Development of a Quasi-Exact Dynamic Finite Element (QDFE) Method For the Free Vibration Analysis of Thin Rectangular Multilayered Plates

The Dynamic Finite Element (DFE) method is a well-established superconvergent semianalytical method that has been used in the past to investigate the vibration behaviour of various beam-structures. Considered as a viable alternative to conventional FEM for preliminary stage modal analysis, the DFE method has consistently proven that it is capable of producing highly accurate results with a very coarse mesh; a feature that is attributed to the fact that the DFE method uses trigonometric, frequency-dependant shape functions that are based on the exact solution to the governing differential equation as opposed to the polynomial shape functions used in conventional FEM. In the past many researchers have contributed towards building a comprehensive library of DFE models for various line structural elements and configurations, which would serve as the building blocks that would help the DFE method evolve into a fullfledged, versatile tool like conventional FEM in the future. However, thus far a DFE formulation has not been developed for plate problems. Therefore, in this thesis an effort has been made for the first time to develop a DFE formulation for the realm of two-dimensional structural problems by formulating a Quasi-Exact Dynamic Finite Element (QDFE) solution to investigate the free vibration behaviour of thin single- and multi-layered, rectangular plates. As a starting point for this work, Hamiltonian mechanics and the Classical Plate Theory (CPT) are used to develop the governing differential equation for thin plates. Subsequently, a unique quasiexact solution to the governing equation is sought by following a distinct procedure that, to the best of the author‘s knowledge, has never been presented before. Through this procedure, the characteristic equation is re-arranged as the sum of two beam-like expressions and then solved for by applying the quadratic formula. The resulting quasi-exact roots are then exploited to form the trigonometric basis functions, which in turn are used to derive the frequency-dependant shape functions; the characteristic feature of the QDFE method. Once developed, the new QDFE technique is applied to determine the vibration behaviour of thin, isotropic, linearly elastic, rectangular, homogenous plates. Subsequently, it is also employed to formulate a Simplified Layerwise Quasi-Exact Dynamic Finite Element solution for the free vibration of thin, rectangular multilayered plates. In addition, the quasi-exact solution to the plate equation is also utilised to develop a Dynamic Coefficient Matrix (DCM) method to investigate the vibrational characteristics of thin, rectangular, homogeneous plates and thin, rectangular, multilayered plates. The Method of Homogenization is used as an alternative procedure to validate the results from the Simplified Layerwise Quasi-Exact Dynamic Finite Element method and the Simplified Layerwise Dynamic Coefficient Matrix method. The results from both the QDFE and DCM methods are, in general, verified for accuracy against the exact results existing in the open literature and those produced by two in-house developed conventional FEM codes and/or ANSYS® software.

Development of a Quasi-Exact Dynamic Finite Element (QDFE) Method For the Free Vibration Analysis of Thin Rectangular Multilayered Plates

The Dynamic Finite Element (DFE) method is a well-established superconvergent semianalytical method that has been used in the past to investigate the vibration behaviour of various beam-structures. Considered as a viable alternative to conventional FEM for preliminary stage modal analysis, the DFE method has consistently proven that it is capable of producing highly accurate results with a very coarse mesh; a feature that is attributed to the fact that the DFE method uses trigonometric, frequency-dependant shape functions that are based on the exact solution to the governing differential equation as opposed to the polynomial shape functions used in conventional FEM. In the past many researchers have contributed towards building a comprehensive library of DFE models for various line structural elements and configurations, which would serve as the building blocks that would help the DFE method evolve into a fullfledged, versatile tool like conventional FEM in the future. However, thus far a DFE formulation has not been developed for plate problems. Therefore, in this thesis an effort has been made for the first time to develop a DFE formulation for the realm of two-dimensional structural problems by formulating a Quasi-Exact Dynamic Finite Element (QDFE) solution to investigate the free vibration behaviour of thin single- and multi-layered, rectangular plates. As a starting point for this work, Hamiltonian mechanics and the Classical Plate Theory (CPT) are used to develop the governing differential equation for thin plates. Subsequently, a unique quasiexact solution to the governing equation is sought by following a distinct procedure that, to the best of the author‘s knowledge, has never been presented before. Through this procedure, the characteristic equation is re-arranged as the sum of two beam-like expressions and then solved for by applying the quadratic formula. The resulting quasi-exact roots are then exploited to form the trigonometric basis functions, which in turn are used to derive the frequency-dependant shape functions; the characteristic feature of the QDFE method. Once developed, the new QDFE technique is applied to determine the vibration behaviour of thin, isotropic, linearly elastic, rectangular, homogenous plates. Subsequently, it is also employed to formulate a Simplified Layerwise Quasi-Exact Dynamic Finite Element solution for the free vibration of thin, rectangular multilayered plates. In addition, the quasi-exact solution to the plate equation is also utilised to develop a Dynamic Coefficient Matrix (DCM) method to investigate the vibrational characteristics of thin, rectangular, homogeneous plates and thin, rectangular, multilayered plates. The Method of Homogenization is used as an alternative procedure to validate the results from the Simplified Layerwise Quasi-Exact Dynamic Finite Element method and the Simplified Layerwise Dynamic Coefficient Matrix method. The results from both the QDFE and DCM methods are, in general, verified for accuracy against the exact results existing in the open literature and those produced by two in-house developed conventional FEM codes and/or ANSYS® software.

An enhancement of the warping shear functions of Refined Zigzag Theory

The paper presents an enhancement in Refined Zigzag Theory (RZT) for the analysis of multilayered composite plates. In standard RZT, the zigzag functions cannot predict the coupling effect of in-plane displacements for anisotropic multilayered plates, such as angle-ply laminates. From a computational point of view, this undesirable effect leads to a singular stiffness matrix. In this work, the local kinematic field of RZT is enhanced with the other two zigzag functions that allow the coupling effect. In order to assess the accuracy of these new zigzag functions for RZT, results obtained from bending of angle-ply laminated plates are compared to the three-dimensional exact elasticity solutions and other plate models used in the open literature. The numerical results highlight that the enhanced zigzag functions extend the range of applicability of RZT to the study of general angle-ply multilayered structures, maintaining the same seven kinematic unknowns of standard RZT.

DYNAMIC BIAXIAL STRESS ANALYSIS OF FLAT LAYERED CERAMIC COMPOSITES

We study theoretically the biaxial bending of symmetric, flat layered ceramic composites (laminates) due to external loading. We focus on three-layered alumina/zirconia laminates. We compare the principal stresses in the samples in the case of static and harmonic dynamic loading. The dynamic equation within the Kirchhoff theory for thin homogeneous plates is first generalized to the case of multilayered plates. It is solved numerically with the relaxation method, which we have developed for this purpose.