Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 239

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Structural Analysis Methods II: Structural Systems
A structure can be considered to be a system composed of structural elements connected
at joints or nodes . This structural system is analyzed by assembling information gained
from an analysis of each element. In the previous chapter, the beam element was studied
as a representative structural element. In the present chapter, a beam system formed of
an assembly of many beam elements will be treated, and the appropriate global analysis
procedures investigated. The question arises as to how the element characteristics should be
assembled to permit an effective and efficient computational solution of the whole structure.
The correct solution must fulfill the following conditions:
1. Equilibrium. The applied forces and the resulting internal forces must be in equilib-
rium at each node.
2. Force-displacement relationship. The internal forces and deformations in each
element must satisfy the appropriate stress-strain relationship (material law). These
relations will be linear in this chapter.
3. Compatibility or kinematics. The ends of the elements must fit together at the nodes,
and each element must be continuous.
We begin the study of global analysis procedures of structural systems by considering a
mixed method , the transfer matrix method . This is a progressive matrix multiplication scheme
that applies to a system whose geometry is line-like, such as several beam elements placed
end-to-end to form a long beam or curved bar. The transfer matrix method is characterized
by a sequence of matrix multiplications along the line system, a procedure which leads to
the same size final matrix regardless of the number of elements in the system. It is referred
to as a mixed method, since both displacement and force variables, i.e., all of the state
variables, are retained throughout the computations.
Network structures, e.g., a framework, are normally treated using force or displacement
methods . These techniques will be considered following the discussion of the transfer matrix
method. Rather than the matrix multiplication of the transfer matrix method, the system
response using the force and displacement methods is found by assembling the element
response characteristics through addition. Both the force and displacement methods lead
to final matrices which depend in size on the number of elements composing the system.
Frequently, a combination of methods is useful. For example, it is sometimes convenient to
compute the transfer matrix for some of the more complex portions of a system. Using the
transformation of Chapter 4, these transfer matrices are then converted to stiffness matrices
which are placed in a displacement method analysis. Also, the displacement method can
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