344 10 Influence Lines Method
10.2 Construction of Influence Lines
by the Displacement Method
Let us consider a first-degree kinematically indeterminate structure. In case of a
fixedload, the canonical equation of the displacement method is
r 11 Z 1 CR1PD0:Now we need to transform this equation for the case ofmovingload. In this equation
the free termR1Prepresents reaction caused by given actual load. Since moving
load isunitone, then (for the sake of consistency notations) let us replace a free
termR1Pby theunitfree termr1P; this free term presents a reaction in primary
system in the introduced constraint 1 caused by loadPD 1. The primary unknown
Z 1 Dr1P
r 11:The unit reactionr 11 presents reaction in the introduced constraint caused by unit
displacement of this constraint. Therefore,r 11 is some specificnumber, which de-
pends on the type of a structure and its parameters and does not depend on the
position of the acting load. However,r1Pdepends on unit load location. Since load
PD 1 is traveling,r1Pbecomes afunctionof position of this load, and as result,
the primary unknown becomes afunctionas well:
IL.Z 1 /D1
r 11IL.r1P/: (10.17)The function IL.r1P/may be constructed using Tables A.3–A.6; nondimensionless
parametersuand.uCD1/denote the position of loadP.
In case offixedload, a bending moment at any sectionkmay be calculated by
formula
MkDMkZ 1 CMk^0 :whereZ 1 is the primary unknown of the displacement method;Mkis the bending
moment at sectionkin a primary system due to unit primary unknownZD 1 ;and
Mk^0 is the bending moment at sectionkin a primary system due to given load.
Now we need to transform this equation for the case ofmovingload. The bending
momentMkat any sectionkpresents thenumber, because this moment is caused by
the unit primary unknownZ 1 ; the second componentZ 1 presents a function. The
last component momentMk^0 is caused by the moving load, therefore the bending
momentMk^0 also becomes afunctionof the position of unit load. As a result, the
bending moment at any sectionkbecomes afunction, which can be presented as
follows
IL.Mk/DMkIL.Z 1 /CIL
Mk^0
: (10.18)