Introduction to Elasticity/Solution strategy for Prandtl stress function
Solution strategy using the Prandtl stress function
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
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.