Annular Sector Plates Research Articles

A simplified approach for the thermoelastic large deflection in the thin clamped annular sector plate

ABSTRACTThe present article is concerned with analysis of large deflection of a heated thin annular sector plate with clamped edges under transient temperature distribution using Berger’s approximate methods. The prescribed surface temperature is at the top face of the plate whereas the bottom face is kept at zero temperature. In this study, the Laplace transform as well as the classical method have been used for the solution of heat conduction equation. The thermal moment is derived on the basis of temperature distribution, and its stresses are obtained using resultant bending moment and resultant forces per unit length. The calculations are obtained for the aluminium plate in the form of an infinite series involving Bessel functions, and the numerical results for temperature, deflection, resultant bending moments, and thermal stresses have been illustrated by graphs.

In-plane and shear buckling analysis of FG-CNTRC annular sector plates based on the third-order shear deformation theory using a numerical approach

The buckling analysis of thick composite annular sector plates reinforced with functionally graded carbon-nanotubes (CNTs) is presented under in-plane and shear loadings based on the higher-order shear deformation theory. It is considered that the plate is resting on the Pasternak-type elastic foundation. The overall material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are estimated through the micromechanical model. The governing equations are derived on the basis of the higher-order shear deformation plate theory, and the quadratic form of the energy functional of the system is presented. An efficient numerical method is presented in the context of variational formulation to obtain the discretized version of stability equations. The validation of the present study is demonstrated through comparisons with the results available in the literature and then comprehensive numerical results are given to investigate the impacts of model parameters on the stability of CNT-reinforced composite annular sector plates.

Shear post buckling analysis of FGM annular sector plates based on three dimensional elasticity for different boundary conditions

In this paper, shear post-buckling analysis of functionally graded annular sector plates is considered. In-plane shear loads have been applied to either radial, circumferential, or all edges of annular sector plates. Moreover, post-buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Material properties are graded asymmetrically in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while the Poisson’s ratio is assumed to be constant. The governing equations are based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor instead of the approximate plate theories and Von Karman assumptions. The governing equations are developed based on the principle of virtual work and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio and four different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to weak coupling between applied shear loads and the resulting twisting moments, shear post-buckling behavior of simply supported FGM plates is of the bifurcation-type buckling following a stable post-buckling path.

Thermal buckling analysis of functionally graded perforated annular sector plates using 3D elasticity theory

ABSTRACTThermal buckling of annular sector plates with circular cutouts made of functionally graded material is analyzed in this article. Graded material is considered with two parts of ceramic structure made of zirconia and metal structure which is aluminum. Unlike most conducted research, the direction of property change is observed in three main directions. Thermal loading is assumed as a uniform increase in temperature influencing the whole sector. 3D-finite element method based on elasticity theory is used in this analysis, which resets first and second variations of potential energy of the sector to zero to find equilibrium and stability equations, respectively. Green’s nonlinear strain–displacement relations are used to obtain geometrical stiffness matrix. In the finite element method, unlike most studies, a 3D eight-nodded element is used, which has nodes in the direction of thickness. The circular cutouts of the sector have added to the complexity of the analysis. The finite element formulation is coded in MATLAB. Finally, the effect of different parameters such as dimension and number of cutouts, power law index, and orientation of graded material on the critical thermal buckling temperature is studied.

Postbuckling analysis of smart FG-CNTRC annular sector plates with surface-bonded piezoelectric layers using generalized differential quadrature method

The purpose of the present study is to introduce the application of carbon nanotubes (CNTs) and piezoelectric layers in suppressing the postbuckling deflection of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular sector plates. To achieve this objective, a structural model is developed based on the first-order shear deformation theory (FSDT) along with the von Karman geometrical nonlinearity. The distribution of electric potential across the thickness of piezoelectric layers is modeled by a combination of linear and sinusoidal functions and the closed circuit electrical boundary condition is taken into account for the top and bottom surfaces of the piezoelectric layers. Generalized differential quadrature method (GDQM) is implemented to discretize the nonlinear stability equations, boundary conditions and Maxwell equation. The nonlinear system of equations is solved via a direct iterative method. A detailed parametric study is conducted to explore effects of geometrical parameters, boundary conditions, piezoelectric materials, thickness of the piezoelectric layers, external electric voltage, CNT distribution, and volume fraction on the postbuckling responses of FG-CNTRC annular sector plates with surface-bonded piezoelectric layers. Results indicate that both volume fraction and distribution of CNTs play a key role in enhancing the postbuckling strength of CNTRC annular sector plates. It is found that the type of piezoelectric materials, thickness of piezoelectric layers and the external electric voltage have a significant effect on suppressing the postbuckling deflection of FG-CNTRC annular sector plates. It is also found that the distribution of CNTs plays a pivotal role in changing buckling mode shapes of CNTRC annular sector plates. Besides, the proposed solution procedure has shown clear advantages over existing methods and demonstrated itself as a general, stable and accurate numerical method in solving strongly coupled nonlinear partial differential equations.

A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates.

A generalized solution procedure is developed for in-plane free vibration of rectangular and annular sectorial plates with general boundary conditions. For the annular sectorial plate, the introduction of a logarithmic radial variable simplifies the basic theory and the expression of the total energy. The coordinates, geometric parameters and potential energy for the two different shapes are organized in a unified framework such that a generalized solving procedure becomes feasible. By using the improved Fourier–Ritz approach, the admissible functions are formulated in trigonometric form, which allows the explicit assembly of global mass and stiffness matrices for both rectangular and annular sectorial plates, thereby making the method computationally effective, especially when analysing annular sectorial plates. Moreover, the improved Fourier expansion eliminates the potential discontinuity of the original normal and tangential displacement functions and their derivatives in the entire domain, and accelerates the convergence. The generalized Fourier–Ritz approach for both shapes has the characteristics of generality, accuracy and efficiency. These features are demonstrated via a few numerical examples.

Three-dimensional free vibration analysis of annular sector plates with arbitrary boundary conditions

The objective of this investigation is to apply the spectro-geometric method to study the vibration behavior of the annular sector plates with arbitrary boundary restraints on the basis of the three-dimensional theory of elasticity. All the classical homogeneous boundary conditions can be universally considered as the special cases when the stiffness for each of restraining springs is set equal to either zero or infinity in this study. Under the current solution framework, each displacement of the annular sector plate is, regardless of boundary conditions, invariantly expanded as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series and all the series expanded coefficients are determined with the Rayleigh–Ritz procedure. The present solution method can be generally applied to a broad range of annular sector plates, regardless of their thicknesses and sector angles. The current results are compared with those previously published in literature and the finite element analysis data to validate the current formulation. More results for plates with different boundary restraints and geometrical parameters are presented, which may serve as benchmark solutions for others.

Buckling response of moderately thick fluid-infiltrated porous annular sector plates

In the present article, the buckling of a fluid-infiltrated porous plate is investigated using Mindlin plate theory and an analytical procedure. A cosine rule for the pore distribution across the plate thickness is assumed with a coefficient defining porosity level. The governing stability equations are rewritten in terms of four auxiliary functions and decoupled with the aid of some mathematical manipulations. The decoupled partial differential equations are solved analytically by assuming simply supported radial edges for the plate. The critical buckling loads are calculated by considering fluid-saturated and fluid free conditions for the interconnected network of pores for different sector angles, thickness–radius ratios, coefficients of plate porosity, aspect ratios, and boundary conditions. It is found that the pore fluid compressibility affects the buckling load significantly.

An analytical study on the free vibration of moderately thick fluid-infiltrated porous annular sector plates

An exact analytical approach based on Mindlin plate theory is considered for free vibration analysis of fluid-saturated porous annular sector plates. The interconnected network of pores is saturated by inviscid fluid and the fluid is trapped in the network. The plate’s radial edges are considered to be simply supported and four auxiliary functions are used to evaluate the natural frequencies of porous plates under undrained condition. The mechanical properties of the material are considered to vary through the thickness by expressing shear modulus and density in terms of a simple cosine rule in case of plates with pores free of fluid. The present method is validated by comparing it with the results of other accurate solutions found in the literature. The influence of the coefficient of plate porosity, geometrical parameters as well as the effect of fluid on natural frequency response of porous annular sector plates under various boundary conditions are comprehensively investigated. It is found that the presence of fluid in the interconnected network of pores causes the fundamental natural frequency to increase. The method proposed in this paper may provide useful information for the future assessments of the dynamic response of porous structures when fluid–solid interaction effects are fully taken into account.

An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

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.

Unified approach for vibration analyses of annular sector and annular plates with general boundary conditions

This paper presents a unified approach for predicting the free and forced (steady-state and transient) vibration analyses of annular sector and annular plates with various combinations of classical and non-classical boundary supports. In spite of the types of the boundary restraints and the shapes of the plates, the admissible displacement function is described as a modified trigonometric series expansion, and four sine terms are introduced to overcome all the relevant discontinuities or jumps of elastic boundary conditions. Mathematically, the unification of various boundary value problems for annular sector and annular plates is physically realized by setting a set of coupling springs to ensure appropriate continuity conditions along the radial edges of concern. Numerous examples are presented for the free vibration analyses of annular sector and annular plates with different boundary restraints. With regard to the forced vibration analysis, annular sector and annular plates with different external excitations are examined. The accuracy, convergence and numerical robustness of the current approach are extensively demonstrated and verified through numerical examples which involve plates with various shapes and boundary conditions.

On an extended Kantorovich method for the mechanical behavior of functionally graded solid/annular sector plates with various boundary conditions

Based on the first-order shear deformation plate theory, two approaches within the extended Kantorovich method (EKM) are presented for a bending analysis of functionally graded annular sector plates with arbitrary boundary conditions subjected to both uniform and non-uniform loadings. In the first approach, EKM is applied to the functional of the problem, while in the second one EKM is applied to the weighted integral form of the governing differential equations of the problem as presented by Kerr. In both approaches, the system of ordinary differential equations with variable coefficients in r direction and the set of ordinary differential equations with constant coefficients in $$\theta $$ direction are solved by the generalized differential quadrature method and the state space method, respectively. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified with the available published works in the literature. It is observed that the first approach is applicable to all supports with an excellent accuracy while the second approach does not give acceptable results for a plate with a free edge. Furthermore, various response quantities in FG annular sector plates with different boundary conditions, material constants, and sector angles are presented which can be used as a benchmark.

Vibration of laminated composite panels and curved plates with different types of FGM composite constituent

Abstract Free vibration analysis of truncated conical panels and annular sector plates with functionally graded materials (FGM) is carried out. Governing equations of motion are obtained based on two different shell theories such as Love's shell theory and first-order shear deformation theory (FSDT). The resulting governing differential equations have been solved using the differential quadrature (DQ) and discrete singular convolution (DSC) methods. As the FGM cases two different material properties of structures are assumed to change continuously in the thickness direction according to the volume fraction power law and the general four-parameter power law distributions in terms of the volume fractions of constituents. Accuracy, convergence and reliability of these two methods have been validated by comparing the obtained results with the existing results available in the open literature. Furthermore, the effects of the grid numbers on results for each method have been also investigated for different boundary conditions and mode numbers for conical panel vibrations. Then, using the DQ and DSC methods, the frequencies values have been calculated for different material and geometric parameters, modes and boundary cases for truncated conical panels with FGM. The effects of material power-law distribution are also discussed. The convergence, advantages and accuracy of the present two methodologies are examined in conjunctions with the vibration problem of truncated conical panels with functionally graded materials (FMG). Some results for annular sector plates and circular cylindrical panels have also been obtained via conical panel equations.

Elasticity solutions for a transversely isotropic functionally graded annular sector plate

This paper presents three-dimensional elasticity solutions for an annular sector plate made of transversely isotropic functionally graded material (FGM) subjected to concentrated forces \(\left( X,Y,0 \right) \) or couples \(\left( M_X,M_Y,M_Z\right) \) applied at one of its radial edges. The elastic coefficients can vary arbitrarily through the plate thickness. The analysis was based on the assumed forms of displacements for bending of an FGM plate (Mian and Spencer in J Mech Phys Solids 4:2283–2295, 1998), in which the four analytical functions were constructed properly. Appropriate boundary conditions and end conditions similar to those in the classic plate theory were employed to determine the unknown constants contained in the analytical functions so as to accomplish the analysis. When the material coefficients are all constant, the obtained analytical solutions can be degenerated into those for a homogeneous transversely isotropic annular sector plate, which have never been reported before. The solutions may be further reduced to those for a homogeneous isotropic annular sector plate, among which the ones for concentrated couples \(\left( M_X ,M_Y,0 \right) \) are also new to the literature.

Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory

In the present paper, a non-classical model for functionally graded annular sector microplates under distributed transverse loading is developed based on the modified couple stress theory and the first-order shear deformation plate theory. The model contains a single material length scale parameter which can capture the size effect. The material properties are graded through the thickness of plates according to a power-law distribution of the volume fraction of the constituents. The equilibrium equations and boundary conditions are simultaneously derived from the principle of minimum total potential energy. The system of equilibrium equations is then solved using the generalized differential quadrature method. The effects of length scale parameter, power-law index and geometrical parameters on the bending response of annular sector plates subjected to distributed transverse loading are investigated.

Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers

The objective of this research is to scrutinize large amplitude vibration response of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular sector plates with surface-bonded piezoelectric layers. A nonlinear formulation is derived based on the first-order shear deformation theory (FSDT), von Karman geometrical nonlinearity along with the Hamilton principle. The distribution of electric potential through the thickness of the piezoelectric layers is simulated by a sinusoidal function. The closed circuit electrical boundary condition is taken into consideration for the top and bottom surfaces of the piezoelectric layers. The nonlinear dynamic equations, boundary conditions and Maxwell equation are discretized using the generalized differential quadrature method and direct iterative method is then employed to solve the nonlinear system of equations. The variation of nonlinear frequency versus the vibration amplitude is highlighted considering various influential parameters such as distribution and volume fraction of the CNTs, geometrical parameters, boundary conditions and the thickness of the piezoelectric layers. It is found that the dynamic responses of the CNTRC sector plate may be noticeably enhanced by adjusting values of the CNT volume fraction and distribution. The numerical results reveal that the increase in the nonlinear frequency descents at certain vibration amplitude owing to vibration mode redistribution.

Free vibration analysis of annular sector plates via conical shell equations

Abstract In this paper, free vibration analysis of annular sector plates has been presented via conical shell equations. By using the first-order shear deformation theory (FSDT) equation of motion of conical shell is obtained. The method of discrete singular convolution (DSC) and method differential quadrature (DQ) are used for solution of the vibration problem of annular plates for some special value of semi-vertex angle via conical shell equation. The obtained numerical results based on the two numerical approaches for annular sector plates compare well with the results available in the literature. The effects of some geometric parameters, grid numbers and types of the grid distribution have been discussed for curved plates.

A modified Fourier solution for vibration analysis of moderately thick laminated annular sector plates with general boundary conditions, internal radial line and circumferential arc supports

Abstract In this paper, a modified Fourier solution based on the first-order shear deformation theory is developed for the free vibration problem of moderately thick composite laminated annular sector plates with general boundary conditions, internal radial line and circumferential arc supports. In this solution approach, regardless of boundary conditions, the displacement and rotation components of the sector plate are written in the form of the trigonometric series expansion in which several auxiliary terms are added to ensure and accelerate the convergence of the series. Each of the unknown coefficients is taken as the generalized coordinate and determined using the Raleigh- Ritz method. The accuracy and reliability of the present solution are validated by the comparison with the results found in the literature, and numerous new results for composite laminated annular sector plates considering various kinds of boundary conditions are presented. Comprehensive studies on the effects of elastic restraint parameters, layout schemes and locations of line/arc supports are also made.New results are obtained for laminated annular sector plates subjected to elastic boundary restraints and arbitrary internal radial line and circumferential arc supports in both directions, and they may serve as benchmark solutions for future researches.

Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate

This paper is motivated by the lack of studies in the technical literature concerning to the influence of carbonnanotubes (CNTs) waviness and aspect ratio on the vibrational behavior of functionally graded nanocomposite annular sector plates resting on two-parameter elastic foundations. The carbon nanotube-reinforced (CNTR) plate has smooth variation of CNT fraction based on the power-law distribution in the thickness direction, and the material properties are also estimated by the extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes\' effective aspect ratio as the filler content increases. Parametric studies are carried out to highlight the influence of CNTs volume fraction, waviness and aspect ratio, boundary conditions and elastic foundation on vibrational behavior of FG-CNT thick sectorial plates. The study is carried out based on three-dimensional theory of elasticity and in contrary to two-dimensional theories, such as classical, the first- and the higher-order shear deformation plate theories, this approach does not neglect transverse normal deformations. The annular sector plate is assumed to be simply supported in the radial edges while any arbitrary boundary conditions are applied to the other two circular edges including simply supported, clamped and free. For an overall comprehension on 3-D vibration of annular sector plates, some mode shape contour plots are reported in this research work.