Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 341

Unit state and corresponding internal forces in all members of the truss are shown
in Fig.10.10d. These internal forces will be used for calculation ofıP1andı 11.

Calculation ofıP1Since the lower chord of the truss is loaded by traveling load,
we have to find the vertical displacementsof the joints of the lower chord. The most
effective procedure for calculation ofıP1is the elastic load method.

Elastic loadW 1 According to this method we need to apply two unit couples of
opposite directions to the members, which are located to the left and to the right from
joint 1. Let the left couple rotate counterclockwise and right clockwise; the couples
are not shown. The each couple should be presented as two forces at joints 0 and 1, 1,
and 2. These forces have a vertical direction and their values are1=dD1=4D0:25.
The group of four loads at joints 0, 1, 2 is shown in Fig.10.10e. These four forces
present a self-equilibrate set of forces, therefore the reactions of supports are zeros.
Corresponding internal forces in each element of the truss in the primary system
caused by these four applied forces 0.25 each are shown in diagramN 1.
The first elastic load is determined by formula

W 1 D

XN 1 Nl

EA

; (10.16)

whereNis the internal force in each element of the truss in the primary system
caused by the unit primary unknownX 1 D 1 ; these forces are shown in Fig.10.10d.
Summation is performed by all elements of the truss. The first elastic load according
to (10.16) becomes

W 1 D 2





1

3



.0:666/ 4

1

EA

members 0 1; 1 2

C

5

12

0:833 5 

1

EA

member 0  10

C

5

12

.0:833/ 5 

1

EA

member 2  10

D

1:776

EA

Elastic loadW 2 The loads1=dD1=4D0:25are applied at joints 1 and 2, 2 and 3
as shown in Fig.10.10f; corresponding internal forces are shown on the diagramN 2.
The second elastic load becomes

W 2 D

XN 2 Nl

EA

D 2 

1

3

1:333 4

1

EA

members 10  20 ;2^0  30

C





5

12



.0:833/ 5 

1

EA

members 10  2

C





5

12



0:833 5 

1

EA

members 2  30

D

3:555

EA

:

Elastic loadW 3 Similar procedure leads to the following result for third elastic load
W 3 D8:525=EA.