Elastic Edge Supports Research Articles

On the vibration of functionally graded annular plate with elastic edge supports and resting on Winkler foundation

ABSTRACT The current paper deals with the exact solution of vibration analysis of functionally graded (FG) annular plate elastically restrained boundary conditions and resting on Winkler foundation. Hamilton’s principle has been applied to derive the motion equations of FG annular plate and associated boundary conditions based on the classical plate theory (CPT). It is assumed that the material properties have been graded through the direction of plate thickness according to power law distribution. The effects of some key system parameters such as normalised inner radius, non-dimensional foundation stiffness, non-dimensional transverse and rotational spring constants of the elastic boundary, volume fraction index and number of nodal diameters on the frequency parameter have been assessed in tabular and graphical forms. The accuracy of the present work has been validated by comparing the present results with those computed by previously published literatures.

An approximate closed-form buckling solution for the local skin buckling of stiffened plates with omega stringers: The case of antisymmetric cross-ply and angle-ply laminations

Estimating the buckling resistance of stiffened plates with detailed numerical models can provide high fidelity results, but for the preliminary design phase of aircraft structures this is impractical, since numerous iterations or sensitivity analyses are required. Analytical solutions become preferable for this design phase. To this end, an approximate closed-form solution is developed to estimate the critical skin local buckling load of stiffened plates with omega stringers with antisymmetric cross-ply and angle-ply lamination. This is obtained using an innovative energy-based homogenization method that yields to an equivalent bending stiffness matrix. The research further examines the influence of two key parameters to the analytical estimation of the buckling load of the fore mentioned partially anisotropic skin. The first parameter concerns the selection of the appropriate boundary conditions that must be assumed for the skin-stringer junction. The mathematical model considers the part of the skin between two consecutive omega stringers. The rest of the stiffened plate is replaced by equivalent transverse and rotational springs, which act as elastic edge supports. The elastic restraints are determined with two distinctive approaches considering the bending and torsional stiffness of the omega stringers. As a second parameter two different deflection functions (trigonometric and polynomial) are considered for the Rayleigh-Ritz method. Aim is to determine the appropriate combination of elastic restrain stiffness and deflection function that yield to more accurate buckling results for the cases of antisymmetric cross-ply and angle-ply skin laminations. Finally, the obtained analytical results are evaluated by comparing them with the respective numerical. The comparison showed that a satisfactory correlation can be achieved, for a certain combination of boundary conditions and deflection functions. Discussion and conclusions regarding the accuracy, discrepancies and limitations of the derived buckling solution are highlighted.

Elastic buckling analysis of polygonal thin sheets under compression

In this work, we have studied elastic buckling of regular hexagonal thin sheets made of homogeneous and isotropic materials under in-plane hydrostatic and uniaxial compression, with internal supports, translational and rotational elastic edge supports, and a combination of free, simple-support and clamped boundary conditions. An energy method called Rayleigh–Ritz was utilized and the kinematic hypotheses of the classical thin plate buckling theory were applied. Displacement conditions were applied to edges and internal supports by employing proper basic P-Ritz functions. This approach gives an approximate analytical response that was calculated with Maple software and validated by previous researches. Moreover, the results were compared to numerical results obtained from ANSYS software. The findings revealed that besides having the advantages of other analytical methods, the Rayleigh–Ritz method is easy to use and straightforward to apply different constraints on boundaries and internal points; however, increasing the number of edges and constraints adds significantly the burden of calculation.

Experimentally Investigating Annealed Glazing Response to Long-Duration Blast

This paper investigates the response of annealed (float) glazing during long-duration blast as a function of glazing thickness, area, aspect ratio and edge support conditions. With positive phase durations in excess of 100ms, long-duration blasts produce substantial impulse and pronounced dynamic pressures. Transient dynamic response of annealed glazing during these events is a complex phenomenon dependent upon explosive proximity, structural arrangement and material properties. In particular, breakage time and initial fracture location are dictated by maximum principal stress exceedence at randomly distributed micro-flaws. As part of a larger research study, twelve full-scale air-blast trials employing 24 annealed glazing panels were conducted with ~14kPa peak static overpressure and ~110ms positive phase duration. Results are reported, where it is shown that notionally elastic edge supports can prevent glazing breakage versus rigidly clamped arrangements when suitable panel dimensions are employed. Fragmentation modes are also demonstrated to be a strong function of edge conditions with elastically supported panels producing large, angular fragments. In contrast, rigid arrangements are shown to induce localised impulsive stress transmission at clamped edges, leading to significant cracking and small fragments. Substantially different fragment masses and geometries demonstrate the need to accurately quantify edge supports when appraising fragment hazard. Quantification of peak panel deflection, breakage time and applied breakage impulse is then presented where results indicate the influence of edge supports and aspect ratio on glazing response to be dependent on proximity to the threshold area for a particular thickness.

Frequencies of circular plate with concentric ring and elastic edge support

Exact solutions for the flexural vibrations of circular plates having elastic edge conditions along with rigid concentric ring support have been presented in this paper. Values of frequency parameter for the considered circular plate are computed for different sets of values of elastic rotational and translation restraints and the radius of internal rigid ring support. The results for the first three modes of plate vibrations are computed and are presented in tabular form. The effects of rotational and linear restraints and the radius of the rigid ring support on the vibration behavior of circular plates are studied over a wide range of non-dimensional parametric values. The values of the exact frequency parameter presented in this paper for varying values of restraint parameters and the radius of the rigid ring support can better serve in design and as benchmark solutions to validate the numerical methods obtained by using other methods of solution.

Vibrations of Elastically Restrained Circular Plates Resting on Winkler Foundation

This paper deals with the study of the vibration characteristics of thin circular plates with elastic edge support and resting on an elastic foundation. The Winkler model describes the foundation. The exact analytical method is used to derive the frequency equation of the plate with elastic edge support and resting on an elastic foundation. The basic equations are those which are obtained from classical thin plate theory. Parametric investigations on the vibrations of circular plates with elastic edge support and resting on an elastic foundation have been carried out with respect to various values of translational stiffness ratio, rotational stiffness ratio, foundation stiffness ratio and boundary conditions. The results for 12 modes of vibration are presented for each case considered, and extensive data are tabulated so that pertinent conclusions can be arrived at on the influence of translational edge restraint, rotational edge restraint and the foundation stiffness ratio of the Winkler foundation on the natural frequencies of uniform isotropic circular plates.

Free vibrations of arbitrary quadrilateral thick plates with internal columns and uniform elastic edge supports by pb-2 Ritz method

Free vibration analysis of arbitrary quadrilateral thick plates with internal columns and elastic edge supports is presented by using the powerful pb-2 Ritz method and Reddy\'s third order shear deformation plate theory. The computing domain of arbitrary quadrilateral planform is mapped onto a standard square form by coordinate transformation. The versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial and a basic function are taken to be the admissible functions. Substituting these displacement functions into the energy functional and minimizing the total energy by differentiation, leads to a typical eigenvalue problem, which is solved by a standard eigenvalue solver. Stiffness and mass matrices are numerically integrated over the plate by using Gaussian quadrature. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. A lot of numerical results for reasonable natural frequency parameters of quadrilateral plates with different combinations of elastic boundary conditions and column supports at any locations are presented, which can be used as a benchmark for future studies in this area.

Free Vibration of Circular Plates with Elastic Edge Support and Resting on an Elastic Foundation

An analytical study is presented for the vibration characteristics of thin circular plates with rotational elastic edge support and resting on an elastic foundation. The Winkler model was used to describe this foundation. The exact analytical method is used to derive the frequency equation of the plate with elastic support and resting on this foundation. Parametric investigations of the vibration of circular plates with an restrained elastically edge against rotation and resting on an elastic foundation have been carried out with respect to various rotational stiffness parameters, foundation stiffness parameters, and boundary conditions. Twelve vibration modes are presented. The location of the stepped region with respect to rotational restraint changed very little and moved from the range of 0.014 26† [10.006 204 8‡]–1 134.227 [10.025 94] to 0.017 58 [14.001 133]–6 352.958 2 [14.658 356]. Here, † represents the rotational spring stiffness ratio, and ‡ represents the foundation stiffness ratio. Extensive data are tabulated so that pertinent conclusions can be arrived at on the influence of rotational edge restraint and the foundation parameters of a Winkler foundation on the natural frequencies of uniform thin isotropic circular plates.

Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb-2 Ritz method

Free vibration analysis of rectangular plates with internal columns and elastic edge supports is presented using the powerful pb-2 Ritz method. Reddy's third order shear deformation plate theory is employed. The versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial and a basic function are taken as the admissible functions. Substituting these displacement functions into the energy functional and minimizing the total energy by differentiation, leads to a typical eigenvalue problem, which is solved by a standard eigenvalue solver. Stiffness and mass matrices are numerically integrated over the plate using the Gaussian quadrature. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. Many numerical results for reasonable natural frequency parameters of rectangular plates with different combinations of elastic boundary conditions and column supports at any locations are presented, which can be used as a benchmark for future studies in this area.

Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels

A methodology for the analytical assessment of local buckling and post-buckling behavior of isotropic and orthotropic stiffened plates is presented. The approach considers the stiffened panel segment located between two stiffeners, while the remaining panel is replaced by equivalent transverse and rotational springs of varying stiffness, which act as elastic edge supports. A two-dimensional Ritz displacement function (pb-2 Ritz) is utilized in the solution of the local buckling problem of isotropic and laminated symmetric composite panels with arbitrary edge boundary conditions. The buckling analysis of the segment provides an accurate and conservative prediction of the panel local buckling behavior. Consequently, the developed methodology is extended in the prediction of the post-buckling response of stiffened panels of which the skin has undergone local buckling. Of high importance for the calculation of the post-buckling behavior is the selection of appropriate boundary conditions for the structural members analyzed. A comparison of the present methodology results to respective finite element (FE) results has shown a satisfactory agreement.

Free in-plane vibration analysis of rectangular plates with arbitrary elastic boundaries

Abstract A Ritz method using a set of trigonometric functions is developed to obtain accurate in-plane modal properties of rectangular plates with arbitrary nonuniform elastic edge restraints. Reliability of the current approach is first assured by comparison against exact and analytical-type solutions for plates with classical boundary conditions and uniform elastic boundaries. For the first time to the author’s knowledge, the problem of free in-plane vibration of plates having triangularly and parabolically varying elastic edge supports is then considered. Accurate upper-bound solutions are tabulated to provide valuable benchmark data against which the findings of other researchers can be compared in the future. Effects of nonuniform elastic spring stiffness on the in-plane natural frequencies and modal shapes are also presented.

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading. This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.

System Identification of Partially Restrained Composite Plates Using Measured Natural Frequencies

A nondestructive evaluation technique established on the basis of a global minimization method is presented for the system identification of laminated composite plates partially restrained by elastic edge supports. Six natural frequencies extracted from the vibration data of the flexibly restrained plate are used to identify the system parameters of the plate. In the identification process, the trial system parameters are used in the Rayleigh–Ritz method to predict the theoretical natural frequencies of the plate, an error function is established to measure the sum of the differences between the experimental and theoretical predictions of the natural frequencies, and the global minimization method is used to search for the best estimates of the parameters by making the error function a global minimum. The accuracy and efficiency of the proposed technique in identifying the parameters of several flexibly supported plates made of different composite materials are studied via both theoretical and experimental approaches. The excellent results obtained in this study have validated the applicability of the proposed technique.

Free vibration analysis by the superposition method of rectangular Mindlin plates with internal columns resting on uniform elastic edge supports

Free vibration analysis of rectangular Mindlin plates is carried out via the superposition method and is shown to produce accurate results. The analytical method proposed by the authors in this paper solves the problem for the case where the plate has simultaneous elastic edge and internal supports. The conditions of edge supports are uniform lateral, rotational and torsional elastic supports, whereas, the internal supports are column supports with finite area. Compatibility between the plate and column is achieved by requiring that the column and plate rotations be equal. The results presented herein are verified through comparison with results presented by others. Numerical examples presented in this study confirm that the analytical method is able to model the property of the elastic edge supports while simultaneously considering the effect of column restraint, an effect which increases with column number and area.

A “soft table” for the natural frequencies and modal parameters of uniform circular plates with elastic edge support

The paper introduces a “soft table” for the natural frequencies and modal parameters of uniform circular plates with elastic edge support. In contrast to conventional tabulations on paper, the “soft table” allows to save space and display parameters for required number of vibration modes. The “soft table” is realized in a collection of Matlab ® codes that calculate natural frequencies and mode shape parameters for a given set of translational and rotational constraints and modifiable Poisson ratio. Calculations of natural frequencies and modal parameters for a wide range of constraints revealed distinctive regions where the frequencies and parameters could change considerably. Knowing position and extension of these regions could be essential for design and vibration control of structures incorporating circular plates.

Free in-plane vibration analysis of rectangular plates with elastic support normal to the boundaries

The superposition method is employed to analyze the effects of elastic edge support on the in-plane free vibration frequencies and mode shapes of rectangular plates. The elastic edge support is considered to be uniformly distributed along the plate edges and symmetrically distributed with respect to the plate central axes. In addition, exact solutions are also obtained for a family of simply supported plates whose eigenvalues constitute upper limits for those of the elastically supported plates. An array of highly accurate eigenvalues for the completely free plate is also computed in order to provide lower limit eigenvalues. Eigenvalues are computed and tabulated for elastically supported plates of various aspect ratios and dimensionless elastic support coefficients.

Stability of a Rotating Heated Circular Plate With Elastic Edge Support

Stability of a Rotating Heated Circular Plate With Elastic Edge Support

Buckling analysis of laminated composite plates with arbitrary edge supports

A buckling analysis for a rectangular laminated composite plate with arbitrary edge supports subjected to biaxial compression loading was presented. The higher-order shear deformation theory was employed and a special displacement function, which could express an arbitrary edge support, was introduced into the Rayleigh–Ritz method. For the application of this analysis, the laminated composite plates with various edge supports were analyzed, and the buckling loads with these edge supports were obtained accurately. During these edge supports, particularly elastic edge support was simulated by three springs and reasonable results were obtained. Furthermore, the buckling modes were determined. The results show that the present method can be widely used to analyze the buckling behavior of composite plates with arbitrary edge conditions.

Buckling of triangular plates with elastic edge constraints

This paper presents the first-known investigation on the buckling behavior of triangular plates with both translational and rotational elastic edge constraints. The energy functional of a general triangular plate with elastic edge supports is derived, and thep-Ritz method is utilized to derive the governing eigenvalue equation for the buckling problem. Convergence and comparison studies are carried out to verify the validity and accuracy of the solution method. Extensive buckling factors are presented for several selected isosceles and right-angled triangular plates of various edge support conditions and subjected to isotropic inplane compressive load. The influence of elastic edge supports on the buckling factors for triangular plates of various vertex angles (aspect ratios) and boundary conditions is examined.

A TIP-TILT ADAPTIVE OPTICS SYSTEM FOR AMATEUR ASTRONOMERS AND OPTIMUM PLACEMENT OF ACTUATORS

A major concern in the design of a tip-tilt adaptive optics system, for amateur astronomical telescopes, is to find optimum positions of the actuators that result in least distortions of the mirror surface. A semi-analytical approach has been used, wherein the mirror is modelled as a thin circular plate with a peripheral ring mass and an elastic edge support. Modal analysis is performed to determine the natural frequencies and mode shapes of the system. The results of the modal analysis are incorporated in the subsequent harmonic analysis, where the response of the system to harmonic forces, applied by the three actuators, is expressed in terms of the Green functions. For various positions of the actuators, the maximum distortions on the mirror surface are evaluated, and from these results, the optimum positions of the actuators are located. The semi-analytical results are vertified by purely numerical results obtained from finite element analysis.