Free vibration of generally supported rectangular Kirchhoff plates: State‐space‐based differential quadrature method

Abstract

SUMMARYThe title problem is investigated using the differential quadrature method based on the state-space formalism. The plates, with mixed boundary conditions, may cross over one-way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed.

Semi‐analytical analysis for multi‐directional functionally graded plates: 3‐D elasticity solutions

“…Accordingly, serious numerical instabilities will be encountered when solving Equation (17) using computational program. Similar phenomenon was intensively discussed by Lü et al [38,43,44] when treating thermomechanical behavior of FGM beams and vibration of multi-span continuous plates. To overcome this possible numerical difficulty, the joint coupling matrices method [45] will be introduced in the numerical analysis.…”

“…To overcome this possible numerical difficulty, the joint coupling matrices method [45] will be introduced in the numerical analysis. The details are the same as that outlined in [38,43,44], and is omitted here for brevity.…”

“…Owing to the superior boundary conditions using the -points [27], some near-boundary inner points indispensable in the conventional DQM. The hybrid method of DQM and state-space method (SSM) were pioneered by Chen et al [40][41][42] for composite laminated beams and plates and later applied to FGM beams and plates [37,38,43] as well as multi-span thin plates [44]. The newly developed hybrid method is extended here to the analysis of elasticity problems of multi-directional FGM rectangular plates.…”

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

“…In detail, numerical instabilities will occur when performing the inverse manipulation of the matrix in (14) if the plate is in thick configurations. This problem was well clarified and successfully removed by Lü et al [23] when treating continuous plates and generally supported thick laminated plates [25] . In the current work, we will also use the combined transfer matrix method and the joint coupling matrix proposed in [23] to obtain numerical results with satisfying accuracy for thick plates.…”

“…Fortunately, the state space method (SSM) was successfully combined with the generalized differential quadrature method (DQM), the so-called state-space based differential quadrature method (SSDQM), by Chen and his coworkers [21][22][23][24][25] for laminated beams and plates, functionally graded beams, and continuously multi-span plates, etc. In this paper, the SSDQM is used to obtain the semi-analytical threedimensional elasticity solutions for generally supported functionally graded thick plates.…”