Strategies for Elastic Analysis and Design

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.