In this chapter the one-dimensional theory of beams will be developed from strain energy, kinetic energy, and virtual work per unit length. Although a form of Hamilton s extended principle is used, the usual approach of expressing all variations in terms of displacement and rotation variables is set aside in favor of obtaining an intrinsic formulation for those parts of the theory in which it is feasible. An intrinsic formulation is one that is not tied to a specific choice of displacement or rotation variables, which implies that the variations of one-dimensional strains must be developed in terms of virtual displacement and virtual rotation quantities.

The previous chapter shows that in- and out-of-plane Saint-Venant warping displacements are fully coupled for nonhomogeneous, anisotropic beams. However, Saint-Venant s principle implies that any constraints on the warping displacement at an end of a beam should die out within a short distance from that end. The cross-sectional analyses imply that Saint-Venant s principle holds only for beams of class and class , and that warping need only be addressed explicitly in the cross-sectional analysis for such beams. On the other hand, for class beams one must also take the warping into account in the one-dimensional equations and boundary conditions.

In this chapter we will consider the classical theory and a generalized Timoshenko refined theory for class and class beams, and a generalized Vlasov refined theory for class beams. The generalized Timoshenko theory will be developed in detail, and the classical theory...