A study of parametric instability of eccentrically stiffened rectangular plates
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The purpose of the investigation was to determine the onset of parametric instability for a simply-supported rectangular stiffened plate subjected to periodic in-plane loads. The static buckling and the free vibration of the eccentrically stiffened plate were also studied as special cases of the parametric instability problem. Two different mathematical models, the discrete element model and the orthotropic model, were used to represent the stiffened plate. The Donnell-type nonlinear strain-displacement relations were employed in the derivation of the kinetic and potential energies of the system. The mass effects associated with rib and stringer eccentrically stiffened plates included the transverse inertia, the in-plane in-ertias, and the rotatory inertia of the stiffeners. The governing differential equations and the associated boundary conditions were derived from Hamilton's principle by means of the calculus of variations. A membrane prestress solution was chosen for this investigation. For the discrete element model the equations of motion were reduced by means of the Galerkin method; while the equations of motions for the orthotropic model were reduced by employing the method of separation of variables. The resulting equations, which were time-dependent ordinary differential equations, were then expressed in matrix notation. The solution of the resulting matrix equation, which was assumed in the form of a Fourier series, showed that the boundary of the regions of parametric instability could be determined from the eigenvalues of four matrices for each of the stiffened plate systems. The theory developed for the orthotropic model was used to study the in-plane inertia effect on the parametric instability regions and the natural frequencies of the plate stiffened with a varying number of ribs. The effect of in-plane inertia on the natural frequencies of the stiffened plate was found to be negligible when the plate was stiffened with a large number of ribs. The effect of in-plane inertia on the boundaries of the parametric instability regions of a stiffened plate was found to be negligible when the effect of in-plane inertia on the natural frequencies of the stiffened plate was less than 0.5%. A study of the relative validity of the orthotropic model with respect to the discrete element model was conducted for the case where the cross-sectional area, the depth ratio (depth/width), and the number of ribs attached to the plate were taken to be variables. The theory developed for the orthotropic model was shown to predict the magnitudes of the buckling loads and the natural frequencies of a stiffened plate having a large number of depth ribs within an 8% difference from the corresponding value predicted by the theory developed for the discrete element model. However, the degree of accuracy of the magnitudes of the buckling loads and the natural frequencies for the stiffened plate was increased when the magnitude of the depth ratio and the cross-sectional area of the stiffeners was decreased. The theory developed for the discrete element model was employed to examine the effects of eccentricities, cross-sectional properties and location of the stiffeners on the critical buckling load, the natural frequencies and the boundaries of the parametric instability regions of the stiffened plate. The theory developed for the discrete element model to predict the static buckling loads, the natural frequencies and the boundaries of the parametric instability regions of the stiffened plate allows for an arbitrary choice in the shape of the cross-section (unsymmetrical or symmetrical), size, location and number of rib and stringer stiffeners. The eccentricity (with respect to the plate mid-surface) and size of the stiffeners can have a significant effect on the magnitudes of the critical buckling load and the natural frequencies, and the location of the boundaries of the parametric instability regions. The variations of the loading on a stiffener have a pronounced effect on the location of the boundaries of some of the temporal mode instability regions. For the cases studied, the greatest single effect on the location of the boundaries of the parametric instability was caused by the increase of the rigidity of either a rib or stringer to the point that the lowest value of critical buckling parameter corresponds with a spatial mode other than the first parametric spatial mode. The widths of the parametric instability regions connected with the spatial modes other than the one associated with the fundamental static stability mode were also wide if the values of the critical buckling parameter associated with these modes were close to the lower value of the critical buckling parameter.
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