Advanced Methods of Structural Analysis

Chapter 10

Influence Lines Method

This chapter is devoted to construction of influence lines for different statically
indeterminate structures. Among them are continuous beams, frames, nonuniform
arches, and trusses. Analytical methodsbased on the force and the displacement
methods are applied. Also, kinematical method of construction of influence lines is
discussed. This method allows tracing themodelsof influence lines.

10.1 Construction of Influence Lines by the Force Method

Let us consider a first-degree statically indeterminate structure. In case of afixed
load, the canonical equation of the force method is

ı 11 X 1 C1PD0: (10.1)

In this equation, the free term1Prepresents displacement caused by given fixed
load. Now we need to transform (10.1) for the case ofmovingload. Since moving
load isunitone, then (for the sake of consistency notations) let us replace free term
1Pby theunitfree termı1P; this free term presents adisplacement in primary
system in the direction ofX 1 causedbyloadPD 1. The primary unknown

X 1 D

ı1P

ı 11

:

Unit displacementı 11 presents displacement caused by primary unknownX 1 D 1.
Therefore,ı 11 is somenumber, which depends on the type of a structure, its param-
eters and chosen primary system and does not depend on the position of the acting
load. However,ı1Pdepends on unit load location. Since loadPD 1 is traveling,
ı1Pbecomes afunctionof position of this load, and as result, the primary unknown
becomes afunctionas well:

IL.X 1 /D

1

ı 11

IL.ı1P/: (10.2)

I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-910,cSpringer Science+Business Media, LLC 2010

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