Part 1Given a beam in elastic equilibrium whose mean fibre is a continuous curve and which is acted upon by couples of arbitralily directed axes, the problem which we set is to determine:I. The system of forces acting on any section normal to the mean fibre.II. The stresses existing in this section and the limiting of those stresses.I. Determination of the forces acting on a section. We first consider a section ABC of a plane curved beam represented in Fig. 1 by its mean fibre, with a torsional moment applied Fig. 1 at A in a plane normal to the mean fibre. The moment applied at A is represented by the vector AA' following the ordinary *Read by title at the meeting of the National Academy of Sciences, Nov. 10, 1924. Abstract in The Proc. Nat. Acad. Sc., Jan. 1925. tMassachusetts Institute of Technology, Cambridge, Mass., U. S. A. tUniversity of Ghent, Ghent, Belgium.
Given a beam in elastic equilibrium whose mean fibre is a continuous curve and which is acted upon by couples of arbitrarily directed axes, the problem which we set is to determine: I. The system of forces acting on any section normal to the mean fibre. II. The stresses existing in this section and the limiting of those stresses. Determination of the forces acting on a section.-We first consider a section ABC of a plane curved beam represented in Fig. 1, by its mean fibre, with a torsional moment applied at A in a plane normal to the mean fibre. The moment applied at A is represented by the vector AA' following the ordinary conventions of mechanics. To determine the forces acting on a Section B normal to the mean fibre at B, we prolong AA' until it intersects the trace BD of the section. At D we carry DD' = AA' which we decompose, in conformity with the de St. Venant principle of elastic equivalence of statically equivalent systems, into its components DD' and DD"', tangential and normal, respectively, to the trace BD. Then the vectors
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