P.K. Datta

Research Advances in the Dynamic Stability Behavior of Plates and Shells: 1987–2005—Part I: Conservative Systems

This paper reviews most of the recent research done in the field of dynamic stability/ dynamic instability/ parametric excitation /parametric resonance characteristics of structures with special attention to parametric excitation of plate and shell structures. The solution of dynamic stability problems involves derivation of the equation of motion, discretization and determination of dynamic instability regions of the structures. The purpose of this study is to review most of the recent research on dynamic stability in terms of the geometry (plates, cylindrical, spherical and conical shells), type of loading (uniaxial uniform, patch, point loading ….), boundary conditions (SSSS, SCSC, CCCC ….), method of analysis (exact, finite strip, finite difference, finite element, differential quadrature and experimental ….), the method of determination of dynamic instability regions (Lyapunovian, perturbation and Floquet’s methods ), order of theory being applied (thin, thick, 3D, nonlinear….), shell theory used (Sanders’, Love’s and Donnell’s), materials of structures (homogeneous, bimodulus, composite, FGM….) and the various complicating effects such as geometrical discontinuity, elastic support, added mass, fluid structure interactions, non-conservative loading and twisting etc. The important effects on dynamic stability of structures under periodic loading have been identified and influences of various important parameters are discussed. Review on the subject for non-conservative systems in detail will be presented in Part-2.

Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading

The dynamic instability characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element analysis. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The method of Hills infinite determinants is applied to determine the dynamic instability regions. Numerical results are presented to study the effects of various parameters, such as static load factor, aspect ratio, boundary conditions, stiffening scheme and load parameters on the principal instability regions of stiffened plates using Bolotins method. The results show that location, size and number of stiffeners have a significant effect on the location of the boundaries of the principal instability region.

Buckling and vibration of stiffened plates subjected to partial edge loading

The buckling and vibration characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element method. The initial stresses are obtained considering the pre-buckling conditions. Buckling loads and vibration frequencies are determined for different plate aspect ratios, edge conditions and different partial non-uniform edge loading cases. The non-uniform loading may also be caused due to the supports on the edges. The analysis presented determines the stresses all over the region for different kinds of loading and edge conditions. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The vibration characteristics are discussed and the results are compared with those available in the literature. Buckling results show that the stiffened plate is less susceptible to buckling for position of loading near the supported edges and near the position of stiffeners as well.

Dynamic stability of a rectangular plate on non-homogeneous winkler foundation

Abstract The dynamic stability of a rectangular plate on non-homogeneous foundation, subjected to uniform compressive in-plane bi-axial dynamic loads and supported on completely elastically restrained boundaries is studied. The non-homogeneous foundation consists of two regions having different stiffnesses but symmetric about the centre lines of the plate. The equation governing the small amplitude motion of the system is derived by a variational method. The use of Galerkin method with reduced beam eigenfunctions transforms the system equations in matrix form. The system of coupled Mathieu-Hill equations thus obtained, are analysed by the method of multiple scales which yields the stability boundaries for different combinations of the excitation amplitude and frequency. The effects of stiffness and geometry of the foundation, boundary conditions, static load factor, in-plane load ratio and aspect ratio on the stability boundaries of the plate for first- and second-order simple and combination resonances are studied.

Parametric instability characteristics of rectangular plates subjected to localized edge loading (compression or tension)

Abstract The parametric instability behaviour of a plate subjected to localized in-plane compressive or tensile periodic edge loading is studied, considering the effects of transverse shear deformation and rotary inertia. Different edge loading cases have been considered. Compression and tension buckling results are discussed. The vibration analysis for this problem shows that for certain parameters of tensile loading, the frequency of the plate initially rises with the load, but then begins to decrease with increasing tension, showing the onset of tension buckling. Dynamic instability behaviour under compressive periodic edge loading shows that the instability regions are influenced by the load band width and its location on the edge. Parametric instability behaviour under tensile periodic edge loading shows that the instability regions appear early for positions of the load near the corner of the edge. Instability behaviour due to different load parameters is discussed.

Vibration and static stability characteristics of rectangular plates with a localized flaw

Abstract The free vibration and static stability behaviour of a rectangular plate with localized zones of damage is studied using a finite element analysis. The influence of parameters such as size of damage, extent of damage, position of damage, plate boundary conditions and aspect ratio of the plate on vibration and instability behaviour are discussed. It is observed that the presence of damage has a marked effect on the static buckling load and natural frequency of the plate. A flaw in the plate near its centre usually renders the natural frequency and buckling load of the plate to a minimum value.

Parametric Resonance Characteristics of Laminated Composite Doubly Curved Shells Subjected to Non-Uniform Loading

The parametric resonance characteristics of laminated composite doubly curved panels subjected to various in-plane static and periodic compressive edge loadings, including partial and concentrated edge loading are studied using finite element analysis. The first order shear deformation theory is used to model the doubly curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to the Sander’s first approximation for doubly curved laminated shells, which can be reduced to Love’s and Donnell’s theories by means of tracers. The effects of number of layers, static load factor, side to thickness ratio, shallowness ratio, boundary conditions, degree of orthotropy, ply orientations and various load parameters on the principal instability regions of doubly curved panels are studied in detail using Bolotin’s method. Quantitative results are presented to show the effects of shell geometry, lamination details and load parameters on the stability boundaries. Results of plates and cylindrical shells are also presented as special cases and are compared with those available in the literature.

Dynamic instability of laminated composite rectangular plates subjected to non-uniform harmonic in-plane edge loading

Abstract The dynamic instability behaviour of isotropic, cross-ply and angle-ply laminated composite plates under combined uniaxial and harmonically varying in-plane point or patch loads is investigated using finite element analysis. The first-order shear deformation theory is used to model the composite laminates, considering the effects of transverse shear deformation and rotary inertia. The effects of various geometrical parameters, boundary conditions, lamination and load parameters on the principal dynamic instability regions of composite plates are studied in detail. The preferential orientations depending on the instability regions for simply supported patch and point loaded plates have been suggested for angle-ply plates. The dynamic instability region has been influenced by the restraint provided at the edges.

Vibration characteristics of a rotating blade with localized damage including the effects of shear deformation and rotary inertia

Abstract The flexural vibration characteristics of a uniform rotating bar with localized zones of damage are studied by using a finite element analysis. The effects of shear deformation and rotary inertia have been taken into consideration. The influences of parameters such as the extent, position of damage and rotational speed on the frequencies of oscillation of the blade are discussed. It is observed that the presence of damage has a marked effect on the natural frequencies. A flaw in the beam near the tip of the blade has substantial effects on the frequencies as rotational speed increases. As the rotary inertia and shear deformation parameters increase, the frequencies increase for all damage parameters.

Buckling and vibration of a thin tensioned sheet with an elliptical hole

The out-of-plane deflection which can be developed in a thin tensioned sheet containing a central opening is discussed. It is noted that this behavior is of practical importance because it can alter the stress concentration and fatigue properties.The results of an experimental and analytical study of the buckling and vibration behavior of tensioned sheets with an elliptical opening are presented. Data from buckling and vibration experiments for several opening shapes ranging between a crack and a circle are presented. A generalized Galerkin method for solving the governing differential equations is used and the results obtained are presented in terms of buckling loads and natural frequencies.The experimental-buckling-stress estimates are correlated with the analytical results. The data from the vibration experiments indicate that the load-frequency behavior is dependent on the size of the test specimen and the opening shape. The analytical results display a load-frequency behavior which is experimentally observed for a narrow elliptical hole; i.e., for a small value of the ratio of minor axis to major axis. Some features of the analytical and experimental vibration results differ substantially, however. The analysis indicates that the frequency decreases to zero as the applied load approaches the buckling load. The experimental results do not exhibit this behavior. The sources of this difference in behavior are discussed.