Page 119 - Aircraft Stuctures for Engineering Student

4.10 The reciprocal theorem 103

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An extremely useful principle employed in the analysis of linearly elastic structures is
that of superposition. The principle states that if the displacements at all points in an
elastic body are proportional to the forces producing them, i.e. the body is linearly
elastic, the,effect on such a body of a number of forces is the sum of the effects of
the forces applied separately. We shall make immediate use of the principle in the
derivation of the reciprocal theorem in the following section.

The reciprocal theorem is an exceptionally powerful method of analysis of linearly
elastic structures and is accredited in turn to Maxwell, Betti and Rayleigh. However,
before we establish the theorem we first consider a useful property of linearly elastic
systems resulting from the principle of superposition. The principle enables us to
express the deflection of any point in a structure in terms of a constant coefficient
and the applied loads. For example, a load PI applied at a point 1 in a linearly elastic
body will produce a deflection AI at the point given by

Ai = QllPl
in which the influence orflexibility coefficient all is defined as the deflection at the
point 1 in the direction of PI, produced by a unit load at the point 1 applied in the
direction of P1. Clearly, if the body supports a system of loads such as those
shown in Fig. 4.24, each of the loads PI, Pz, . . . , P,, will contribute to the deflection
at the point 1. Thus, the corresponding deflection A, at the point 1 (i.e. the total deflec-
tion in the direction of Pi produced by all the loads) is then

AI =allPl +a12P2+~~~+al,P,
where aI2 is the deflection at the point 1 in the direction of PI, produced by a unit load
at the point 2 in the direction of the load P2 and so on. The corresponding deflections

.
Fig. 4.24 Linearly elastic body subjected to loads f,, fz, f3,. , f,,.
.