Chapter 2
General Theory of Influence Lines
Construction of influence lines for one span simply supported and cantilevered
beams in case of direct and indirect load applications are considered. All influence
lines are constructed using analytical expressions for required factor. Applications
of influence lines for fixed and moving loads are discussed. This chapter forms the
set of concepts which creates a framework for comprehensive analysis of different
statically determinate structures.
2.1 Analytical Method for Construction of Influence Lines
The engineering structures are often subjected to moving loads. Typical exam-
ples of moving loads on a structure are traveling cars, trains, bridge cranes, etc.
In classical structural analysis, the term “moving load” requires one additional
comment: this concept means that only the loadpositionon the structure may be
arbitrary.
It is obvious that internal forces and displacements in any section of a beam
depend on the position of a moving load. An important problem in analysis of struc-
tures is the determination of maximum internal forces in a structure caused by a
given moving load and the corresponding most unfavorable position of this load.
This problem may be solved using influence lines. Influence line is a fundamental
and very profitable concept of structural analysis. Their application allows perform
a deep and manifold analysis of different types of structures subjected toanytype
of fixed and moving loads. Influence lines method becomes especially effective tool
analysis if a structure is subjected todifferent groupsof loads.
Definition: Influence line is a graph, which shows variation of some particular functionZ
(reaction, shear, bending moment, etc.) in thefixedcross section of a structure in terms of
positionof unit concentrated dimensionless loadPD 1 on the structure.Each ordinate of influence line means the value of the function, for which in-
fluence line is constructed, if the unit load is located on the structure above this
ordinate. Therefore, the unit loadP, which may have different positions on the
structure, is called a moving (or traveling) load. The term “moving load” implies
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-92,cSpringer Science+Business Media, LLC 2010
15