We first analyze the bending of a rectangular beam by surface tractions applied only at its end faces. The resultant of these surface tractions vanishes, but they have a nonzero moment about an axis perpendicular to the plane of deformation. To delineate differences between predictions from the linear and the nonlinear elasticity theories, the material of the beam is first assumed to be Hookean and then neo-Hookean. The two problems are studied by the semi-inverse method.
Infinitesimal deformations of a rectangular beam deformed by tangential surface tractions applied on its top surface are analyzed by using an Airy stress function. This approach is also similar to using the semi-inverse method, as the postulated Airy stress function is shown to satisfy the boundary conditions. The displacement field induced by stresses derived from the Airy stress function is also determined.
We assume that a uniform rectangular beam shown in Fig. 8.1a is deformed by surface tractions applied at the end faces X 3 = L that have a null resultant force but a nonzero moment about the X 2-axis. Because of the symmetry of the geometry and the loading, we assume that deformations of the beam are symmetrical about the plane X 3 = 0, and analyze deformations of the right half of the beam for which 0 ? X 3 ? L. The symmetry conditions on the plane X 3 = 0 require that points ...
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