Solid Circular Plate Research Articles

Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

On a Winkler foundation, solid circular plate vibration is examined using a higher-order finite element in polar co-ordinate system. The present formulation has developed a Mat-lab code to handle any boundary conditions. Validation of the code is carried out after the convergence studies. The results are compared to other researchers and show excellent conformity. Furthermore, a parametric analysis gave the first 10 natural frequency characteristics in tabular and graphical form. The authors conclude that the present formulation is straightforward, behaves exceptionally well for thin solid circular plates on elastic foundations with reasonable convergence rate and accuracy, and requires less computational effort, resources, and time.

Finite Element Investigation of Anisotropic Circular Plates

The mathematical modeling and finite element analysis of solid circular plates, analyzed using continuum finite elements under conditions of axisymmetry, is investigated in this paper. Both isotropic and anisotropic material characterizations are considered. Once a suitable “baseline” model is developed, parametric analyses are performed so as to gain insight into the behavior of anisotropic solid circular plates.

The Impact of the Cut-Out Shape on the Dynamic Behavior of Composite Thin Circular Plates

This study measures the impact of the cut-out shape on the fundamental frequency and the harmonic response of thin composite circular plates under fixed boundary conditions. For this purpose, the Finite Element Method has been employed using the SHELL181 element of ANSYS Workbench 18.2 software. Nine different circular plates having central equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon, regular octagon, regular nonagon, regular decagon, and circular cut-outs have been designed to understand the changes in the vibrational behavior of a structure as the cut-out geometry approximates to circular. Additionally, a solid circular plate has been modelled for comparison purposes. To investigate the effect of the cut-out shape more deeply, all structures have been designed as cross-ply and angle-ply composites. The free vibration analysis has been performed to obtain the fundamental frequency of each structure. The harmonic response analysis problem has been solved by employing the Mode Superposition Method and considering a constant damping ratio. The results of the study have been interpreted by considering the displacement response, stress, and phase shifts at a certain frequency range. The results indicate that the shape of the cut-out considerably affects the fundamental frequency and harmonic response of the circular structure regardless of its fiber orientation.

Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic–hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.

A 3D Free Vibration Analysis of the Horn-Gear System through Chebyshev–Ritz Method in Ultrasonic Gear Honing

Applying the ultrasonic machining in gear honing can improve honing speed, reduce cutting force, and avoid blocking. There are two problems leading to the decrease of calculation accuracy in the traditional nonresonant theory of the ultrasonic gear honing. One is that one-dimensional longitudinal vibration theory and two-dimensional theory cannot reflect the vibration characteristics of ultrasonic horn and gear comprehensively. And, the other one is that the difference of the analysis dimension between the two theories leads to mismatch of the coupling condition dimension between ultrasonic horn and gear. A free vibration analysis through Chebyshev–Ritz method based on three-dimensional elasticity theory was presented to analyze the eigenfrequencies of the horn-gear system in ultrasonic gear honing. In the method, the model of the horn-gear system was divided into four parts: a solid circular plate, an annular plate, a solid cylinder, and a cone with hole. The eigenvalue equations were derived by using displacement coupling condition between each part under completely free boundary condition. It was found that the eigenfrequencies were highly convergent through convergence study. The hammering method for a modal experiment was used to test the horn-gear systems’ eigenfrequencies. And, the finite element method was also applied to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, it showed that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of the horn-gear system in ultrasonic gear honing.

3-D elasticity numerical solution for magneto-hygrothermal bending of FG graphene/metal circular and annular plates on an elastic medium

Three-dimensional (3-D) asymmetric and axisymmetric bending analyses of solid circular and annular plates lying on an elastic medium with different boundary conditions are presented in this article. The present nanocomposite circular and annular plates are made of aluminum (Al) matrix reinforced with graphene platelets (GPLs) that uniformly distributed or functionally graded (FG) through the thickness. This model is exposed to transverse external load, thermal load and humid conditions as well as in-plane magnetic field. Lorentz magnetic force is obtained by applying Maxwell's equations. While, the temperature and moisture, along the transverse direction, are deduced by solving the one-dimensional heat conduction and moisture diffusion equations, respectively. The governing equations are established based on the three-dimensional elasticity theory. By applying the differential quadrature method (DQM) along both radial and thickness directions, the solution of magneto-hygrothermal bending of the circular and annular plates is obtained. Convergence and comparison investigations are introduced to illustrate the precision and validity of the present solution. The present paper aims to deduce the displacements and stresses of the FG graphene/metal circular and annular plates employing the DQM. Moreover, various parametric studies are presented to investigate the effects of outer-to-inner radius ratio, thickness to radius ratio, weight fraction of GPLs, boundary conditions, elastic foundation parameters, temperature, moisture concentrations on the stresses and displacements of the present model.

One dimensional thermal shock problem in a thick plate

In this research, investigation carried out for thermo-elastic response of thick solid circular plate subjected to certain boundaries. According to the linear function of temperature, heat source are generated further. For finding the solution of heat equation, the integral transform techniques are used. Hankel transform techniques are incorporated for analysis of the numerical results. For designing of engineering applications of beneficial structures or machines, the temperature, displacement and thermal stresses are investigated. Suitable values are provided to investigate the parameters and the expression for specific case of special interest and the numerical results has been investigated.

Experimental investigation of bolted hollow section splices using unstiffened circular end plate

Moment resisting connections are widely used nowadays in different applications. The use of circular hollow sections (CHS) has increased due to its aesthetic appearance and good mechanical properties. Attention should be given to their connections under different loading conditions to achieve greater flexibility for their use. This research investigates the behavior of bolted CHS splices with solid circular end plates under pure bending moment that allows the use of CHS as long flexural members. Eight connections were tested and corresponding finite element models were constructed. The finite element models were verified against the experimental results and shown to provide acceptable agreement in terms of both ultimate moment capacity and load-displacement curves. Three modes of failure were observed where the first is pure bolt failure, the second is pure end plate yielding while the third is a combination of the two modes where end plate yielding is accompanied by bolt failure. Stiffness was also observed and was found to be greatly affected by the thickness of the end plate. Increasing the number of bolts in the end plate did not have a substantial effect on the connection stiffness but it increased its moment capacity. Contact forces between the two end plates were observed and the axis of rotation can be conservatively considered to be tangent to the CHS on its compression side. This finding can be used for end plate yielding analysis.

Solid circular plate unilaterally supported along two antipodal edge arcs and deflected by a central transverse concentrated force

A solid circular plate unilaterally supported along two antipodal edge arcs and deflected by a static central transverse concentrated force is considered. It is clarified that two distinct mechanical responses are possible, depending on the angular extent of the supports; in the first kind of response, valid for small support angular widths, the plate rotates about the support lateral sides, lifting from the supports along their central zone. The second response is valid for high support angular extents, and according to this response the plate additionally beds over the central portion of the support upper faces. Only the first kind of response is examined in this paper. This plate contact problem is classifiable as receding, and the plate deflection is described in terms of a plate Kirchhoff model. The plate deflection is analytically expressed together with the transitional value of the support angular width that describes the passage from the first to the second mechanical response.

A unified solution for the vibration analysis of two-directional functionally graded axisymmetric Mindlin–Reissner plates with variable thickness

In this paper, an effective unified numerical approach is utilized to analyze the free and forced vibration response of two-directional functionally graded (2D-FG) thick circular and annular plates with variable thickness. The material properties are assumed to be graded continuously both in thickness and radial directions. The governing equations are converted to a set of ordinary differential equations (ODEs) based on the first order shear deformation theory. Obtained canonical equations are solved numerically by the Complementary Functions Method (CFM) combined with the Laplace transform for the first time. Using an effective and suitable inverse numerical Laplace transform method, the results are transferred back to time space. The damping model of Kelvin is used in the damped forced vibration analysis. The novelty of this paper is to infuse this efficient and accurate method to the dynamic analysis of a wide range of solid circular or annular plates, with linear or non-linear thickness profiles, radially Functionally Graded (FG), FG in the thickness direction or 2D-FG, without any restrictions. The influence of material variation exponents and thickness profiles on the considered problems are investigated. Several examples are presented and results are verified with those obtained by finite element method and available published literature. Excellent agreement is observed.

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress – resultants – displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (h/r0).

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material properties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England’s method, the problem can be solved by determining the expressions of four analytic functions. Expanding the transverse load in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.

A new modified higher-order shear deformation theory for nonlinear analysis of macro- and nano-annular sector plates using the extended Kantorovich method in conjunction with SAPM

In this research, the nonlinear local and nonlocal analysis of an annular sector plate is studied and solved based on a new modified higher-order shear deformation theory. Due to the shortcomings of HSDT in the two-dimensional nonlinear analysis, it is modified by eliminating the defects, and a comprehensive theory is presented for analyzing the mechanical behavior of an annular sector sheet in general form. The strain field is developed by considering the von Karman assumptions and also the nonlocal theory of Eringen from which the classical local analysis can be deduced conveniently by neglecting the small-scale effects. Whereas the annular sector plate is assumed, the sector, annular/circular, rectangular and solid circular plates can be simulated. Afterward, the nonlocal constitutive equations are derived and solved by using the two-dimensional SAPM (Dastjerdi et al. in Compos B Eng 98, 78–87, 2016). Moreover, a combination of the extended Kantorovich method and one-dimensional SAPM is applied. Since the presented theory is relatively new and similar studying was not available in order to compare the results, a comparison is done with the results of lower-order theories. Finally, the effect of various parameters, such as boundary conditions, different theories, nonlocal and local analyses, loading and the size of the plate, on the mechanical behavior of an annular sector plate are investigated.

Exact Solutions for Nonaxisymmetric Vibrations of Radially Inhomogeneous Circular Mindlin Plates With Variable Thickness

The nonaxisymmetric transverse free vibrations of radially inhomogeneous circular Mindlin plates with variable thickness are governed by three coupled differential equations with variable coefficients, which are quite difficult to solve analytically in general. In this paper, we discover that if the geometrical and material properties of the plates vary in generalized power form along the radial direction, then the complicated governing differential equations can be reduced into three uncoupled second-order ordinary differential equations which are very easy to solve analytically. Most strikingly, for a class of solid circular Mindlin plates with absolutely sharp edge, the natural frequencies can be expressed explicitly in terms of elementary functions, with the corresponding mode shapes given in terms of Jacobi polynomials. These analytical expressions can serve as benchmark solutions for various numerical methods.

Thermo elasticity solution of functionally graded, solid, circular, and annular plates integrated with piezoelectric layers using the differential quadrature method

ABSTRACTStatic analysis of functionally graded (FG) solid circular/annular plates imbedded in piezoelectric layers under thermo-electro mechanical load is investigated using the differential quadrature method. The plate has various edge boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. The method is validated by comparing numerical results with the results obtained in the literature. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the thermoelastic behavior of FG solid circular and annular plates are investigated.

Optimal location of ring support for heavy Mindlin plates under axisymmetric loading

Bending of an annular thick plate resting on a ring support is analyzed under the action of power-law axisymmetric loading. A single governing differential equation for the Mindlin plate theory is derived. By solving associated boundary value problem, the optimal support location is determined to achieve minimizing the maximum deflection of a moderately thick circular or annular plate. The minimum sag of a heavy solid circular plate with or without the center support under self-weight is also analyzed. In addition to applied loading and the restraint of plate's rims, the optimal location of the ring support is also related to Poisson's ratio and the ratio of inner-to-outer radius. Auxetic plates with negative Poisson's ratio require larger ring support's radius, and conventional plates require smaller ring support's radius. Usually, the optimal support location is closer to the outer rim and far to the inner rim for a plate of self-weight. The obtained results are useful in safety design of circular or annular plates under complicated loading.

Closed-form solution for circular microstructure-dependent Mindlin plates

In this paper, we derive a general closed-form solution for the static, axisymmetric bending of solid and annular microstructure-dependent Mindlin plates based on the modified couple-stress theory. The solution is developed by employing a suitable change in displacement variables. The couple-stress solution contains modified Bessel functions which do not appear in the classical case. The stress distributions of the annular microstructure-dependent plate are obtained in terms of load resultants by using the general closed-form solution. Analytical and numerical examples considering solid and annular circular plates are presented. The examples include simply supported and clamped solid and annular plates subjected to a uniformly distributed load and annular plates attached to a point-loaded rigid shaft.

The Influence of Flexibility Coefficient on the Size of Internal Forces and Deformations in Circular Plates on Elastic Medium

Abstract This paper presents an analytical study which deals with the behavior of the circular plates in bending theory, considering the soil-structure interaction under Winkler's hypothesis. It was intended to illustrate the variation of internal forces and deformations according to the flexibility coefficient of plates considering three models: a fixed solid circular plate subjected to a uniformly distributed load, a fixed solid circular plate acted by a displacement applied on the exterior contour and a solid plate subjected to a temperature gradient. For this study the computation relations were written as a product between a dimensional and a non-dimensional factor, the last one indicating the variation of internal forces and deformations. For each type of action there are presented results obtained using the finite element method to illustrate the differences between this method and the analytical computation.

Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory

This study presents the buckling analysis of radially loaded solid circular plate made of porous material. Properties of the porous plate, where pores are assumed to be saturated with fluid, vary across its thickness. The boundary condition of the plate is assumed to be clamped and the plate is assumed to be geometrically perfect. The higher order shear deformation plate theory (HSDT) is employed to derive the governing equations. The equilibrium and stability equations, derived through the variational formulation and based on the Sanders non-linear strain–displacement relation, are used to determine the prebuckling forces and critical buckling loads. The results are compared with the buckling loads of circular plates made of porous material and reported in the literature based on the classical plate theory (CPT) and the first order shear deformation plate theory (FSDT).

A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects

A new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is developed using a modified couple stress theory, a surface elasticity theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. The new plate model contains a material length scale parameter to capture the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical plate model includes the plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved by directly applying the general formulas derived. The numerical results reveal that the deflection of the clamped circular plate predicted by the current Kirchhoff plate model is smaller than that predicted by the classical elasticity-based model, but the former approaches the latter with the increase of the plate thickness. It is also found that the difference between the plate deflection predicted by the model incorporating the surface energy effect and that predicted by the classical model is significant when the plate thickness is sufficiently small. However, the difference is diminishing as the plate thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.