TABLE OF CONTENTS In this chapter we start with the kinematical foundation of the last chapter and create from this general three-dimensional kinematical description a reduced-order model of the type of a one-dimensional or beam model. Before delving into the actual analysis, however, we will give a short description of the VAM followed by an illustrative example. Then, asymptotically correct beam models of two types are developed: classical and refined. Transformation of the refined theory to the form of a common engineering model, a generalized Timoshenko theory, will be undertaken and recovering relations derived. The relationship of the refined theory to a generalized Vlasov approximation will also be presented. Finally, an asymptotic treatment of the trapeze correction is given. These sections form, collectively, the theoretical basis for the computer program VABS. A brief tutorial for VABS, written by Wenbin Yu, is presented in Appendix A.
The VAM is a very useful mathematical methodology to simplify the procedure for finding the stationary points of a functional depending on one or more small parameters. It is applicable to any problem that can be posed in terms of seeking the stationary points of a functional with some inherently small parameters. It is therefore especially the right tool for building accurate models for dimensionally reducible structures, structural members that are amenable to a reduction in dimensionality because of the presence of one or more small parameters (e.g., beams, plates, and shells). This is because the original elasticity problem can be stated as...
