TABLE OF CONTENTS Symmetric bending of a beam occurs when:
Its cross section has an axis of symmetry
Applied forces lie in the symmetry plane formed by this axis and by the neutral axis
The vectors of the applied moments are perpendicular to this plane
This case of bending, occurring entirely in the symmetry plane, was covered in an introductory course of strength of materials. Here we consider the bending of a homogeneous, linearly elastic prismatic beam with an arbitrary cross section (Fig. 8-13) carrying forces and moments that do not necessarily lie in the same plane. It is assumed, however, that every plane of loads passes through a centroidal axis. [*] We introduce the arbitrary, but centroidal, coordinates ? and ? and make the same assumptions as are made in the case of symmetric bending:
Deformations are infinitesimal
Plane cross sections of the beam remain plane after deformation
The latter implies that the normal strain in the axial direction is a linear function of the cross-sectional coordinates ?, ?. Since for a linearly elastic solid stress is a linear function of strain, it depends linearly on the coordinates ? and ? or on any other coordinates obtained from ? and ? by rotation and translation. The expression for the normal stress ? xx takes on a particularly simple form when referred to the principal axes of inertia. Thus our...
