Laminated composite plates and shells are being increasingly used in aerospace, automotive, and civil engineering as well as in many other applications of modern engineering structures. Tailoring ability of fiber-reinforced polymer composite (FRPC) materials for the stiffness and strength properties with regard to the reduction of structural weight made them superior compared with metals in such structures. In some specific applications such as aircraft wing skins composite structures need to be stiff at one location and flexible at another location. It is desirable to tailor the material and structural arrangements so as to match the localized strength and stiffness requirements by dropping the plies in laminates. Such laminates are called as tapered laminates. 
In the dynamic instability that occurs in the structures subjected to harmonic in-plane loading not only the amplitude of the harmonic in-plane load but also the forcing frequencies make the structures fail at load amplitude that is much less than the static buckling load and over a range of forcing frequencies rather than at a single value. In this case, the bending deformations, the rotations and the strains are not small enough in comparison with unity, so the linear theory just provides an outline about the dynamically-unstable regions and is not capable to determine the amplitude of the steady-state vibration in these instability regions.
The main objective of this dissertation is to develop a geometric non-linear formulation and the corresponding solution method for uniform and internally-thickness-tapered laminated composite plates. This Ph.D. research work is completed by extension of this developed geometric non-linear formulation to the uniform laminated composite cylindrical shells as well. The novel parts of this Ph.D. dissertation are the geometric non-linear formulations and corresponding displacement-based solutions obtained using approximate analytical methods, for dynamic instability analysis of internally-thickness-tapered laminated composite plates and cylindrical panels. To the best of author knowledge, there is no non-linear dynamic instability study on internally-thickness-tapered laminated composite plates and cylindrical panels in literature. There is only one study on the linear dynamic instability of internally-thickness-tapered flat plates using the FEM and Ritz method.  Here the developed analytical geometric nonlinear formulation not only is capable of predicting the instability regions but also is capable of determining both stable- and unstable-solutions amplitudes of steady-state vibrations of such internally-thickness-tapered laminated composite plates and cylindrical panels in these dynamically-unstable regions. Furthermore, the effect of the influential parameters on the non-linear dynamic instability of laminated plates and cylindrical shells is extensively studied. These parametric studies were carried out on cross-ply laminated composite uniform plates, flat and cylindrical tapered plates, and uniform cylindrical shells. In this study, the non-linear von Karman strains associated with large deflections are considered. Considering the simply supported boundary condition the Navier’s double Fourier series with the time-dependent coefficient is chosen to describe the out-of-plane displacement function. For the uniform laminated composite rectangular plates and uniform laminated composite cylindrical shells, a combination of displacement and a stress-based solution is considered while for the internally-thickness-tapered laminated composites plates and cylindrical panels a displacement-based solution is considered to solve the equations of motion. Then the general Galerkin method is used for the moment-equilibrium equation of motion to satisfy spatial dependence in the partial differential equation of motion to produce a set of non-linear Mathieu-Hill equations. These equations are ordinary differential equations, with time-dependency. Finally, by applying the Bolotin’s method to these non-linear Mathieu-Hill equations, the dynamically-unstable regions, stable-, and unstable-solutions amplitudes of the steady-state vibrations in these dynamically-unstable regions are obtained for both the uniform and the internally-thickness-tapered laminated composites plates and uniform cylindrical shells.
A comprehensive parametric study on the non-linear dynamic instability of these simply supported cross-ply laminated composite uniform plates, flat and cylindrical internally-thickness-tapered plates and uniform cylindrical shells are carried out to examine and compare: the effects of the orthotropy in the laminated composite uniform plates, number of layers for symmetric and antisymmetric uniform cross-ply laminated composite plates and cylindrical shells, different taper configurations and  taper angles in both flat tapered plates and tapered cylindrical panels, magnitudes of both tensile and compressive axial loads in the uniform and tapered plates and uniform cylindrical shells, aspect ratios of the loaded-to-unloaded widths of the uniform plates, flat and cylindrical internally-thickness-tapered panels and length-to-radius ratio of the cylindrical shells, length-to-average-thickness ratio of the flat plates and cylindrical panels and radius-to-thickness ratio of the cylindrical shells, and curvature of the tapered cylindrical panels i.e. radius-to-loaded widths ratio on the instability regions and the parametric resonance particularly the steady-state vibrations amplitudes of cross-ply laminated composite uniform plates, flat and cylindrical internally-thickness-tapered plates and uniform cylindrical shells. The present results show good agreement with those available in the literature.