Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 313

TABLE 5.4
Comparison of the Transfer Matrix, Displacement, and Force Methods
Transfer Matrix Displacement

Method Method Method Force Method

Unknowns Displacement and Displacement Force variables

force variables variables

Transfer matrix U i Stiffness matrix k i Flexibility matrix f i To characterize the i th

element
To characterize the

i

U = U i K = i k i = a T ka F = b T fb

system
Conditions fulfilled — Compatibility (of the Equilibrium (of the

at the outset geometrically statically determinate

determinate system) system)

Resulting relations Equilibrium and Equilibrium Compatibility

satisfy compatibility

In contrast to the force and displacement methods, the transfer matrix approach does not
involve the assembly of a system matrix whose size increases with the DOF of the system.
Rather, the system matrix for the transfer matrix method is characterized by progressive
element matrix multiplications instead of by superposition. As a result, the system matrix
for the transfer matrix method is the same size as the element matrix. Of course, the transfer
matrix is suitable primarily for the solution of line-like systems. The three methods are
compared in Table 5.4.

5.5.1 Indeterminacy

In the development of the force method, a distinction was made between determinate and
indeterminate systems. Static indeterminacy is usually defined in terms of the number of
equations in addition to the equations of equilibrium that are necessary for an analysis. This
extra number of equations is employed as a measure of the degree of static indeterminacy,
which is equal to the number of redundant forces. It should be noticed, however, that it was
not necessary to distinguish between statically determinate and indeterminate structures in
setting up a displacement method solution. However, to understand better the relationship
between the displacement and force methods, it is helpful to define the concept of kinematic
indeterminacy.
Kinematic indeterminacy may appear to be a significantly different concept than static
indeterminacy, although it can be considered to be analogous. The number of DOF (dis-
placements) necessary to provide the response of a structure is a measure of kinematic
indeterminacy. More specifically, the degree of kinematic indeterminacy or redundancy is
equal to the number of nodal displacements that would have to be constrained, in addition
to those displacements constrained by the boundary conditions, in order to impose a value
of zero for each DOF. A system so constrained is said to be kinematically determinate. In
the displacement method, the degree of kinematic indeterminacy is equal to the number
of equations needed for an analysis, i.e., it is the number of columns or rows in a stiffness
matrix in which the boundary constraints have been taken into account.