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be used to compute displacement and forces at the nodes, followed by the transfer matrix
method to calculate the state variables between the nodes.
The force method is also referred to as the
flexibility
or
influence coefficient method
and
occasionally as the
compatibility method
. This method is based on the principle of comple-
mentary virtual work, which provides global compatibility conditions that lead to a system
flexibility matrix relating redundant forces (forces in excess of those that can be determined
using the equations of equilibrium) to applied loadings.
The dominant method in use today in the practice of structural analysis is the displace-
ment method which is also called the
stiffness method
or, somewhat less often, the
equilibrium
method
. The displacements (at the joints or nodes of the elements into which the structure
is idealized) essential for describing the deformed state of the structure are selected as the
unknowns. The principle of virtual work, which is the basis for this method, provides global
equilibrium conditions and corresponding global stiffness equations. These equations are
solved for the nodal displacements in terms of applied forces. This method is normally
considered to be much easier to automate than the force method for the solution of large
structural systems [Argyris, 1954; Przemieniecki, 1968].
A number of considerations determine the most suitable method for a particular problem.
These considerations will be discussed as the various methods are presented.
5.1
Transfer Matrix Method
A transfer matrix
U
i
relating the state variables
z
at point
a
to the state variables at point
b
of a structure can be written as
U
i
z
a
+
z
i
z
b
=
(5.1)
in which
U
i
for a beam element is given by Chapter 4, Eq. (4.8c) for Sign Convention 1. The
bar over
z
i
indicates that these are applied loads.
Note from Chapter 4, Eq. (4.9) that all of the basic relations, i.e., conditions of equilib-
rium, geometry, and material law, are included in the transfer matrix. All three will be
satisfied simultaneously as the element matrices are placed together to represent a whole
system.
From Chapter 4, Eqs. (4.86) and (4.87) (Sign Convention 1), the transfer matrix for a simple
beam element appears as
b
a
i
3
2
b
w
V
M
1
−
−
/
6
EI
−
/
2
EI
w
V
M
w
2
b
V
b
M
b
01
/
2
EI
/
EI
θ
=
+
(5.2)
00
1
0
00
1
U
i
z
i
z
b
=
z
a
+
It is often useful to incorporate the loading terms in the transfer matrix. In so doing, an
extended state vector
z
and an
extended transfer matrix
U
i
are defined.
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