Advanced Methods of Structural Analysis

18 2 General Theory of Influence Lines

Influence Line forRB

Equilibrium equation in form of moments of all the external forces about supportA
leads to the following expression for reactionRBin terms of positionx

RB!

X

MAD 0 W RBlPxD 0 !RBD

Px

l

:

The last equation leads to the following equation of influence line:

IL.RB/D

x

l

: (2.2)

Ifx D 0 (at supportA), then ordinate of influence line IL.RB/D 0 .IfxD l
(at supportB), then ordinate of influence line IL.RB/D 1. Influence line forRB
is presented in Fig.2.1. This graph can be used for analysis of reactionRBonly.
If loadP D 1 is located above pointA,thenreactionofRBis equal to zero. It
means that loadPdoes not get transmitted on to the supportB, when the loadP
is situated directly over the left-hand support. If loadPD 1 is located above point
B,thenreactionofRBis equal to 1. If loadPD 1 has, for example, coordinate
xD0:25l, then reaction ofRBis equal to 0.25.

2.1.1.2 Simply Supported Beam with Overhang (Fig.2.2)

The equilibrium equations and corresponding equations for influence lines of
reactions are

RA!

X

MBD 0 W

RAlP.xl/D 0 !RADP

xl

l

!IL.RA/D

xl

l

;

RB!

X

MAD 0 WRBlPxD 0 !RBDP

x

l

!IL.RB/D

x

l

: (2.3)

Fig. 2.2 Simply supported
beam with overhang.
Influence lines for reactions

1

Inf. line RA

−

+

1

+ Inf. line RB

P= 1

x

RA l

RB d

A D

B