Advanced Methods of Structural Analysis

6.7 Reciprocal Theorems 191

denoted by letterımn. The first indexmindicates the direction of the displacement
and the second indexndenotes the unit load, which causes this displacement.
Thus,ı 11 andı 12 are displacements in the direction of loadP 1 due to loadP 1 D 1
andP 2 D 1 , respectively;
ı 21 andı 22 are displacements in the direction of loadP 2 due to loadP 1 D 1 and
P 2 D 1 , respectively.
In case of unit loads, the theorem of reciprocal worksP 2  21 DP 1  12 leads to
the following fundamental relationshipı 12 Dı 21. In general,

ınmDımn: (6.30)

This equation shows thatin any elastic system, displacement alongnth load
caused by unitmth load equals to displacement alongmth load caused by unit
nth load. The term “displacement” refers to linear or angular displacements, and
the term “load” means force or moment.
This theorem is demonstrated by the following example. Simply supported beam
is subjected to unit loadPin the first condition and unit momentMin the sec-
ond condition (Fig.6.28). Displacementsı 11 andı 12 are linear displacements along
forcePin the first and second states; displacementsı 21 andı 22 are angular dis-
placements along momentMin the first and second states.

Fig. 6.28 Theorem of
reciprocal unit displacements

State 2

State 1

l/2 l/2

P =1

M =1

d 21

d 12 d 22

d 11

In the first state, displacement due to loadPD 1 along the load of the state 2 is

ı 21 DD

1 l^2

16 EI

:

In the second state, displacement due to loadMD 1 along the load of the
state 1 is

ı 12 DyD

1 l^2

16 EI

:

Theorem of reciprocal displacements will be widely used for analysis of statically
indeterminate structures by the Force method. Theorem of reciprocal unit displace-
ments was proved byMaxwell( 1864 ) before the more general Betti theorem;
however, Maxwell’s proof was unoticed by scientists and engineers. Mohr proved
this theorem in 1864 independently from Betti and Maxwell.