Shear Deformation Plate Theory Research Articles

Asymptotically accurate and locking-free finite element implementation of first order shear deformation theory for plates

A formulation of the asymptotically exact first-order shear deformation theory for linear-elastic homogeneous plates in the rescaled coordinates and rotation angles is considered. This allows the development of its asymptotically accurate and shear-locking-free finite element implementation. As applications, numerical simulations are performed for circular and rectangular plates, showing complete agreement between the analytical solution and the numerical solutions based on two-dimensional theory and three-dimensional elasticity theory.

Natural frequencies and modal shapes of folded sandwich plates made of porous core and FG-CNTRC coating layers resting on two parameters elastic foundation

Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type.

Bending of Sandwich FGM Plates with a Homogeneous Core Either Hard or Soft Via a Refined Hyperbolic Shear Deformation Plate Theory

Bending of Sandwich FGM Plates with a Homogeneous Core Either Hard or Soft Via a Refined Hyperbolic Shear Deformation Plate Theory

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

Nonlinear aero-thermo-elastic flutter analysis of stiffened graphene platelets reinforced metal foams plates with initial geometric imperfection

Due to its lightweight design and high load-bearing capacity, the stiffened plate structure is extensively applied in the passive control of aircraft flutter. However, the evolution mechanism of the nonlinear response of stiffened plates within the unstable region remains unclear. In particular, there is a lack of research on the nonlinear aero-thermo-elastic properties of stiffened plates with initial geometric imperfection. Inspired by this, this article investigates the stability and post-instability behavior of stiffened geometrical imperfect graphene platelets reinforced metal foams (GPLRMF) plates under combined thermal load and aerodynamic pressure. Graphene platelets (GPLs) and foam are distributed evenly or unevenly in the plates, while the material properties are calculated by modified Halpin-Tsai model and rule of mixture. Considering initial geometric imperfection, the models of plates and stiffeners are established by first-order shear deformation (FOSD) plate and Timoshenko beam theories, respectively. Using the linear piston theory to simulate supersonic aerodynamic pressure. The stability boundary of stiffened plates is determined by eigenvalue analysis. The Newton-Raphson approach is used to simulate the aero-thermo-elastic static response of stiffened plates, and the dynamic post-flutter behavior of the stiffened plate is determined through the Runge-Kutta method. Finally, the effects of initial geometric imperfection, material parameters (foam distribution type, porosity coefficient, GPLs weight fraction and GPLs distribution type), and stiffeners (size, position, and stiffening scheme) on the thermal aeroelastic behavior of stiffened GPLRMF plates are revealed. It can be found that the presence of initial geometric imperfection will significantly change the nonlinear flutter behavior of the stiffened plates.

Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents.

Transient Response of Sandwich Plates with Corrugated Core Under Mechanical-Thermal Loads

The nonlinear dynamic responses of the corrugated sandwich plate under mechanical-thermal loads are studied and analyzed. Based on hyperbolic parabola shear deformation plate theory (HPSDT), the partial differential equation of the sandwich plate with the corrugated core is established. Using the Galerkin truncation method, the nonlinear motion equation is derived. By the solution, the response of the corrugation plate with simply supported four edges at different temperatures is obtained and validated. Then the transient responses of the corrugated sandwich plate under different impact loads are analyzed, and the effects of the base materials, the corrugation types and the structural parameters of the corrugated plate on the transient responses are discussed in detail. The research provides a reference for improving the impact resistance of the sandwich plate in practical application.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

Panel flutter analysis of nano-hybrid laminated composite quadrilateral plates presuming curvilinear fibers

In the present article, the panel flutter behavior of nanocomposite hybrid laminated quadrilateral panel is studied. A three-phase carbon-nanotube (CNT)/polymer/fiber laminated composite is considered where the probable agglomeration of the CNT phase in the nano-matrix mixture is also addressed. Diverse use of fiber phase material in different plies is assumed in the context of hybrid laminated composite. A combination of nanocomposite polymer matrix, tow-steered curvilinear reinforcing fibers within the plies, and hybrid multi-material laminate is considered. The panel flutter behavior of the arbitrary quadrilateral hybrid laminated nanocomposite plate is formulated by a non-uniform rational B-splines-type isogeometric analysis method based on the first-order shear deformation plate theory. The nonideal mixture of the nanocomposite is also concerned about through applying a micromechanics approach. Representative results are obtained and compared with those available in the literature to demonstrate the quality of the proposed formulation. The effects of different parameters including the layup, mixture quality, boundary condition, geometry, and flow direction on the flutter behavior are investigated by performing parametric studies.

Thermal buckling in multi-directional porous plates: The effects of material grading and aspect ratio

In the present study, a trigonometric shear deformation plate theory was employed to perform a thermal buckling analysis of multi-directional functionally graded (FG) plates. During the manufacturing of the multi-directional graded plate, the formation of pores is abounded. Hence, the effect of porosity on the buckling performance was investigated by considering the variation of porosity in the plate for power law gradation variation of material properties. The adverse effect of porosity on the material properties was taken into account by employing the rule of mixture relation. Finite element results show that the thermal expansion coefficient is unaffected by the presence of porosity. For simply supported boundary conditions, the non-linear governing equations are solved for different thermal loads such as uniform, linear, and non-linear. A parametric study was performed in which the effect of grading parameters, aspect ratio, and side-to-thickness ratio under variable temperature change was studies. Critical material grading indices for multi-directional plates have been identified that help researchers and industry personnel in fabrication planning.

Higher-order shear deformation theory for accurate prediction of vibration behavior of thick piezoelectric disks and design of efficient surface electrodes

A modified higher-order shear deformation plate theory (MHSDT) is proposed to predict the resonance frequencies and related mode shapes of piezoelectric disks of various thicknesses. The theory explicitly accounts for the effects of the electrical potential on the displacement of the disk. The equations of motion for the disk are derived using Hamilton's principle in conjunction with a variational approach. It is shown analytically that the distribution of the electrical field on the surface of the disk is quantitatively similar to the distribution of the sum of the total stresses within the disk. Thus, the deformation of the disk can be approximated directly from the stress distribution on the disk, thereby greatly simplifying the solution process. The validity of the proposed MHSDT is confirmed by comparing the solutions for the natural frequencies, displacement fields, and electric field distributions of the disk for various radius-to-thickness ratios and boundary conditions with those obtained using Reddy’s third-order plate theory and COMSOL finite element simulations, respectively. The results show that the solutions obtained from the MHSDT method are consistently closer to the exact solutions than those determined using Reddy’s third-order method. Finally, the linear relationship between the electric field and the flexural displacement of the disk is leveraged to design surface electrodes capable of exciting specific flexural mode shapes of the disk under a free boundary condition. The efficiency of the designed electrodes is experimentally demonstrated using an amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI).

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Transient analysis of functionally graded plates using extreme gradient boosting

This paper is aimed at quickly predicting the dynamic behavior of functionally graded plates using nontraditional computational approaches consisting of artificial neural networks (ANN) and extreme gradient boosting (XGBoost). Through the use of ANN and XGBoost, the dynamic behavior of the plate can be directly predicted based on optimal mapping, which is found by learning the relationship between input and output data from a data set during the training process. A data set including 1000 data pairs (input and output) is generated by using a combination of isogeometric analysis (IGA) and third-order shear deformation plate theory through iterations. In this model, a power index that controls the plate’s material distribution is regarded as input, and output consists of 200 values of deflection versus time. In order to demonstrate the effectiveness of XGBoost in terms of accuracy and computational time, results obtained by the optimal XGBoost model are compared to those obtained by the optimal ANN model and IGA.

Application of Hashin–Shtrikman bounds homogenization model for frequency analysis of imperfect FG bio-composite plates

Despite abundant theoretical investigations on the dynamic behavior of functionally graded (FG) structures, the study on frequency analysis of FG bio-composite structures is limited. FG bio-composite materials due to their biocompatibility potentials and good material properties can be applied in biomedical applications, especially dental implants. In this investigation, a natural frequency response of the FG bio-composite plate is analyzed within the framework of the newly developed refined higher-order shear deformation plate theory. Additionally, the imperfection impact on frequency behavior is evaluated while three imperfection distribution patterns are taken into account. The constitutive materials of FG bio-composite plate are Hydroxyapatite and Titanium. The effective material properties of the structure are determined with the help of the upper Hashin–Shtrikman bounds homogenization model. In continuation, to solve the derived governing equations of imperfect FG bio-composite plate, Galerkin's analytical method is employed. Also, the precision of the used theory is validated, the obtained outcomes are compared and an acceptable matching is found. Later, the sensitivity of different considerable variables is comprehensively assessed and discussed.

Buckling and Post-buckling of Sandwich Plate with Functionally Graded Graphene Origami Reinforced Core Layer in Hygrothermal Environment

The buckling and post-buckling of magneto-electric-elastic (MEE) sandwich plate are presented. The core layer consists of epoxy matrix reinforced with graphene origami auxetic metal-material in which the volume fraction of graphene origami is assumed to vary the thickness direction in three different distributions. For MEE face sheets, the proportions of ferroelectric and ferromagnetic components are chosen equally. The Reddy's higher order shear deformation plate theory forms the basis for establishing the governing equations. The axial compressive loading- deflection amplitude relation and the expression of critical buckling loading are obtained through the application of Galerkin method. The method's accuracy is validated by comparing its results with those published by other authors. The influences of various parameters on the buckling and post-buckling of the sandwich plate are investigated in details. 

Static Buckling and Free Vibration Analysis of Aligned CNTs Reinforced Composite Plates

This work introduces an analysis of the nonlinear buckling and free vibration behavior of polymer plates reinforced with aligned carbon nanotubes using Reddy's third-order shear deformation plate theory and incorporating Theodore von Kármán's geometric nonlinearity. The polymer plates were enhanced with single-walled carbon nanotubes assumed to exhibit either uniform distribution or functionally graded distribution across the thickness. The equations of motion were established through Hamilton’s principle and then solved by the Galerkin method and Airy’s stress function for the composite plates with fully simply supported edges. The investigation focused on assessing the effects of carbon nanotube distribution, volume fraction, and geometrical parameters on the buckling load and fundamental frequency parameters of composite plates through numerical results.

Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

This research addresses analytical solutions for vibration performance of simply supported sandwich plate subjected to coupled mechanical-electric-magnetic-thermal loads. The plate is made of graphene reinforced metal matrix composite (GRMMC) core and magneto-electro-elastic (MEE) face sheets. Three graphene distribution patterns, FG-X, FG-O, and UD are considered for evaluating the reinforcement efficiency of graphene. The Reddy’s higher order shear deformation plate theory and Hamilton’s principle are employed to formulate the equations of motion of the sandwich plate. Then, using double trigonometric functions for variables and applying Bubnov–Galerkin procedure, the system of nonlinear second-order differential equations is constructed to reveal the natural frequency, frequency ratio, and dynamic response of the sandwich plate. The influences of graphene distribution patterns and volume fraction, magnetic and electric potentials, temperature change, and volume fraction of piezoelectric phase are discussed in details. This research provides basis for the design of sandwich composite structures, especially smart structures and devices.

Free Vibration Response of Functionally Graded Porous Metallic Plates Embedded with Piezoelectric Layers

The objective of this study is to determine the natural frequencies of Functionally Graded (FG) metallic plates comprising piezoelectric layers on both the top and bottom surfaces. The material characteristics of the FG plates are expected to exhibit a gradual variation along the thickness direction in accordance with a power-law model along with porosity. The governing equations are derived using the principle of virtual displacements, taking into account the first-order shear deformation plate theory. A commercial finite element programme in ANSYS Parametric Design Language (APDL) is developed to compute the natural frequency and mode shapes of functionally graded porous plates embedded with piezoelectric layers. The obtained natural frequencies results are used for different boundary condition to show their variations with respect to the constituent volume fractions, boundary condition, and piezoelectric thickness for the parameteric study. The present paper highlights some important characteristics of Functionally Graded Materials (FGM) plate embedded with piezoelectric layers that can be advantageous in the design of smart structures.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

Natural Frequency Characteristics of Stiffened FG Multilayer Graphene-Reinforced Composite Plate with Circular Cutout Resting on Elastic Foundation

In this research, natural frequency response of functionally-graded multilayer graphene-reinforced composite plate with circular cutout reinforced by orthogonal stiffeners is investigated for the first time. The structure is surrounded by Winkler-type elastic support. The plate is composed of polymethyl methacrylate (PMMA) as matrix material and reinforced by graphene platelets (GPLs). The material of the orthogonal stiffeners is the same as that for the matrix. Rule of mixtures and Halpin–Tsai approach are applied to estimate the effective material properties of the composite plate. Third-order shear deformation plate theory and finite element procedure is employed to obtain the element matrices of the structure. Natural frequencies and mode shapes of the stiffened plate are reported for different variables such as nanofillers dispersion patterns, width and height of the stiffeners, aspect ratio of plate, plate thickness ratio, weight fraction of nanofillers, number of stiffeners, boundary conditions, elastic foundation stiffness parameter and size of circular cutout. The obtained results denote that with the addition of a set of stiffeners, fundamental frequency enhanced up to 32.3% with just about 10% increase of mass.