Advanced Methods of Structural Analysis

6.7 Reciprocal Theorems 193

6.7.4 Theorem of Reciprocal Unit Displacements and Reactions

(Rayleigh Second Theorem)

Let us consider two states of elastic structure subjected tounit displacementZ 1 D 1
andunit loadP 2 D 1 (Fig.6.30a). Reactionr 12 arises in constrain 1 due to unit load
P 2. Displacementı 21 occurs in direction of loadP 2 due to unit displacementZ 1.

State 2

P 2 =1

State 1

1 2 d 21

r 12

a Z 1 =1

rBA

a b

B

A

F= 1

a =1

ab

B

A

dAB

b

Fig. 6.30 Theorem of reciprocal of unit displacements and reactions

The theorem of reciprocal work in extended form should be presented as follows
r 12  1 D 1 ı 21 so we get thatr 12 Dı 21. In general,

rjkDıkj: (6.32)

The theorem of reciprocal unit displacements and reactions said thatreaction injth
constrain due to unit load ofkth direction and displacement inkth direction
due to unit displacement ofjth constrain are equal in magnitude but opposite
in sign.
This theorem is illustrated in Fig.6.30b. In order to find a vertical displacement
at the pointAdue to unit rotation of the supportB, apply unit forceFD 1 in the
directionıAB. Moment at fixed support due to forceFD 1 isrBADF.aCb/.
SinceFD 1 , therefore the vertical displacement isıABDaCb.
Theorem of reciprocal reactions and displacements will be used for analysis of
statically indeterminate structures by mixed method. In general form, this theorem
was considered byRayleigh(1873–1875). The form (6.32) was presented by Prof.
A.A.Gvozdev( 1927 ).

6.7.5 Summary

There are two principle approaches to computation of displacements. The first of
them is based on the integration of differential equation of an elastic curve of a
beam. The second approach is based on the fundamental energetic principles. Rela-
tionships between different methods of calculation displacement and their evolution
are presented in Fig.6.31.