Page 83 - Aircraft Stuctures for Engineering Student

Energy methods of

st r uct u ra I ana I ys i s

In Chapter 2 we have seen that the elasticity method of structural analysis embodies
the determination of stresses and/or displacements by employing equations of
equilibrium and compatibility in conjunction with the relevant force-displacement
or stress-strain relationships. A powerful alternative but equally fundamental
approach is the use of energy methods. These, while providing exact solutions for
many structural problems, find their greatest use in the rapid approximate solution
of problems for which exact solutions do not exist. Also, many structures which
are statically indeterminate, that is they cannot be analysed by the application of
the equations of statical equilibrium alone, may be conveniently analysed using an
energy approach. Further, energy methods provide comparatively simple solutions
for deflection problems which are not readily solved by more elementary means.
Generally, as we shall see, modern analysis' uses the methods of total comple-
mentary energy and total potential energy. Either method may be employed to solve
a particular problem, although as a general rule deflections are more easily found
using complementary energy, and forces by potential energy.
Closely linked with the methods of potential and complementary energy is the
classical and extremely old principle of virtual work embracing the principle of virtual
displacements (real forces acting through virtual displacements) and the principle of
virtual forces (virtual forces acting through real displacements). Virtual work is in fact
an alternative energy method to those of total potential and total complementary
energy and is practically identical in application.
Although energy methods are applicable to a wide range of structural problems and
may even be used as indirect methods of forming equations of equilibrium or
compatibility'>2, we shall be concerned in this chapter with the solution of deflection
problems and the analysis of statically indeterminate structures. We shall also include
some methods restricted to the solution of linear systems, viz. the unit loadmethod, the
principle of superposition and the reciprocal theorem.

Figure 4.l(a) shows a structural member subjected to a steadily increasing load P. As
the member extends, the load P does work and from the law of conservation of energy