Formulation of problems in the Bernoulli–Euler theory of anisotropic inhomogeneous beams

A procedure of reducing the three-dimensional problem of the elasticity theory for a rectilinear rod made of a nonuniform anisotropic material to a one-dimensional problem on the rod axis is studied. The rod is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the rod's part from its end to any cross section. The internal forces are related to the characteristics of the deformed axis under the prior assumptions on the distribution of displacements across the cross section of the rod. To regulate these assumptions, the displacements of the rod's points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli–Euler, Timoshenko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of nonuniform anisotropic rods on the basis of the Bernoulli–Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.