Calculating the stresses in a member, and determining if they are within it’s capacity, is one of the main tasks of a structural engineer. Often this is accomplished by making simplifying assumptions that allow the use of relatively simple mathematical methods. But for anything other than the simplest shapes under the simplest loading conditions, these approximations don’t work. In these situations, the difficulty of calculating stresses ratchets upward, and requires solving second-order partial differential equations.
Today, thanks to modern computers and tools such as finite element analysis, solving these sorts of problems has become relatively trivial. But before these aids existed, complex stress problems were simply too difficult to solve mathematically. Because of this, alternative methods had to be devised for working out the stresses, methods that didn’t rely on solving intractable equations. One of the more ingenious of these was the soap-film method for calculation torsion stresses.