St. Venant Formulae for Generalized St. Venant Problems P. PODIO-GUIDUGLI Communicated by E. STERNBERG 1. Introduction In a paper which just appeared in print [1], W. A. DAY addresses himself to the following problem proposed by C. TRUESDELL [2], [3, 4]" For an isotropic, linearly elastic cylinder subject to end tractions equipollent to a torque M, define the "twist" ~ in such a way that M = (/~R) 3, (1.1) where/zR, the torsional rigidity of the cylinder, is resolved into the factors/~, the shear modulus, and R, a geometric quantity depending only on the cross- section. As remarked by TRUESDELLand recalled by DAY, such a twist would generalize ST. VENANT'S notion of twist so as to apply also to solutions of the torsion prob- lem corresponding to distributions of end tractions different from the one assumed by ST. VENANT. DAY offers the following elegant and exact solution of TRUESDI3LL'Sproblem. For any displacement field u defined over the cylinder, he considers the real function o~~ IIu -- o~tlilt, (1.2) where t is the St. Venant torsion field with unit twist, and ll'[J~ is the Lz-norm of strain. He then calls generalized twist z(u) the value of ~ at which this function attains its minimum, and he gives an explicit formula for z(u). Finally, by the use of BETTFS reciprocal theorem applied to the pair (u, t), he shows that (1) holds exactly, with "r = ~(u), whenever u is a solution of the torsion problem. On reading [1] I immediately wondered whether, were TRUESDELL'Sproblem rephrased in the obvious manner, DAY'S procedure would apply to further gener- alized St. Venant problems.