Page 92 - Aircraft Stuctures for Engineering Student

4.6 Application to deflection problems 77

the externally applied virtual forces. From the principle of virtual forces, i.e. Eq. (4.9)

- lv0, ydP + kAr6Pr = 0 (4.12)
r= 1
Comparing Eq. (4.12) with Eq. (4.2) we see that each term represents an increment in
complementary energy; the first, of the internal forces, the second, of the external
loads. Thus Eq. (4.12) may be rewritten
S( ci + C,) = 0 (4.13)
where
P
Ci=[ ydP and Ce=-eArPr (4.14)
VOI 0 r= 1
C, is in fact the complement of the potential energy V of the external loads. We shall
now call the quantity (Ci + C,) the total complementary energy C of the system.
The displacements specified in Eq. (4.12) are real displacements of a continuous
elastic body; they therefore obey the condition of compatibility of displacement so
that Eqs (4.12) and (4.13) are, in exactly the same way as Eq. (4.9), equations of
geometrical compatibility. The principle of the stationary value of the total complemen-
tary energy may then be stated as:
For an elastic body in equilibrium under the action of applied forces the true internal
forces (or stresses) and reactions are those for which the total complementary energy
has a stationary value.
In other words the true internal forces (or stresses) and reactions are those which
satisfy the condition of compatibility of displacement. This property of the total
complementary energy of an elastic system is particularly useful in the solution of
statically indeterminate structures, in which an infinite number of stress distributions
and reactive forces may be found to satisfy the requirements of equilibrium.

4.6 Application to deflection problems

Generally, deflection problems are most readily solved by the complementary energy
approach, although for linearly elastic systems there is no difference between the
methods of complementary and potential energy since, as we have seen, complemen-
tary and strain energy then become completely interchangeable. We shall illustrate
thc method by reference to the deflections of frames and beams which may or may
not possess linear elasticity.
Let us suppose that we require to find the deflection A2 of the load P2 in the simple
pin-jointed framework consisting, say, of k members and supporting loads
PI, P2,. . . , Pn, as shown in Fig. 4.7. From Eqs (4.14) the total complementary
energy of the framework is given by
k rF n
Xi
c=CJ,' dFi - ArP, (4.15)
i= 1 r= 1