There are then, within our structural blob, four interrelated fields flowing between the boundaries according to certain rules (the field equations) but controlled by conditions specified on the boundaries.

**Strategies for Elastic Analysis and Design**

**Boundary Conditions **

There are then, within
our structural blob, four interrelated fields flowing between the boundaries
according to certain rules (the field equations) but controlled by conditions
specified on the boundaries. Besides the shape of the boundary, the conditions
may include:

a. Displacements
at 'points' or surfaces, and

b. Applied
loads, often called 'tractions,' at 'points' or distributed over an area.

This includes unloaded sections of the
boundary where the normal and tangential tractions are zero. Traction boundary
conditions are given by Equation (2.11) in 3D which, in two dimensions, reduce
to:

which can be seen to be simply the equilibrium
requirement.

Thus
there are basically three types of boundary value problems:

1. Only
tractions specified

2. Only
displacements specified

3. Mixed

The last is the most
common in practice and, if the conditions are specified 'properly,' there will
be a unique solution. For the first type, tractions only, the solution for the
stress field will involve an arbitrary constant translating to arbitrary linear
displacements. The question of what is a 'proper' specifi-cation of boundary
conditions for a unique solution will be discussed later. Obviously a solution
for real physical problems must exist, but in rare cases uniqueness is
questionable. From the design perspective however, existence of an inverse
solution is certainly not guaranteed.

The yield condition is also, in a sense,
an 'internal' boundary condition sepa-rating, along surfaces of unknown
geometry, the sections of the structure that have 'gone plastic' from those
that remain elastic. Determining the geometry of the plastic zones is difficult
and is obviously related to the inverse problem of design. If the plastic zones
can be isolated, however, then the elastic field theory can be used outside
plastic zones to determine structural behavior even past first yield. This
difficult challenge is the subject of the concluding chapters of this text.

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Civil : Principles of Solid Mechanics : Strategies for Elastic Analysis and Design : Strategies for Elastic Analysis and Design |

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