To review once again, the basic field equations in terms of stress are: Equilibrium:

**Airy’s Stress
Function in Cartesian Coordinates **

To review once again, the basic field equations in
terms of stress are: Equilibrium:

Once the stress field
is determined, the elastic strains and rotations are found from the
stress'strain and conjugate relationships giving the components of the relative
deformation vector *du*, *dv*, *dw* which can then be integrated
for the displacement field.

In 1862, G.B. Airy introduced his potential function
defined as:

satisfied “by definition” and the compatibility
equation for plane stress becomes

is substituted for v. If the body forces are given by a harmonic function (e.g.,
constant, linear, magnetic fields, seep-age, etc.), Equation (6.2) reduces to
the homogeneous elliptic form known as the biharmonic equation:

With this ingenious approach, Airy reduces the
two-dimensional prob-lem to solving one equation.* Generally, if body forces
are significant, their effect is solved for separately as in Chapter 5 and
introduced by superposi-tion. Therefore, unless specifically stated to be
otherwise, body forces will be neglected and Equation (6.3) used in this and
succeeding pages.

**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.

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

Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : Airy’s Stress Function in Cartesian Coordinates |

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