TABLE OF CONTENTS This chapter has presented an asymptotic cross-sectional analysis for thinwalled, anisotropic beams. In contrast to the treatment in Chapter 4, which results in a finite element analysis over the cross-sectional plane, the treatment here is analytical and the results are obtained in closed form. CLPT and CLST, depending on whether the contours are straight or curved, have been shown to be suitable starting points for development of an analytical cross-sectional theory for thinwalled beams. This results in a simpler development than would be obtained with three-dimensional elasticity as a starting point, and it has no effect on the resulting theory as long as h ? a. Moreover, the use of refined plate and shell theories makes no difference in the final result either, as long as h ? a.
Because we have a general thin-walled theory, it was feasible to confirm Vlasov s theory for isotropic I-beams asymptotically. However, it is shown to be asymptotically incorrect to assume that the cross-section is rigid in its own plane. One cannot consistently neglect both stresses and strains associated with local plate membrane and bending effects. Indeed, the asymptotically correct theory for isotopic I-beams can be derived by neglecting the stresses. This is especially important for asymptotically correct recovery of three-dimensional field variables.
We also consistently extended Vlasov s theory to anisotropic I-beams. As in the isotropic case, the asymptotically correct theory can only be derived by neglecting stresses associated with local plate membrane and bending effects. A theory in which asymptotic...
