State-Space-Based Differential Quadrature Method And Its Applications

When applied to elasticity problems, the traditional state space method (SSM) is confined to beams, plates and shells with simple or roller supports. To resolve this difficulty, a new semi-analytical method, the state-space-based differential quadrature method (SS-DQM), is developed by introducing the DQM into the state space formalism. Firstly, based on the equations of elasticity, the state equation is derived in the transfer domain, perpendicular to which the remaining domain(s) is/are discretized by DQM, and hence, the partial differential equation is translated into the one at an arbitrary discrete point and only with respect to the coordinate of the transfer domain. State variables are used to express the boundary conditions at the end points, without the necessity of using the Saint-Venant principle, and the behavior in the near-boundary zone can be precisely modeled. The present SS-DQM was applied to analyze the bending and free vibration of beams resting on elastic foundations, composite laminated beams and plates with one pair of simply-supported opposite edges. Numerical examples with a comprehensive comparison with known results indicate that SS-DQM converges very well and can give accurate predictions.Numerical instability is always encountered when using the conventional transfer matrix method (CTMM) based on the state space formalism, in cases of higher order frequency computation, high ratio of the global transfer length to the length of the discretized domain, and large number of sampling points in the DQ solution. The inherent cause is explored by looking into the property of the determinant of exponential matrices. By decomposing the structure into several substructures, the transfer distance is reduced, and the joint coupling matrices (JCMs) areintroduced to develop the coupling relations at arbitrary inner joints. Then, the CTMM and JCMs are combined to establish the global analysis (TM-JCM) of the structure, which can remove the aforementioned numerical instability.SS-DQM is employed to multiple domains, enabling us to obtain the semi-analytical solution of bending and free vibration of laminated plates with general boundary conditions. The use of SSM along the thickness direction avoids the discretization of all spatial domains, and hence, the total number of discrete points is greatly reduced, the formulation simplified, and the computation efficiency enhanced. The SS-DQM procedure is then used to derive the semi-analytical solution of FGM beams. The approximate exponential laminate model is proposed to reflect the continuous nature of the variation of material properties of FGMs.Free vibration of continuous Kirchhoff plates is studied using the present method. At the internal rigid line supports and the inner artificial substructuring lines, JCMs are obtained according to the equilibrium and compliance conditions, making it possible to treat strip-type plates or plates with large spans. Meanwhile, SS-DQM is successfully applied to the analysis of thin-walled structures such as box-girders based on the theory of folded plates.The present hybrid method is extended to analysis of thin plates with discontinuous boundary conditions, as well as beams in plane stress state. All the state variables at the boundary points can be derived from the original boundary conditions with the use of state equation and induced variables. This allows the variable transfer for problems with different supporting conditions along the same edges. Effects of the mixed portion on the natural frequencies of the beams are discussed, and the numerical results can render a reference for future numerical analyses.