Advanced Methods of Structural Analysis

186 6 Deflections of Elastic Structures

Fig. 6.24 Unitstate for

calculation of elastic load

at jointn n

dn– 1 dn

Fn–1 =1/dn–1 Fn–1 Fn Fn =1/dn

M=1 M=1

5.Construct the bending moment diagram for fictitious beam on the tensile fibers.
This diagram will present the precise displacements of all joints of the entire real
truss.

Example 6.17.Design diagram of the truss is presented in Fig.6.25a. The truss is

carrying two equal forcesP. The axial stiffness of the upper and lower chords is 2EA

and diagonal and vertical members have axial stiffnessEA. Compute displacements

of the joints of the lower chord.

Solution.Calculation of internal forces in all members of the truss due to applied

loadsPmay be performed by analytical method. Corresponding diagramNpof

internal forces is presented in Fig.6.25b.

Since it is required to find displacements ofthe joints of the lower chord, therefore

the elastic loads should be applied to the joints of the same chord.

Elastic loadW 1 This load is related to joint 1 and represents the mutual angle of

rotation of members 0-1 and 1-2. In order to find the elastic loadW 1 , we need to

apply two unit couples (consisting of forces1=dD1=4) to members 0-1 and 1-2

and compute the internal forces in all members of the truss induced by these four

forces. Corresponding distribution of internal forces is presented in diagramNN 1

(Fig.6.25c).

Magnitude of elastic loadW 1 is obtained by “multiplication” of two axial force

diagramsNN 1 andNp

W 1 D

XNN 1 Npl

EA

: (a)

The summation should be extended over all members of the truss. For members 0 a

anda 2 , internal forces in actual condition are equal5P =3and have opposite signs,

while in unit conditions are equal5=12for both members. Therefore summation

within these members equals to zero. Summation within members 0-1and 1-2 of the

truss leads to the following result

W 1 D

XNN 1 Npl

EA

D

1

2 EA

4P

3



1

3

 4  2 D

64

36

P

EA

: (b)

This procedure explains why the elastic load method is very effective. The four

forces present the self-equilibrated set offorces. Therefore the reactions of supports