It was relatively easy to do a complete study of the plastic behav-ior of a thick ring because symmetry reduces the equilibrium requirement to one separable, first-order differential equation.

**Theorems of Limit Analysis**

In Previous Section , it was
relatively easy to do a complete study of the plastic behav-ior of a thick ring
because symmetry reduces the equilibrium requirement to one separable,
first-order differential equation. However, most structures do not have
symmetry, simple loadings, or ideal supports. In these situations even with
mod-ern computational power, a full range elastic-plastic analysis can be
nearly impossible or of dubious value.

Fortunately, there is
an alternative approach where we forget about obtaining information between
first yield and collapse and concentrate on finding the col-lapse state
directly. This strategy, called limit analysis, is reasonable if, as is
usu-ally the case, we are primarily interested in the strength of a structure.
We may lose all information about deformations or even the stress distribution
in the plastic range but, after all, most structures are, or should be,
designed to operate under normal working loads in the elastic range well below
yield. All we really need is a good estimate of the “collapse mechanism” and
the load at which it occurs to allow design for safety under extreme loading
(earthquake, hurricane, collision, etc). For these rare events, we usually want
to also control the failure mode to prevent sudden collapse and to localize
residual plastic deformations, whatever they may be, so that the structure can
be repaired.

Application of the so-called “bound theorems” and in
particular the “upper-bound theorem” of limit analysis provide a way to do this
with a straightforward strategy. Limit analysis from either a lower-bound or
upper-bound perspective is an intuitive and interactive process. We shall find
that in most cases we are free to specify certain variables arbitrarily, make
edu-cated guesses as to failure modes or stress distribution, and use
trial-and-error methods to converge on good answers. It is this freedom and the
rela-tively simple calculations involved that makes limit analysis so valuable.
In fact, graphical analysis is very often the best method giving the simplest
approach and greatest insight to the interplay of variables. In an important
sense, limit analysis is more of an art than elastic analysis. Limit analysis
requires, and in turn develops, a physical feel for structural behavior that
may be even more important for a creative engineer than the simplicity of the
method itself.

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Civil : Principles of Solid Mechanics : Concepts of Plasticity : Theorems of Limit Analysis |

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