bar under axial load
Consider a bar of elastic material fixed at one end and subjected to a steadily increasing
force
P
, Fig. 5.2.1.
The force is applied slowly so that kinetic energies are negligible.
The initial length of the bar is
L
.
The work
dW
done in extending the bar a small
amount
Δ
d
is
1
Δ
=
Pd
dW
(5.2.1)
Figure 5.2.1: a bar loaded by a constant stress
It was shown in §4.3.2 that the force and extension
Δ
are linearly related through
EA
PL
/
=
Δ
, Eqn. 4.3.5, where
E
is the Young’s modulus and
A
is the cross sectional
area.
This linear relationship is plotted in Fig. 5.2.2.
The work expressed by Eqn. 5.2.1 is
the white region under the force-extension curve (line).
The total work done during the
complete extension up to a
final
force
P
and
final
extension
Δ
is the total area beneath
the curve.
The work done is stored as elastic strain energy
U
and so
EA
L
P
P
U
2
2
1
2
=
Δ
=
(5.2.2)
If the axial force (and/or the cross-sectional area and Young’s modulus) varies along the
bar, then the above calculation can be done for a small element of length
dx
.
The energy
stored in this element would be
EA
dx
P
2
/
2
and the total strain energy stored in the bar
would be
1
the small change in force during this small extension may be neglected
P
L
Δ
d
Δ

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