Advanced Methods of Structural Analysis

6.7 Reciprocal Theorems 189

Summary

The elastic load method is based on conjugate beam method. According to this

method, displacement of any point of a realstructure is proportional to bending

moment at the same point of fictitious beam (factor 1/EI). Based on this, the elastic

load method uses external resemblance of displacements of elastic structure and

bending moment diagrams.

Elastic load at jointnpresents the mutual angle of rotation of two members of

the truss chord, which is adjacent to jointn.

Elastic load method presents very effective way for computation of displace-

ments forsetof the joints of a truss chord. For trusses this method leads to the

precise results. The advantage of the method is that the procedure (6.26) is not re-

lated to all members of a truss, but only to small subset of the members.

Elastic load method is also applicable for beams and arches, but for such struc-

tures this method is approximate and cumbersome. However, it has advantages over

exact methods (for example, Initial Parameters method); this fact can be seen in case

of beam with variable cross section.

6.7 Reciprocal Theorems

Reciprocal theorems reflect fundamental properties of any linear statical deter-

minate or indeterminate elastic systems. These theorems will be extensively

used for analysis of statically indeterminate structures. Primary investigations

were performed byBetti( 1872 ),Maxwell( 1864 ),Lord Rayleigh(1873–1875),

Castigliano( 1872 ), andHelmholtz( 1886 ).

6.7.1 Theorem of Reciprocal Works (Betti Theorem)

Let us consider elastic structure subjected to loadsP 1 andP 2 separately; let us call

it as first and second states (Fig.6.27). Set of displacementsmnfor each state are

shown below. The first indexmindicates the direction of the displacement and the

second indexndenotes the load, which causes this displacement. Thus

 11 and 12 are displacements in the direction of loadP 1 due to loadP 1 and

P 2 , respectively

 21 and 22 are displacements in the direction of loadP 2 due to loadP 1 and

P 2 , respectively.

Let us calculate the strain energy of the system by considering consequent appli-

cations of loadsP 1 andP 2 , i.e., state 1 isadditionallysubjected to loadP 2 .Total

work done by both of these loads consists of three parts:

1.Work done by the forceP 1 on the displacement 11. Since loadP 1 is applied
statically(from zero toP 1 according to triangle law), thenW 1 D.1=2/P 1  11 :