Vibrations of rectangular plates with internal cracks or slits

This work applies the famous Ritz method to analyze the free vibrations of rectangular plates with internal cracks or slits. To retain the important and useful feature of the Ritz method providing the upper bounds on exact natural frequencies, the paper proposes a new set of admissible functions that are able to properly describe the stress singularity behaviors near the tips of the crack and meet the discontinuous behaviors of the exact solutions across the crack. The validity of the proposed set of functions is confirmed through comprehensive convergence studies on the frequencies of simply supported square plates with horizontal center cracks having different lengths. The convergent frequencies show excellent agreement with published accurate results obtained by an integration equation technique, and are more accurate than those obtained by a previously published approach using the Ritz method combined with a domain decomposition technique. Finally, the present solution is employed to obtain accurate natural frequencies and mode shapes for simply supported and completely free square plates with internal cracks having various locations, lengths, and angular orientations. Most of the configurations considered here have not been analyzed in the previously published literature. The present results are novel, and are the first published vibration data for completely free rectangular plates with internal cracks and for plates with internal cracks, which are not parallel to the boundaries.     

The influence of corner stress singularities on the vibration characteristics of rhombic plates with combinations of simply supported and free edges

This paper offers accurate flexural vibration solutions for rhombic plates with simply supported and free edge conditions. A cornerstone here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, hinged–hinged and/or hinged–free corners having obtuse angles of the rhombic plates. These singularities become significant to the vibration solution as the rhombic plate becomes highly skewed (i.e. the obtuse angles increase). The classical Ritz method is employed with the assumed normal displacement field constructed from a hybrid set of (1) admissible and mathematically complete algebraic polynomials, and (2) comparison functions (termed here as “corner functions”) which account for the bending stress singularities at the obtuse hinged–hinged and/or hinged–free corners. It is shown that the corner functions accelerate the convergence of solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75° (i.e. obtuse angles of 165°), so that the influence of the stress singularities is large. Frequencies and mode shapes of isosceles triangular, hinged–free plates are also available from the data presented.     

Free vibrations of square plantes with stiffened square openings

A grid frame work model is extended to obtain the natural frequencies of square plates with stiffened square openings. This technique eliminates the use of fictitious points at the re-entrant corner point, junction point, etc. on the stiffened opening. Numerical results are presented for the first time for the cases of plates with simply supported and clamped boundary conditions in order to show the convergence and versatility of the method. Also, this investigation provides a general feeling about the changes in fundamental frequencies that occur when stiffeners are introduced.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Buckling of symmetrically laminated rectangular plates under parabolic edge compressions

Buckling analysis of symmetrically laminated rectangular plates with parabolic distributed in-plane compressive loadings along two opposite edges is performed using the Rayleigh-Ritz method. Classical laminated plate theory is adopted. Stress functions satisfying all stress boundary conditions are constructed based on the Chebyshev polynomials. Displacement functions for buckling analysis are constructed by Chebyshev polynomials multiplying with functions that satisfy either simply supported or clamped boundary condition along four edges. Methodology and procedures are worked out in detail. Buckling loads for symmetrically laminated plates with four combinations of boundary conditions are obtained. The proposed method is verified by comparing results to data obtained by the differential quadrature method (DQM) and the finite element method (FEM). Numerical example also shows that the double sine series displacement for simply supported symmetrically laminated plates having bending-twisting coupling may overestimate the stiffness, thus providing higher buckling loads.     

Effect of notch geometry on the fatigue strength and critical distance of TC4 titanium alloy

Studying the effect of different geometric features of machined notch on the fatigue strength and critical distance has an important guiding role to understand the critical distance size effect and to predict the HCF strength of turbine engine fan blades after FOD. Systematically experimental investigations of geometrical characteristic effects on the 10⁶ cycle fatigue strength and critical distance for TC4 machined notched plates at stress ratios of R = 0.1 have been conducted. 123 specimens, including unnotched plates and three different types of notched plates (V-notches, U-notches and C-notches) with various notch root radii, depths and angles have been considered. The results indicate that the notch with small radius can significantly lead to high stress concentration and greatly reduce the HCF strength, while the notch angle and notch depth can affect the HCF strength to a certain extent. The K _t related model does not apply to describe the critical distance size effect perfectly. The critical distance has linear relationship with the notch root radius but no significant correction with the notch depth or notch angle. The findings of this study are beneficial for the size effect modeling and later fatigue strength evaluating of TC4 notched components.     

Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.     

Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method

The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh-Ritz method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the boundary characteristic orthogonal polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigenvalues, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel functions, where the exact solution is available. Moreover, the eigenvectors, which are the unknown coefficients of the Rayleigh-Ritz method, are used to present the mode shapes of the plate. To validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh-Ritz method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes.     

Postbuckling behavior of rectangular orthotropic plates with two free side edges

The postbuckling behavior of rectangular orthotropic plates with two free side edges is studied based on a perturbation method. Both exact and approximate solutions of the second-order perturbation equations are given. Numerical results for plates having various elastic properties are also presented.     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

3-D vibration analysis of skew thick plates using Chebyshev–Ritz method

The free vibration characteristics of skew thick plates with arbitrary boundary conditions have been studied based on the three-dimensional, linear and small strain elasticity theory. The actual skew plate domain is mapped onto a basic cubic domain and the eigenvalue equation is then derived from the energy functional of the plate by using the Ritz method. A set of triplicate Chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometric boundary conditions of the plate is developed as the trial functions of the displacement components. The vibration modes are divided into antisymmetric and symmetric ones in the thickness direction and can be studied individually. The convergence and comparison studies show that rather accurate results can be obtained by using this approach. Parametric investigations on rhombic plates with fully clamped edges and completely free edges are performed in detail, with respect to the thickness-span ratio and skew angle. Some results known for the first time are reported, which may serve as the benchmark values for future numerical technique research.     

Free vibration analysis of multi-directional functionally graded circular and annular plates

This paper addresses the free vibration of multi-directional functionally graded circular and annular plates using a semianalytical/ numerical method, called state space-based differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semi-analytical/numerical method. This method gives an analytical solution along the thickness direction, using a state space method and a numerical solution using differential quadrature method. Some numerical examples are presented to show the accuracy and convergence of the method. The most of simulations of the present study have been validated by the existing literature. The non-dimensional frequencies and corresponding displacements mode shapes are obtained. Then the influences of thickness ratio and graded indexes are demonstrated on the non-dimensional natural frequencies.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Levy solutions for vibration of multi-span rectangular plates

This paper presents the Levy method to investigate the vibration behaviour of multi-span rectangular plates. The Levy method is applicable and analytical for rectangular plates with at least two parallel simply supported edges. The continuity at an interface between two spans is maintained by imposing both the essential and natural boundary conditions along the interface. The impact of the internal line supports on the vibration behaviour of the plates is investigated by varying both the number of internal lines and the line positions. Results for the vibration of two- and three-span rectangular plates are presented, in which the first-known exact solutions for plates involving free edges are included. The present results may serve as benchmark solutions for such plates.     

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.     

Natural frequencies of shear deformable rhombic plates with clamped and simply supported edges

The natural vibrations of thick and thin rhombic plates with clamped and simply supported edges are analyzed, using assemblages of nine-node Lagrangian isoparametric quadrilateral C⁰ continuous finite elements based on a higher-order shear deformable thick plate theory. Here, additional nodal displacement degrees of freedom are derived by retaining higher-order powers of the thickness coordinate in the in-plane displacement fields, which in turn allows for the proper representation of the transverse shear strains of thick plates. Essential rotary inertia terms are derived and included in the present analysis. Nondimensional frequencies are calculated for thick and thin rhombic plates having various combinations of clamped and simply supported edge conditions, and skew angles. The efficacy of using higher-order shear deformable plate finite elements for predicting the in-plane vibration modes of rhombic plates is found to increase as the span-to-thickness ratio decreases and the skew angle increases. The present work shows that higher-order shear deformable finite elements essentially eliminate the transverse shear over-correction of thick rhombic plate frequencies that is produced when finite elements based on the widely used first-order Reissner-Mindlin plate theory are utilized.     

Modal characteristics of symmetrically laminated composite plates flexibly restrained at different locations

A method for determining modal characteristics (natural frequencies and mode shapes) of symmetrically laminated composite plates restrained by elastic supports at different locations in the interior and on the edges of the plates is presented. The classical lamination theory together with an appropriate set of characteristic functions are used in the Rayleigh-Ritz method to formulate the eigenvalue problem for determining the modal characteristics of the flexibly supported laminated composite plates. Sweep-sine vibration testing of several laminated composite plates flexibly restrained at different locations on the plates is performed to measure their natural frequencies. The close agreement between the experimental and theoretical natural frequencies of the plates has verified the accuracy of the proposed method. The effects of elastic restraint locations on the modal characteristics of flexibly supported laminated composite plates with different lamination arrangements and aspect ratios are studied using the present method. The usefulness of the results obtained for predicting sound radiation behavior of flexibly supported laminated composite plates is discussed.     

Analytical buckling solutions for mindlin plates involving free edges

This paper investigates the buckling behaviour of rectangular Mindlin plates having two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The proper boundary conditions at free edges subjected to in-plane loads have been examined. The buckling analysis is performed by applying the concept of state space to the Levy-type solution method to obtain the closed-form critical loads from the governing differential equations. The results, where possible, are compared with existing solutions to verify the validity of the solution method. The differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are reported. Several design charts representing the essential features of the critical load characteristics of rectangular plates with two opposite edges simply supported at least one free edge are obtained. The critical loads can be determined from the design charts without difficulty.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.