Elasticity/Solution strategy for Prandtl stress function

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Solution strategy using the Prandtl stress function[edit | edit source]

The equation is a Poisson equation.

Since the equation is inhomogeneous, the solution can be written as

where is a particular solution and is the solution of the homogeneous equation.

Examples of particular solutions are, in rectangular coordinates,

and, in cylindrical co-ordinates,

The homogeneous equation is the Laplace equation , which is satisfied by both the real and the imaginary parts of any analytic function of the complex variable

Thus,

Suppose . Then, examples of are

where , , , are constants.

Each of the above can be expressed as polynomial expansions in the and coordinates.

Approximate solutions of the torsion problem for a particular cross-section can be obtained by combining the particular and homogeneous solutions and adjusting the constants so as to match the required shape.

Only a few shapes allow closed-form solutions. Examples are

  • Circular cross-section.
  • Elliptical cross-section.
  • Circle with semicircular groove.
  • Equilateral triangle.