Saint-Venant’s problem

Abstract

A prismatic rod is the body obtained by translating a plane figure S along a straight line which is perpendicular to the plane of the figure. In this case the plane figure S presents the cross-section of the rod. The axis Oz of the rod is the straight line which is the locus of the centres of inertia of the cross-sections whereas axes Ox and Oy lying in the cross-sectional plane are directed along the principal axes of inertia of the cross-section. the origin O of the system of axes Oxy lies in a cross-section (in the cross-section z=const). The cross-sections z=0 and z=l are referred to as the end faces, their centres of inertia being respectively denoted as O⁻ and O⁺. Let I _x and I _y designate the moments of inertia of the cross-section about the corresponding axis of Tthis cross-section and S senote its cross-sectional area. Then

$$ \left. \begin{gathered} S = \iint\limits_S {do,}\iint\limits_S {xdo = 0,}\iint\limits_S {ydo = 0,} \hfill \\ I_x = \iint\limits_S {y^2 do,}I_y = \iint\limits_S {x^2 do,}\iint\limits_S {xydo = 0(do = dxdy),} \hfill \\ \end{gathered} \right\} $$

((1.1.1))

for all z ⊂ [0,l].