Advanced Methods of Structural Analysis

192 6 Deflections of Elastic Structures

6.7.3 Theorem of Reciprocal Unit Reactions

(Rayleigh First Theorem)

Let us consider two states of elastic structure subjected tounit displacementsof
supports. They areZ 1 D 1 andZ 2 D 1 (Fig.6.29a). Reactions caused by unit
displacements are called theunit reactionsand denoted by letterrmn. The first index
mindicates constrain where unit reaction arises and the second indexndenotes
constrain, which is subjected to unit displacement.
Thusr 11 andr 12 are reactions in the constrain 1 due to displacementZ 1 D 1
andZ 2 D 1 , respectively andr 21 andr 22 are reactions in the constrain 2 due to
displacementZ 1 D 1 andZ 2 D 1 , respectively.

State 2

Z 2 =1

Z 1 =1

State 1

r 11 r 21

r 22

r 12

1

a

Z 2 =1

Z 1 =1

r 21

r 12

State 2

1 State 1 2

12

b

Fig. 6.29 Theorem of reciprocal unit reactions

The theorem of reciprocal worksr 11  0 Cr 21  1 Dr 12  1 Cr 22  0 leads to the
following relationshipr 21 Dr 12. In general,

rnmDrmn: (6.31)

The theorem of reciprocal reactions said thatin any elastic system reactionrnm,
which arises innth constrain due to unit displacement of constrainm, equals
reactionrmn, which arises inmth constrain due to unit displacement of con-
strainn.
This is demonstrated by the following example (Fig.6.29b). Unit displacements
of the clamped-pinned beam areZ 1 D 1 is a unit angle of rotation of the clamped
support andZ 2 D 1 is a vertical linear displacement of the pinned support. Unit
reactions are as follows:r 21 is vertical reaction in constrain 2 caused by unit angular
displacement of support 1 andr 12 is moment in constrain 1 caused by unit vertical
linear displacement of support 2.
Using TableA.3(pos. 1 and 2), unit reactions may be written asr 21 Dr 12 D 3 EI=l^2 :
Theorem of reciprocal reactions will be widely used for analysis of statically inde-
terminate structures by the displacement method.