The solution of plane stress problems by an electrical analogue method

It is shown that the biharmonic equation ∇4χ = 0, which Airy's stress function has to satisfy, can be solved by an electrical analogue comprising two resistance networks in cascade. Apart from the requirement of high accuracy of the solution for χ, as the stresses are to be obtained by numerical differentiation of the measured analogue data, difficulties arise through the unfavourable nature of the boundary conditions which have to be satisfied in many practical problems. These difficulties have been overcome by an experimental iteration method. Details are given of the technique of solving the biharmonic equation in this way for prescribed shape of the stressed body and prescribed forces acting on it. The solution is of the same type as that found by numerical methods, e.g. by relaxation, but it is obtained with much greater speed and facility. An accuracy in the evaluation of the stresses of 1 to 2% can be obtained; agreement with photoelastic measurements is very good. Examples are given of several problems which have been solved by the new method. The "cascaded network" technique can also be applied to the solution of other problems in elasticity, described by equations of the type ∇4w = f(x, y), e.g. the bending and extension of flat plates.