Overview

Although the cross-sectional analysis and the finite element code VABS, described in Chapter 4, are applicable to thin-walled beams, closed-form expressions for the cross-sectional elastic constants are sometimes advantageous because of computational efficiency demanded in, for example, preliminary design and optimization. A simple analytical beam theory can be quite beneficial for several reasons. First, the approach taken allows all but the essential variables to be eliminated. Having more variables in the analysis than necessary can obscure a clear understanding of the phenomena being studied. Moreover, preliminary calculations may span a vast design space or interface with other disciplines, which may necessitate keeping the information about the elastic deformation in a maximally compressed form (such as in dynamics, control, or aeroelastic analysis of rotor-craft). Contrary to widespread belief, we demonstrate that the simplest classical beam theories, which contain only the four classical beam variables, can provide sufficiently accurate models for long-wavelength static and low-frequency deformation of thin-walled composite beams with strip-like or closed cross-sections. However, only when the simplest beam theory is free from internal flaws can its comparison with more complex theories truly attest to the need for the additional complexity in specific situations.

Let us first consider prismatic beams, for which the three-dimensional constitutive law and strain-displacement relationships can be considered linear. Beam theories are associated with the introduction of variables that depend only on x ₁, the coordinate along the beam axis. For a general type of deformation, at least four such one-dimensional variables have to...