Civil Engineering Reference
In-Depth Information
roller support. The same result would be achieved if one support remained fixed and
the other support was removed entirely. Also, in the truss in Fig. 4.7(b), removing
a diagonal, vertical or horizontal member would result in the truss becoming stat-
ically determinate. Releasing a structure in this way would produce displacements that
would not otherwise be present. These displacements may be calculated by analysing
the released statically determinate structure; the force system required to eliminate
them is then obtained, i.e. we are employing a compatibility of displacement condition.
This method is generally termed the
flexibility
or
force method
; in effect this method
was used in the solution of the propped cantilever in Fig. 13.22.
The alternative procedure, known as the
stiffness
or
displacement method
is analogous
to the flexibilitymethod, themajor difference being that the unknowns are the displace-
ments at specific points in the structure. Generally the procedure requires a structure
to be divided into a number of elements for each of which load-displacement rela-
tionships are known. Equations of equilibrium are then written down in terms of the
displacements at the element junctions and are solved for the required displacements;
the complete solution follows.
Both the flexibility and stiffness methods generally result, for practical structures
having a high degree of statical indeterminacy, in a large number of simultaneous
equations which are most readily solved by computer-based techniques. However, the
flexibility method requires the structure to be reduced to a statically determinate state
by inserting releases, a procedure requiring some judgement on the part of the analyst.
The stiffness method, on the other hand, requires no such judgement to be made and
is therefore particularly suitable for automatic computation.
Although the practical application of the flexibility and stiffness methods is generally
computer based, they are fundamental to 'hand' methods of analysis as we shall see.
Before investigating these handmethods we shall examine in greater detail the indeter-
minacy of structures since we shall require the degree of indeterminacy of a structure
before, in the case of the flexibility method, the appropriate number of releases can
be determined. At the same time the
kinematic indeterminacy
of a structure is needed
to determine the number of constraints that must be applied to render the structure
kinematically determinate in the stiffness method.
16.2 D
EGREE OF
S
TATICAL
I
NDETERMINACY
In some cases the degree of statical indeterminacy of a structure is obvious from
inspection. For example, the portal frame in Fig. 16.1 has a degree of external statical
indeterminacy of 3, while the truss of Fig. 4.7(b) has a degree of internal statical
indeterminacy of 1. However, in many cases, the degree is not obvious and in other
cases the internal and external indeterminacies may not be independent so that we
need to consider the complete structure, including the support system. Amore formal
and methodical approach is therefore required.
