Refined Shear Deformation Theory Research Articles

A simple refined plate theory for buckling problems of in-plane bi-directional functionally graded plates with porosity under various boundary conditions

In this article, a simple refined shear deformation theory that eliminates the use of a shear correction factor was presented for buckling analysis of in-plane bi-directional functionally graded (IBFG) porous plates under various boundary conditions. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. Material properties of IBFG are assumed to vary continuously along with two different directions simultaneously, that is, the longitudinal and transversal ones, respectively. Governing equations and boundary conditions are derived. Analytical solutions were obtained for buckling analysis of (IBFG) porous plates. Several numerical examples are presented to demonstrate the performance and effectiveness of the proposed theory. The effects of material gradations, aspect ratios, and porosity on IBFG plate responses are examined in detail as well.

Natural frequency analysis of bidirectional functionally graded plates based on a refined shear theory using nonconforming finite elements

In this study, we investigate the free vibrations of functionally graded plates using the refined shear deformation theory. These plates have material properties that vary in two directions: the z-axis and the x-axis. The variation follows a bidirectional gradation described by a power law based on the Voigt rule of mixture. The displacement field is characterized by four variables, chosen in such a way that shear strain and stress vanish, and traction is free at the upper and lower surfaces of the plate. This eliminates the need for a shear correction factor, as required in the first-order shear deformation theory. We derive the equations of motion and related boundary conditions using Hamilton’s principle and a finite element approach. The proposed element ensures C1-continuity by including the first derivative of the transverse deflection in the kinematic field. For spatial discretization, we employ a rectangular element with four nodes and two interpolation schemes. Lagrangian shape functions are used for in-plane displacement, while a Hermitian scheme is employed for transverse deflection. This enables us to determine the natural frequencies of the 2D-FG plates. We demonstrate the validity, convergence, and accuracy of the proposed element through illustrative examples and compare it with previous literature. Furthermore, we conduct numerical analyses to examine the effects of transverse shear function, boundary conditions, plate aspect ratio, slenderness ratio, and gradation indices on the natural frequencies. The developed model provides insights into the significance of natural frequencies in the design of various structural elements, including 2D-FG plates.

A refined shear deformation theory of an asymmetric sandwich beam with porous core: Linear bending problem

This paper is devoted to a sandwich beam with an asymmetric structure under uniformly distributed load. The first end of this beam is simply supported, while second end is clamped. The individual nonlinear deformation function of a plane beam cross section – the shear effect with consideration of the classical shear stress formula is analytically developed. Displacements, strains and stresses taking into account the individual nonlinear deformation function for successive layers of this beam are elaborated in detail. The system of two differential equation of equilibrium of this beam, based on the principle of stationary potential energy, is obtained. This system of equations is precisely solved analytically, according to the theory of differential equations. The individual nonlinear deformation functions, shear stresses and deflections of two cases of sample beams are determined in detail: the first with variable core porosity in the direction of the beam depth, the second with constant core porosity – standard sandwich structure. The calculations results specified in Tables and graphically presented in Figures.

Thermal Mechanical Bending Response of Symmetrical Functionally Graded Material Plates

This paper investigates the thermal mechanical bending response of symmetric functionally graded material (FGM) plates. This article proposes a thermodynamic analysis model of both the FGM plate and FGM sandwich plate, and the model only involves four control equations and four unknown variables. The control equation is based on the refined shear deformation theory and the principle of minimum potential energy. The Navier method is used to solve the control equation. According to the method, numerical examples are provided for the thermo-mechanical bending of the symmetric FGM plate and FGM sandwich plate under a simply supported boundary condition, and the accuracy of the model is verified. Finally, parameter analysis is conducted to investigate the effects of the volume fraction index, side-to-thickness ratio, thermal load, and changes in core thickness on the thermal mechanical bending behavior of the symmetric FGM plate and FGM sandwich plate in detail. It was found that the deflection of the FGM plate is greater than that of the FGM sandwich plate, while the normal stress of the FGM plate is smaller than that of the FGM sandwich plate. Moreover, the FGM plate and FGM sandwich plate are sensitive to nonlinear temperature changes.

Buckling and vibration characteristic of anisotropic sandwich plates with negative Poisson’s ratio based on isogeometric analysis

This article performs the buckling and vibration studies of the sandwich plates with a negative Poisson’s ratio honeycomb core. Based on the isogeometric analysis in conjunction with the refined shear deformation theory, the discrete equilibrium equations of the sandwich plate are established. Numerical results show that the plate with negative Poisson’s ratios of the honeycomb core layer has a higher load-bearing capacity than that with positive Poisson’s ratios. In addition, the optimal inclined angles of honeycomb core for negative Poisson’s ratios subjected to different loading types are obtained. The normalized natural frequency of the plate almost decreases by 10% when the Poisson’s ratio ν 12 of honeycomb core decreases from −0.3 to −5.5.

Bending, buckling and free vibration analyses of shallow-to-deep FG curved sandwich beams using a global–local refined shear deformation theory

This paper investigates the bending, vibration, and buckling behaviour of functionally graded (FG) curved sandwich beams using a global–local refined shear deformation theory. Material properties of these beams are varied through the thickness according to the power-law distribution. A beam element with a combination of Hermite cubic and quadratic Lagrange shape functions is developed to solve the problems. The obtained results are compared with the results from finite element software (ABAQUS), and other higher-order beam theories. It can be observed that the present model can predict accurately static, vibration, and buckling responses of both shallow and deep FG sandwich beams with arbitrary boundary conditions.

Three-Dimensional Vibrational Characteristics of Functionally Graded Rotating Microbeams Based on the Higher-Order Refined Shear Deformation Theory

This study establishes a novel three-dimensional dynamic model for functionally graded rotating microbeams simultaneously considering flapwise and chordwise shear effects based on a refined shear deformation theory and the modified couple stress theory. Five kinematic variables are introduced to describe the rotating microbeam’s axial, flapwise, and chordwise motions using a higher-order refined shear deformation theory. The related strain energy and kinetic energy functionals are discretized by the assumed mode method. Applying the Euler–Lagrange variational formulation, the governing equations for the rotating microbeam are obtained. Selective examples are provided to demonstrate the convergence and effectiveness of the proposed model. Finally, the impacts of slenderness ratio, rotation speed, material index, material length scale parameter (MLSP), and Poisson’s effect on the vibrational characteristics of the microbeam are examined. Numerical results indicate that the mode shape transition and frequency loci veering phenomena could easily occur under slight changes in the above factors when dense modes appear in the rotating microbeam; moreover, there is a positive/negative competitive relationship among the effects of rotation speed, MLSP, and shear deformation; Poisson’s effect seriously affects the coupling between axial and non-axial motions of the rotating microbeam. The results obtained are expected to provide data accumulation and methodological basis for the failure analysis and vibration shape determination of micro air vehicles.

Accurate mechanical buckling analysis of couple stress-based skew thick microplates

We investigate, for the first time, the elastic buckling of skew thick microplates under combined in-plane shear and compressive loading within the framework of modified couple stress theory. The displacement field of the microplates is described by a two-variable refined shear deformation theory, wherein the transverse deflection is partitioned into bending and shear components, resulting in a simple and universal elastic buckling model. The Euler-Lagrange equation is used to obtain the equations governing the motion of the skew thick microplates. Obtaining an analytical buckling solution for general boundary supported skew microplates is challenging; therefore, a C1-type four-node 32-DOF differential quadrature finite element is developed. The element stiffness and geometric stiffness matrices are derived according to the minimum potential energy principle. Significant parametric studies are conducted with different geometrical dimensions, boundary edges, in-plane loadings, and material length scale parameters. The buckling characteristics of skew microplates are investigated using a combination of unequal biaxial compression loads and in-plane compression and shear loads. Numerical results imply that the buckling modes are influenced by the combination of size effects and skew angle and not by the buckling loads alone.

Programmable deformation of liquid crystal elastomer plates subjected to concentrated light illumination

In this paper, the programmable deformation of liquid crystal elastomer (LCE) plates with arbitrary director orientation subjected to concentrated light illumination is investigated. An improved method is provided by extending the isogeometric analysis (IGA) with consideration of the refined shear deformation theory (RSDT) to take account of the shear strains induced by light. The present numerical method is validated by the finite element method (FEM) and available results. With the director orientation, the position and the intensity of the concentrated light illumination acting as decoders, the deformations of the LCE plates can deliver various kinds of letters such as H, I, T, X and Y. This fact indicates that we could control the deformation of LCE plates to show some information by programming code.

A unified formulation of various shell theories for the analysis of laminated composite spherical shells

This study investigates the static and free vibration responses of orthotropic laminated composite spherical shells using various refined shear deformation theories. Displacement-based refined shear deformation theories are presented herein for the analysis of laminated composite spherical shells via unified mathematical formulations. Equations of motion associated with the present theory are derived within the framework of Hamilton's principle. Analytical solutions for the static and free vibration problems of laminated spherical shells are obtained using Navier's technique for the simply supported boundary conditions. Few higher order and classical theories are recovered from the present unified formulation; however, many other theories can be recovered from the present unified formulation. The numerical results are obtained for symmetric as well as anti-symmetric laminated shells. The present results are compared with previously published results and 3-D elasticity solution. From the numerical results, it is concluded that the present theories are in good agreement with other higher order theories and 3-D solutions.

Size-Dependent Elastic Buckling of Two-Variable Refined Microplates Embedded in Elastic Medium

This study applies a two-variable refined shear deformation theory (TVRSDT) and a modified couple stress theory (MCST) to develop a size-dependent elastic buckling model for microplates under combined axial compression and in-plane shear and resting on Winkler–Pasternak foundation. Our model incorporates two transverse displacement variables and one material length scale parameter (MLSP); it satisfies the zero-traction boundary conditions on the top and bottom surfaces of microplates, thereby circumventing the use of a shear correction factor. The equations of motion are determined via the Euler–Lagrange equation. A closed-form buckling solution is presented for microplates with four edges simply supported. To handle the nonsimply supported microplates, a thirty-two-degree-of-freedom (32-DOF) four-node differential quadrature (DQ) finite element with [Formula: see text]-continuity is proposed. The corresponding element matrices are obtained using the minimum potential energy principle. Comparison studies are conducted to exhibit the validity of the derived formulations. Finally, the present buckling model is employed for predicting the elastic buckling behaviors of microplates embedded in an elastic medium. The effects of in-plane loading patterns, boundary conditions, aspect ratio, length-to-thickness ratio, MLSP and elastic medium parameters on the buckling load and buckling mode are elucidated.

Analysis of Electromagnetic Effects on Vibration of Functionally Graded GPLs Reinforced Piezoelectromagnetic Plates on an Elastic Substrate

A new nanocomposite piezoelectromagnetic plate model is developed for studying free vibration based on a refined shear deformation theory (RDPT). The present model is composed of piezoelectromagnetic material reinforced with functionally graded graphene platelets (FG-GPLs). The nanocomposite panel rests on Winkler–Pasternak foundation and is subjected to external electric and magnetic potentials. It is assumed that the electric and magnetic properties of the GPLs are proportional to those of the electromagnetic materials. The effective material properties of the plate are estimated based on the modified Halpin–Tsai model. A refined graded rule is introduced to govern the variation in the volume fraction of graphene through the thickness of the plate. The basic partial differential equations are provided based on Hamilton’s principle and then solved analytically to obtain the eigenfrequency for different boundary conditions. To check the accuracy of the present formulations, the depicted results are compared with the published ones. Moreover, impacts of the variation in elastic foundation stiffness, plate geometry, electric potential, magnetic potential, boundary conditions and GPLs weight fraction on the vibration of the smart plate are detailed and discussed.

Assessment of four-variable refined shear deformation theory for low-velocity impact analysis of curved sandwich beams

In this paper, the accuracy of the four-variable refined global-local (FRGL) shear deformation theory in the prediction of dynamic responses of curved sandwich beams under low-velocity impact has been investigated. The governing equations of motion of the curved sandwich beams are derived by employing a finite element (FE) model based on FRGL shear deformation theory. By using the method of truncated superimposition of modes, the size of the total dynamic system is firstly reduced. Then, the resulting dynamic system is solved via the state-space (SS) approach. For validation, curved sandwich beams with various deepness ratios and different boundary conditions are analyzed using the proposed model. Different materials and lay-up configurations were assumed for the face-sheets. The obtained results are validated through comparison with the results of ABAQUS simulations and other analytical and numerical results reported in the open literature. The comparisons show that the FRGL shear deformation theory in conjunction with the truncated reduced modal SS approach is a precise and computationally low-cost model for solving the dynamic problems of curved sandwich beams under impact loads.

Analytical Study of Bending Characteristics of an Elastic Rectangular Plate using Direct Variational Energy Approach with Trigonometric Function

In this paper, an analytical three-dimensional (3D) bending characteristic of an isotropic rectangular thick plate with all edges simply supported (SSSS) and carrying uniformly distributed transverse load using the energy technique is presented. The three-dimensional constitutive relations which involves six stress components were used in the established, refined shear deformation theory to obtain a total potential energy functional. This theory obviates application of the shear correction factors for the solution to the problem. The governing equation of a thick plate was obtained by minimizing the total potential energy functional with respect to the out of plane displacement. The deflection functions which are in form of trigonometric were obtained as the solution of the governing equation. These deflection functions which are the product of the coefficient of deflection and shape function of the plate were substituted back into the energy functional, thereafter a realistic formula for calculating the deflection and stresses were obtained through minimizations with respect to the rotations and deflection coefficients. The values of the deflections and stresses obtained herein were tabulated and compared with those of previous 3D plate theory, refined plate theories and, classical plate theory (CPT) accordingly. It was observed that the result obtained herein varied more with those of CPT and RPT by 25.39% and 21.09% for all span-to-thickness ratios respectively. Meanwhile, the recorded percentage differences are as close as 7.17% for all span-to-thickness ratios, when compared with three dimensional plate analysis. This showed that exact 3D plate theory is more reliable than the shear deformation theory which are quite coarse for thick plate analysis. Doi: 10.28991/esj-2021-01320 Full Text: PDF

On the deformation of laminated composite and sandwich curved beams

Abstract Plenty of research articles are available on the static deformation analysis of laminated straight beams using refined shear deformation theories. However, research on the deformation of laminated curved beams with simply supported boundary conditions is limited and needs more attention nowadays. With this objective, the present study deals with the static analysis of laminated composite and sandwich beams curved in elevation using a new quasi-3D polynomial type beam theory. The theory considers the effects of both transverse shear and normal strains, i.e. thickness stretching effects. In the present theory, axial displacement has expanded up to the fifth-order polynomial in terms of thickness coordinates to effectively account for the effects of curvature and deformations. The present theory satisfies the zero traction boundary condition on the top and bottom surfaces of the beam. Governing differential equations and associated boundary conditions are established by using the Principal of virtual work. Navier’s solution technique is used to obtain displacements and stresses for simply supported beams curved in elevation and subjected to uniformly distributed load. The present results can be benefited to the upcoming researchers.

Hygrothermal environment effect on the critical buckling load of FGP microbeams with initial curvature integrated by CNT-reinforced skins considering the influence of thickness stretching

Abstract Due to the need for structures with refined properties to bear against different loading conditions, recently, carbon nanotubes (CNTs) have been used widely to reinforce them. The extremely high stiffness of CNTs makes them significant as one of the best reinforcements to improve the mechanical behaviors of structures. This work focuses on microbeam buckling response with an initial curvature that includes three layers. The mid-layer that is known as the core is constituted of functionally graded porous (FGP) materials and two CNT-reinforced composite skins are bonded to the core to integrate it. The whole structure is affected by the hygrothermal environment and springs and shear layers are put below it. For the first time, for such a structure, a refined shear deformation theory (RSDT) as a higher-order theory that considers thickness stretching effect in polar coordinates is used that presents more accurate results, especially for deeply curved beams. Modified couple stress theory (MCST) in combination with the virtual displacement principle is utilized to establish the governing equations. The obtained results demonstrate the significance of porosity percentage and CNTs’ addition to the skins on the critical nanotubes buckling load. Also, the different behaviors of the microstructure at various temperatures are analyzed and discussed in detail.

Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory

In this article, the well-posedness of several common nonlocal models for higher-order refined shear deformation beams is studied. Unlike the case of classic beams models, both strain-driven and stress-driven purely nonlocal theories lead to an ill-posed issue (i.e., there are excessive mandatory boundary conditions) when considering higher-order shear deformation assumption. As an effective remedy, the well-posedness of strain-driven and stress-driven two-phase nonlocal (StrainDTPN and StressDTPN) models is pertinently evidenced by studying the free vibration problem of nanobeams. The governing equations of motion and standard boundary conditions are derived from Hamilton’s principle. The integral constitutive relation is transformed equivalently to a differential form equipped with two constitutive boundary conditions. Using the generalized differential quadrature method (GDQM), the governing equations in terms of displacements are solved numerically. Numerical results show that both the StrainDTPN and StressDTPN models can predict consistent size-effects of beams with different boundary conditions.

Isogeometric analysis of bending, vibration, and buckling behaviors of multilayered microplates based on the non-classical refined shear deformation theory

Isogeometric analysis of bending, vibration, and buckling behaviors of multilayered microplates based on the non-classical refined shear deformation theory

Functionally graded thick sandwich beams with porous core: Buckling analysis via differential transform method

The purpose of this study is to investigate the buckling characteristics of a sandwich beam consisting of a porous ceramic core (Alumina), two bottom and upper layers which are gradually changed from ceramic (Alumina) to metal (Aluminum) through the thickness direction. A core with three kinds of porosity patterns is considered. Shear deformation effects are taken into account based on third-order shear deformation theory (TSDT) and a two-variable refined shear deformation theory. Based on the physical neutral axis of the beam, the governing equilibrium equations for buckling are obtained using the principle of virtual work and are solved for different end conditions using the differential transform method (DTM). A parametric study on buckling behavior is conducted to focus on the influences of length-to-thickness ratio, volume fraction of FGM in two FG layers, and three porosity patterns, namely, uniform, asymmetric and symmetric distribution of porosity in a ceramic core for clamped-clamped, pinned-pinned and cantilever beams. The convergence and comparison studies are carried out. It is demonstrated that the critical buckling load for an FG three-layer-beam, with an asymmetric pattern in a porous ceramic core, is higher than the two other ones. In addition, as demonstrated, DTM is an efficient and reliable method to be employed in this area of solid mechanics.

Bending response of porous advanced composite plates under thermomechanical loads

The new composite functionally classified materials (FGM) used in plates contain probably a porosity volume fraction which needs taking into account this aspect of imperfection in the thermo-mechanical behavior of such structures. A refined shear deformation theory is developed to investigate the thermo-mechanical bending response of porous advanced composite plates such as functionally graded plates by using Navier solution. This work presents a model that employed a new transverse shear function by decomposing the transverse displacements into two parts, the bending part and shear part, the simplified refined shear deformation theory has less unknowns than the first-order shear deformation theory, the higher-order shear deformation theory, the sinusoidal shear deformation theory and the quasi-3D theory, so its computational expenses are reduced. In addition, the new refined theory needs no shear correction factor. The comparisons of deflection, axial stresses, and transverse shear stresses of the porous FGM plates obtained by the proposed theory, these results are compared with others in the literature.