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

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 σ_x=−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/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

Buckling of composite plates

The buckling of composite plates is a very complicated subject about which at least 200 references are already available. The present paper examines the characteristics and parameters which may need to be considered, both from mathematical and physical points of view, and provides perspective and organization to the subject. The first part deals with classical bifurcation buckling analysis, discusses the relevant plate equations and their solutions, and considers shapes, edge conditions and loadings which may arise. The second part treats classical complicating effects (those still yielding linear eigenvalue problems with bifurcation) including: elastic foundation, variable thickness, shear deformation, hygrothermal effects and inplane heterogeneity. The third part takes up non-classical considerations—postbuckling, geometric imperfections, parametric excitation, follower forces and inelastic material.     

An exact solution for in-plane vibrations of an arch having variable curvature and cross section

An exact solution for in-plane vibration of arches with variable curvature as well as cross section has been developed using the famous Frobenius method combined with the dynamic stiffness method. The effects of shear deformation and rotary inertia are taken into account. A convergent solution is always guaranteed without numerical difficulties. An important by-product of this series solution is that the first known dynamic stiffness matrix for an arch with variable curvature and variable cross section is also explicitly formulated. Some new numerical results are given for non-dimensional frequencies of parabolic arches with a certain type of variation of cross section along the arch that is often used in practical structures. Extensive and accurate (six significant figure ) non-dimensional frequency tables and graphic charts are presented for a series of parabolic arches showing the effects of rise to span length, slenderness ratio, and variation of cross section.     

Vibrations of twisted cantilever plates—a comparison of theoretical results

Previously published literature shows widely different results for the free vibration frequencies of twisted cantilever plates. Inasmuch as it is important to know the vibration characteristics of turbomachinery blades, which may have considerable twist, it would be desirable to have a comprehensive, definitive set of results, and to establish which of the numerous theoretical methods available can adequately analyse such problems. For this purpose, a joint government/industry/university research study was organized. Numerical results were obtained for a set of 20 different twisted plates having various aspect ratios, thickness ratios and pretwist angles. Nineteen distinct theoretical methods were employed, 15 using finite elements, two using shell theory, and two using beam theory. Although some of the best-known computational procedures (especially finite element codes) were used by analysts with great experience, the numerical results obtained showed considerable disagreement. The present paper describes the analytical methods used and exhibits samples of the type of results obtained.     

Bending of plates with circular holes

Summary The method of point matching is used to solve three problems for the bending of a plate having circular holes. The first two problems consist of a uniformly loaded square plate either simply supported or clamped along the outer boundary. Free edge boundary conditions are satisfied exactly along the internal hole and the conditions along the outside contour are satisfied by point matching in the least squares sense. Deflection and bending moment curves along the hole and along the outer edges are presented for various ratios of hole diameter to size of plate. Results for the case when the outer edges are simply supported can be compared with the less extensive results ofDedic.The last problem solved is that of an infinite plate having equally spaced circular holes. The plate is loaded by its own weight and supported at points equidistant from the hole centers. Three different approaches to the problem are used, all satisfying the boundary conditions by point matching. Results for deflections and bending moments for various hole diameters are presented.

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Zusammenfassung Mittles der Methode der Randkollokation werden drei Biegungsprobleme für eine Platte mit kreisförmigen Löchern gelöst. In den beiden ersten Problemen wird eine quadratische Platte unter Gleichlast behandelt, die am Außenrand entweder frei aufliegt oder eingespannt ist. Die Bedingungen am freien Lochrand werden exakt erfüllt, jene am Außenrand durch Kollokation im Sinne der kleinsten Quadrate. Durchbiegung und Biegemoment am Lochrand und am Außenrand werden für verschiedene Verhältnisse von Lochdurchmesser zur Seitenlänge der Platte angegeben. Im Falle des frei aufliegenden Außenrandes können die Ergebnisse mit jenen—allerdings weniger umfassenden—vonDedic verglichen werden.Als letztes wird das Problem einer unbegrenzten Platte mit gleichmäßig verteilten kreisförmigen Löchern gelöst. Die Platte wird durch ihr Eigengewicht beansprucht und wird an Punkten unterstützt, die äquidistant zu den Mittelpunkten der Löcher liegen. Es werden drei verschiedene Näherungsmethoden verwendet, die alle die Randbedingungen mittels Kollokation befriedigen. Durchbiegungen und Biegemomente werden für verschiedene Lochdurchmesser angegeben.

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With 14 Figures

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The results contained in this paper are a partial product of research supported by the U.S. Air Force under Contract No. AF 33 (615) 2504, monitored byG. E. Maddux, FDTR-13, Wright Patterson Air Force Base.     

Vibrations of cantilevered circular cylindrical shells: Shallow versus deep shell theory

Free vibrations of cantilevered circular cylindrical shells having rectangular plan-forms are studied in this paper by means of the Ritz method. The deep shell theory of Novozhilov and Goldenveizer is used and compared with the usual shallow shell theory for a wide range of shell parameters. A thorough convergence study is presented along with comparisons to previously published finite element solutions and experimental results. Accurately computed frequency parameters and mode shapes for various shell configurations are presented. The present paper appears to be the first comprehensive study presenting rigorous comparisons between the two shell theories in dealing with free vibrations of cantilevered cylindrical shells.     

Conditions for laminated plates to remain flat under inplane loading

There is considerable confusion in the existing literature as to whether an unsymmetrically laminated, composite plate will remain flat due to the application of inplane compressive or shear loads. If it does not remain flat, then bifurcation buckling would not normally take place, and transverse displacements would occur no matter how small the inplane loads. This paper investigates the conditions under which arbitrarily laminated and arbitrarily loaded plates remain flat, and therefore when buckling can occur. It is demonstrated that for uniform or linearly varying inplane loads no transverse pressure is necessary to keep a plate flat, although edge moments or transverse forces may be required at the boundaries.     

Effects of edge constraints upon shallow shell frequencies

The effects of changing edge constraints upon the frequencies of shallow shells with rectangular planforms are studied. Attention is focused upon a single edge, with the other three edges remaining completely free. For that edge clamped, simply supported, and free edge conditions are imposed; however, each of these has four possibilities depending upon the existence of either or both types of membrane constraint along the single edge. The Ritz method, assuming algebraic polynomials as displacement functions, is used to obtain accurate results. Frequencies for three types of shallow shells (circular cylindrical, spherical, hyperbolic paraboloidal) are obtained, for two shallowness ratios and two thickness ratios each. Careful attention is paid to the number of rigid body modes (zero frequencies) present, for these enter quite strongly into considerations of the effects of changing edge constraints.     

Buckling studies for simply supported symmetrically laminated rectangular plates

Extensive and accurate numerical results are presented for the critical buckling loads of simply supported, rectangular, laminated composite plates subjected to five types of loading conditions: (1) uniaxial, (2) hydrostatic biaxial, (3) compression-tension biaxial, (4) positive shear and (5) negative shear. Considerably different results are found for the two types of shear loading for angle-ply composites. The Ritz method, along with displacements assumed in the form of a double sine series, is used to solve the problems. Convergence studies are presented to demonstrate the accuracy of the results. Contour plots of the buckled mode shapes are shown for some of the more interesting plate and loading configurations.     

Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloids and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components u_r, u_θ and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.