Most solutions in two dimensions, for other than linear stress fields where direct integration is possible, can be obtained most easily by the semi-inverse method using stress functions.

**The Classic
Stress-Function Approach **

Most solutions in two
dimensions, for other than linear stress fields where direct integration is
possible, can be obtained most easily by the semi-inverse method using stress
functions. Usually Airy’s stress function is employed which, in two dimensions,
reduces the three field equations to one fourth-order partial differential
equation.* This “biharmonic equation” is exact for plane strain and nearly so
for plane stress if the plate is reasonably thin.

For simple geometry and
loadings, a wide range of practical problems in var-ious coordinate systems
have been solved since elasticity was formalized. These results are used
extensively for design since many structures can be ide-alized to plane stress
or plane strain without serious error. Moreover, such two-dimensional solutions
are essential benchmarks to, on the one hand, establish the range where the
simpler strength-of-materials type analysis is adequate and, on the other, to
validate the more complicated numerical or experimental models necessary to
determine fields in specific structures that cannot be safely idealized to a
shape or loading pattern where a closed-form ana-lytic solution is feasible.

Hundreds of solutions
are available and there is space in this and the next two chapters to present
only a limited number* selected on the basis of one or more of the following
criteria:

a. to
best demonstrate the stress-function method,

b. to
give insight as to the fundamental behavior of isotropic and devi-atoric stress
fields,

c. to
present questions, perhaps unanswered, concerning the theory,

d. to
suggest a new problem not yet solved that might excite a student’s interest, or

e. to
display particular utility for design in themselves or by superpo-sition.

The presentation will be in a somewhat diagrammatic
form with frequent ref-erence to figures augmented by sketches and notes. This
approach is intended to emphasize the rational aspects of the semi-inverse
method based on physi-cal reasoning.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : The Classic Stress-Function Approach |

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