Advanced Methods of Structural Analysis

190 6 Deflections of Elastic Structures

P 1

D 11 D 21

D 11 D 22

D 22

D 11 D 21

D 12

D 1 D 2

D 12

D 12 D 22

State 1

State 2

A 1

P 1

P 1

A 3 A 2

P 2

P 2

P 2

Fig. 6.27 Two state of the elastic structure. Computation of work done by the loadP 1 and addi-

tional loadP 2

2.Work done by the forceP 2 on the displacement 22. Since loadP 2 is applied
statically, thenW 2 D.1=2/P 2  22.
3.Work done by the forceP 1 on the displacement 12 ; this displacement is
caused by loadP 2 .TheloadP 1 approached its maximum valueP 1 early (be-
fore application ofP 2 ). CorrespondingP 1 –  1 diagram is shown in Fig.6.27,so
W 3 DP 1  12.
Since potential energyUequals to the total work, then

UD

1

2

P 1  11 C

1

2

P 2  22 CP 1  12 : (6.27)

On the other hand, considering of application of loadP 2 first and thenP 1 ,i.e.,if

state 2 is additionally subjected to loadP 1 , then potential energyUequals

UD

1

2

P 2  22 C

1

2

P 1  11 CP 2  21 : (6.28)

Since strain energy does not depend on the order of loading, then the following

fundamental relationship is obtained

P 1  12 DP 2  21 or W 12 DW 21 : (6.29)

The theorem of reciprocal works (6.29)saidthatin any elastic system the work

performed by load of state 1 along displacement caused by load of state 2 equals

to work performed by load of state 2 along displacement caused by load of

state 1.

6.7.2 Theorem of Reciprocal Unit Displacements

(Maxwell Theorem)

Let us consider two states of elastic structure subjected tounit loads P 1 D 1 and

P 2 D 1. Displacement caused by unit loads is called theunit displacementsand