Advanced Methods of Structural Analysis

360 10 Influence Lines Method

The elastic curve caused byP D 1 in the primary system is presented in
Fig.10.17b;ı1Pis angle of rotation at support 1 caused by moving forcePD 1.
Instead of calculation ofı1Pfor different position ofPD 1 , we will calculateıP1;
this displacement occurs under the forcePcaused by unit primary unknownX 1 ,as
shown in Fig.10.17c.
Elastic curveıP1caused by fixed unit coupleX 1 D 1 may be easily constructed
by the initial parameter method (Fig.10.17d). For the left span

EIyDEIıP1DEIy 0 CEI 0 

R.x0/^3

3Š

;

whereRis the reaction of the left support. SinceRD1= l, and vertical displacement
at initial pointy 0 D 0 , the equation of elastic curve becomes

EIyDEIıP1DEI 0 

1

l

x^3

6

:

Initial parameter 0 may be calculated using boundary condition at the right support
(point 6):

EIy.l/DEIıP1.l /DEI 0 

1

l

l^3

6

D 0 ! 0 D

l

6EI

:

Finally, displacement in the directionP caused by unit primary unknown
X 1 D 1 , may be written as follows

ıP1D

l^2

6 EI

x

l



1 

x^2

l^2



(10.25)

Now we need to compute the unit displacementı 11. Bending moment diagram
caused by unit primary unknownX 1 is shown in Fig.10.18b. Using the graph mul-
tiplication method we get

ı 11 D

M 1 M 1

EI

D 2 

1

2

l 1 

2

3

 1

1

EI

D

2

3

l

EI

:

Equation (10.24) leads to the following equation for influence line for bending
moment at support 1:

IL.X 1 /D

l

4

x

l



1 

x^2

l^2



: (10.26)

Influence line for bending moment at support point 6 is presented in Fig.10.18c. It
is obvious that this influence line is symmetrical.

Discussion. The same influence line had been obtained early in Sect.10.1,
Fig.10.2d. The fundamental difference between both solutions is related to the