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We report herein the asymmetric total synthesis of (+)-engelharquinone and its epoxide by exploiting two key reactions; (1) Rhodium catalyzed asymmetric 1,4-addition and (2) thiolate-mediated reduc...

Based on Hellinger-Reissner's variational principle, a rectangular finite element for plate bending is derived with bilinearly varying deflection and incompletely linear moments. This method is an example of the mixed method first proposed by Herrmann, where both displacement and stress are adopted simultaneously as fundamental variables. Because continuity conditions required on the fundamental variables are much relaxed compared with compatible models, suitable shape functions can be easily derived and calculation of the element matrix can be performed by hand. The explicit expression of the element matrix for general orthotropic plates is presented in this paper. A few simple test problems are computed in order to demonstrate the accuracy of the present model, and it is shown that it gives improved results compared with Herrmann's triangular element. Convergence is also assured when suitably small mesh sizes are adopted, and this element can be utilized in general practical problems.

The finite element method is applied to the axisymmetric buckling of shallow spherical shells under uniform external pressure. In order to analyze the nonlinear behavior of shells with sufficient accuracy and efficiency, the simple load incremental method is modified by introducing the mid-increment stiffness matrix and residual load correction. The analysis is performed on a fairly wide range of the geometric parameter α to obtain pressure-deflection curves in both pre-and post-buckling states. The results appear generally satisfactory, at least up to the early stage of post-buckling. In particular, the upper and lower buckling pressures are obtained in excellent agreement with known analytical solutions.