This paper presents an analytical study on the dynamic characteristics of castellated beams. The study focuses on the effect of web shear on the free vibration frequencies of castellated beams. By using the Hamilton's principle, a simple closed-form solution for determining the free vibration frequencies of simply supported castellated beams is developed. The results show that the shear effect on the free vibration frequencies increases with the cross-sectional area and distance between the centroids of the two tee sections of castellated beams, but decreases with respect to increasing web thickness or increasing beam length. The shear effect is also found to be greater in higher vibration modes.

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As dislocation emission is regarded as the critical point of brittle-to-ductile transition of fracture mechanism and the defects interact with domain walls motions due to their self-stress fields, the investigations on dislocation forces, especially the interaction force and the reaction between domain switching and dislocations, are the most important fundamentals of dislocation slip in the brittle-to-ductile transition and the pinning effect on domain wall motion mechanisms. In this paper, an innovative method base on the strain nucleus simulated by an assembly of four dislocations and the Green's function integration is used to solve generalized stress field arising from domain switching. Then, the accurate expressions of the interaction forces, pinning forces and the image forces on dislocation are obtained and analyzed. At last, the lattice resistance is also discussed. The numerical results show that the dislocation emission is possible when temperature and load are proper, and the domain switching interaction force is the main driving force of dislocation slip in ferroelectric material under negative electric field load. The curve and equation of the lattice resistance correlated with temperature can also be fitted by the results of the proposed analytical method and corresponding experiments.