The solution of the torsion problem for circular shafts of varying radius

When a bar of circular section whose radius is a function ofz , fig. 1, is subjected to terminal couples applied in a suitable way, the stresses and strains may be expressed in terms of a function ψ. This function satisfies the equation ∂2 ψ/∂r 2 - 3/r ∂ψ/∂r + ∂2 ψ/∂z 2 = 0 throughout an axial section, with ψ = constant on the boundary. The stresses are ͡θz = μ/r 2 . ∂ψ/∂r ,r͡ θ = μ/r 2 . ∂ψ/∂z where μ, is the modulus of rigidity of the material. The displacement,v , of any point is directed at right angles to an axial plane passing through the point, and is given byr 3 . ∂/∂r (v /r ) = - ∂ψ/∂z ,r 3 . ∂/∂z (v /r ) = - ∂ψ/∂r . Equation (1) has been solved for certain boundaries by the usual analytical methods; for the general case (including boundaries of non-mathematical form) the only method so far developed is an approximate graphical one due to Willers. The present paper describes an arithmetical trial and error method, applicable to the general case, which may be carried to any desired degree of accuracy. An appendix to the paper mentions other physical problems which can be treated by similar methods.