Chapter 6
Deflections of Elastic Structures
This chapter describes some effective methods for computing deflections of
deformable structures. The following structures are considered: beams, frames,
arches, and trusses subjected to differentactions, such as variety of external loads,
change of temperature, settlements of supports, and errors of fabrication. Compu-
tation of different types of deflections is shown. They are linear, angular, as well
as the mutual linear and angular deflections. Advantages and disadvantages of each
method and field of their effective application are discussed. Much attention is
given to a graph multiplication method which is a most effective method for bend-
ing structures. Fundamental properties of deformable structures are described by
reciprocal theorems; they will be widely used for analysis of statically indeterminate
structures.
6.1 Introduction
Any load which acts on the structure leads to its deformation. It means that a struc-
ture changes its shape, the points of the structure displace, and relative position of
separate points of a structure changes. There are other reasons of the deformation
of structures. Among them is a settlementof supports, change of temperature, etc.
Large displacements could lead to disruption of a structure functioning properly and
even its collapse. Therefore an existing Building Codes establish limit deflections
for different engineering structures. Ability to compute deflections is necessary for
estimation of rigidity of a structure, for comparison of theoretical and actual de-
flections of a structure, as well theoretical and allowable deflections. Beside that,
computation of deflections is an important part of analysis of any statically inde-
terminate structure. Deflections computation is also an integral part of a dynamical
analysis of the structures. Thus, the computation of deflections of deformable struc-
tures caused by different reasons is a very important problem of Structural Analysis.
Outstanding scientists devoted theirs investigations to the problem of calcula-
tion of displacements. Among them are Bernoulli, Euler, Clapeyron, Castigliano,
Maxwell, Mohr, etc. They proposed a number of in-depth and ingenious ideas for the
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-96,cSpringer Science+Business Media, LLC 2010
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