Advanced Methods of Structural Analysis

326 10 Influence Lines Method

Fig. 10.1 Design diagram of

the beam and primary system

a

P= 1

k

l l

0.4l

C

B

A

EI=constant

b P= 1

X (^1) Primary system
Influence Line for Primary UnknownX 1
Equation of influence line for primary unknown is presented by formula IL.X 1 /D
.1=ı 11 /IL.ı1P/,whereı 11 is a mutual angle of rotation in direction ofX 1 due
to primary unknownX 1 D 1 andı1Pis a slope at the middle support in primary
system due to traveling loadPD 1.
1.For calculation of unit displacementı 11 we need to construct the bending
moment diagramM 1 in primary system due to primary unknownX 1 D 1
(Fig.10.2a).
Graph multiplication method leads to the following result:
ı 11 D
M 1 M 1
EI
D 2 
1
2
l 1 
2
3
 1 
1
EI
D
2l
3 EI
:
2.For calculation of displacementı1Pwe need to place the moving load at the left
and right spans. Let the loadPD 1 be located within theleftspan of the primary
system. The angle of rotationı1Pat the right supportBof the simply supported
beam is presented in terms of dimensionless parameteru, which defines the posi-
tion of the load (Fig.10.2b). If the loadPD 1 travels along therightspan, then
the angle of rotationı1Pat the left support of the simply supported beam (i.e., the
same supportB) is presented in terms of dimensionless parameter(Fig.10.2b).
Expressions forı1Pin terms of parametersuandare shown in Fig.10.2b; they
are taken from Table A.9, pinned–pinned beam. Each span is subdivided into
five equal portions and the angle of rotationı1Pis calculated for location of the
PD 1 at the each point (uD0:0, 0.2, 0.4, 0.6, 0.8, 1.0) (Table10.1). Parameter
uis reckoned from the left support of each span, parametersuandsatisfy the
following condition:uCD 1.
Corresponding influence line forı1Pis shown in Fig.10.2c.
3.Influence line of primary unknownX 1 is obtained by dividing the ordinates of
influence line forı1Pbyı 11 D2l=3EI. Corresponding influence line forX 1
is presented in Fig.10.2d; all ordinates must be multiplied by parameterl.
We can see that for any position of the load the bending moment at supportBwill
be negative, i.e., the extended fibers in vicinity of supportBare located above the
neutral line.