Chapter 7
The Force Method
The force method presents a powerful method for analyzing linear elastic statically
indeterminate structures; this method also has a wide application in problems of
stability and dynamics of structures. The method is very attractive because it has
clear physical meaning, which is based on a convenient and well-ordered procedure
of calculation of displacements of deformable structures, and presently, this method
has been brought to elegant simplicity and perfection. In this chapter, the reader
find the numerous examples of application of this method for analysis of different
structures subjected to external loads, temperature change, settlements of supports,
and errors of fabrication.
7.1 Fundamental Idea of the Force Method
Definitions. Statically indeterminate structures are called the structures for whichall
reactions and internal forces cannot be determined solely using equilibrium equa-
tions. Redundant constraints (or excess) are constraints, which are not necessary for
geometrical unchangeability of a given structure.
7.1.1 Degree of Redundancy, Primary Unknowns
and Primary System
Degree of redundancy, or statical indeterminacy, equals to the number of redun-
dant constraints whose elimination leads to the new geometrically unchangeable
and statically determinate structure. Thus, degree of statical redundancy is the dif-
ference between the number of constraints and number of independent equilibrium
equations that can be written for a given structure.
Primary unknowns representreactions(forces and/or moments), which arise in
redundant constraints. That is the reason why the method is called the Force Method;
this method also is called the flexibility method or the method of consistent deforma-
tions. Unknowninternalforces also may be treated as primary unknowns. Primary
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-97,cSpringer Science+Business Media, LLC 2010
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