A.W. Leissa

The free vibration of rectangular plates

This work attempts to present comprehensive and accurate analytical results for the free vibration of rectangular plates. Twenty-one cases exist which involve the possible combinations of clamped, simply-supported, and free edge conditions. Exact characteristic equations are given for the six cases having two opposite sides simply-supported. The existence of solutions to the various characteristic equations is carefully delineated. The Ritz method is employed with 36 terms containing the products of beam functions to analyze the remaining 15 cases. Accurate frequency parameters are presented for a range of aspect ratios (a/b = 0·4, 2/3, 1·0, 1·5, and 2·5) for each case. For the last 15 cases, comparisons are made with Warburtons useful, approximate formulas. The effects of changing Poissons ratio are studied.

Curvature effects on shallow shell vibrations

Abstract The primary objective of the present work is to assess the effect of curvature upon the vibration frequencies of shallow shells. For this purpose, the shell is chosen to have a rectangular boundary supported by shear diaphragms and yields exact, closed form solutions for convenient comparison in the linear, small deflection regime and, at the same time, represents a situation which can readily occur in practical application. The shell is taken to have two independent radii of curvature, Rx and Ry whose planes are parallel to the edges. The linear eigenvalue problem is solved, both when tangential inertia is included and when it is neglected. Extensive tabular and graphical results are presented which show the effect of curvature ratio Rx/Ry in the interval −1·5 ⩽ Rx/Ry ⩽ 1·5 and the “average curvature” upon the frequencies. The analysis is extended into the non-linear, large deflection regime by assuming mode shapes and satisfying the non-linear field equations of motion and compatibility approximately by means of the Galerkin procedure. The effect of large deflections upon frequencies and phase plane plots as the curvature ratio changes is shown.

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Abstract Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σx=−N0[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N0/Ncr=0,0.5,0.8,0.95,1, where Ncr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.

Vibration and buckling of rectangular composite plates with variable fiber spacing

Abstract This is a summary of the first known work which analyzes the structural behavior of composite plates having nonuniformly spaced fibers. The present investigation is limited to single layer composites having parallel fibers. This results in a plate which is macroscopically orthotropic, but nonhomogeneous. The free vibrations and buckling of such plates subjected to inplane boundary loadings are studied. A plane elasticity problem must first be solved to determine the inplane stresses caused by the applied boundary loading, and these stresses become input to the vibration and buckling problem. Both problems are dealt with by the Ritz method. Numerical results are obtained for six nonuniform distributions of E-glass, graphite and boron fibers in epoxy matrices in simply supported, square plates. The redistributions are seen to increase the buckling load by as much as 38%, and the fundamental frequency by as much as 21%.

Vibration studies for simply supported symmetrically laminated rectangular plates

Abstract Accurate and reasonably comprehensive results are provided for the free-vibration frequencies of symmetrically laminated, simply supported plates of rectangular planform. The Ritz method is used for angle-ply plates, and good convergence is obtained by using 144 terms in the double-sine series for the displacement function. Utilizing this accuracy, tabulated results are presented for the first eight non-dimensional frequencies of square and rectangular ( a b = 1,2 and 0·5 ) plates made of E-glass/epoxy, boron/epoxy, and graphite/epoxy materials, for different numbers of layers and for varying fiber orientation angles. Representative contour plots of the corresponding mode shapes are presented. For cross-ply plates comparisons in frequencies are made between square plates having fibers parallel to the edges (‘specially orthotropic’) and those having fibers parallel to the diagonals (‘diagonally orthotropic’). The effects of the various parameters (material, number of layers, fiber orientation) upon the frequencies and mode shapes are discussed.

Natural frequencies for cantilevered doubly-curved laminated composite shallow shells

Abstract The Ritz method with algebraic polynomial displacement functions is used to analyze free vibrations of thin cantilevered laminated plates and shallow shells having rectangular planforms. Convergence studies are made which demonstrate that accurate results are obtained using 192 displacement terms for spherical, circular cylindrical, hyperbolic paraboloidal shallow shells and 64 terms for plates. Results are compared with those obtained experimentally and by finite element methods. It is found that the present method needs considerably less degrees of freedom than the finite element method to obtain the same accuracy and compares well with results from experiment. The effect of various parameters (material, number of layers, fiber orientation, curvature) upon the frequencies is studied.

Vibration studies for laminated composite twisted cantilever plates

Abstract The first known natural frequencies and mode shapes of the title problem are presented. The Ritz method with algebraic polynomial displacement functions is used. Convergence studies are made which demonstrate that accurate results are obtained using 144 displacement terms. The effect of the thickness ratio and fiber orientation angle upon the natural frequencies and mode shapes of three-layer, E-glass/epoxy and graphite/epoxy angle-ply plates is studied. The effect of the angle of twist is also studied.

Natural frequencies of simply supported circular plates

Abstract Although the problem of finding the natural frequencies of free vibration of a simply supported circular plate has a straightforward solution, very few numerical results are available in the literature. In the present work accurate (six significant figure) non-dimensional frequency parameters (λ2) are given for all values of n + s ⩽ 10, where n and s are the numbers of nodal diameters and internal nodal circles, respectively, and for Poissons ratios 0, 0·1, …, 0·5. Simplified formulas for determining additional values of λ2 for large s are derived by the use of asymptotic expansions.

THREE-DIMENSIONAL FREE VIBRATIONS OF THICK SKEWED CANTILEVERED PLATES

Abstract The natural frequencies of skewed cantilevered thick plates are determined by using the Ritz method. The present work is the first known three-dimensional study of the problem. Assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly, and which are mathematically complete. Accurate natural frequencies are calculated for skewed thick plates having arbitrary degrees of skewness. Detailed numerical studies reveal interesting trends concerning the variation of frequencies with increasing skew angle. Results obtained by using the present method are compared with those obtained by using three-dimensional finite elements.

Large amplitude free vibration of thick shallow shells supported by shear diaphragms

Abstract The effect of thickness and curvature upon the large amplitude vibration of shallow shells is studied in this paper. For this purpose, the non-linear governing equations for thick shallow shells which have principle curvatures and rectangular planform are derived by Hamiltons principle using first order shear deformation theory. Applying Galerkins procedure and eliminating variables except for transverse displacement, the governing equations are reduced to an elliptic ordinary differential equation in time. The period of vibration for the shell is calculated by integrating the equation using a Gauss-Legendre integration method. The present method is applied to a shallow shell which has a rectangular boundary supported by shear diaphragms.