Elasticity/Solution strategy for Prandtl stress function

Solution strategy using the Prandtl stress function

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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.