Civil Engineering Reference
In-Depth Information
and
10
6
880
×
1
δ
D,h
=
200 000
×
1800
which gives
δ
D,h
=
2
.
44mm
Both deflections are positive which indicates that the deflections are in the directions
of the applied unit loads. Note that in the above it is unnecessary to specify units for
the unit load since the unit load appears, in effect, on both sides of the virtual work
equation (the internal
F
1
forces are directly proportional to the unit load).
Examples 15.2-15.6 illustrate the application of the principle of virtual work to the
solution of problems involving statically determinate linearly elastic structures. We
have also previously seen its application in the plastic bending of beams (Fig. 9.43),
thereby demonstrating that the method is not restricted to elastic systems. We shall
now examine the alternative energy methods but we shall return to the use of virtual
work in Chapter 16 when we consider statically indeterminate structures.
15.3 E
NERGY
M
ETHODS
Although it is generally accepted that energy methods are not as powerful as the
principle of virtual work in that they are limited to elastic analysis, they possibly find
their greatest use in providing rapid approximate solutions of problems for which
exact solutions do not exist. Also, many statically indeterminate structures may be
conveniently analysed using energy methods while, in addition, they are capable of
providing comparatively simple solutions for deflection problems which are not readily
solved by more elementary means.
Energy methods involve the use of either the
total complementary energy
or the
total
potential energy
(TPE) of a structural system. Either method may be employed to solve
a particular problem, although as a general rule displacements are more easily found
using complementary energy while forces aremore easily found using potential energy.
STRAIN ENERGY AND COMPLEMENTARY ENERGY
In Section 7.10 we investigated strain energy in a linearly elastic member subjected to
an axial load. Subsequently in Sections 9.4, 10.3 and 11.2 we derived expressions for
the strain energy in a linearly elastic member subjected to bending, shear and torsional
loads, respectively. We shall now examine the more general case of a member that is
not linearly elastic.
Figure 15.13(a) shows the
j
th member of a structure comprising
n
members. The
member is subjected to a gradually increasing load,
P
j
, which produces a gradually
