Higher Order Shear Deformation Model Research Articles

Higher-order shear deformation model for free vibration analysis of imperfect functionally graded plates

Higher-order shear deformation model for free vibration analysis of imperfect functionally graded plates

Static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix

A higher-order shear deformable model is extended in this paper for static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix. Principle of virtual work is used to derive governing equations of static bending through the computation of strain energy and external work. A higher-order shear deformable model is used to develop the kinematic relations. The sandwich cylindrical panel is subjected to mechanical and thermal loads. The governing equations are solved using Eigenvalue–Eigenvector method for clamped–clamped boundary conditions. The eigenvalues of the characteristic equation are obtained, and the solution is obtained for real and complex roots. The boundary conditions are applied to obtain integration constants. The axial and radial variation of the displacement, strain, and stress components are presented in terms of significant input parameters such as volume fraction, and folding angle of the graphene origami, various distribution patterns, porosity coefficient, thermal and mechanical loads, and geometric parameters. The results indicate that the stress, strain, and deformation components are increased with an increase in porosity coefficient and folding angle of the graphene. Furthermore, an enhancement in the volume fraction of the graphene leads to a significant decrease in various stress, strain, and deformation components. The results of this work can be used in aerospace science and technology specially in airplanes.

Buckling of constant/variable stiffness curved arbitrary layup composite beams by renewed constitutive relations-based sine finite element with improved in-plane and transverse responses

The elastic stability performance of constant- and variable-stiffness based composite curved beams with general layups is evaluated using an enriched finite element that accounts for three-dimensional structural response features. The variable stiffness is spatially created over the beam by layers with curvilinear fibers. The structural model in the present work is represented by sinusoidal variation based higher-order shear deformable model, augmented with improved in-plane response by zig-zag function and stretching in thickness direction by higher polynomial order. The appropriate constitutive relationships for laminated composite curved beam with arbitrary layup producing stiffness coupling effects are deduced from three-dimensional elasticity relations. The beam governing equations are formed applying the total potential energy minimum principle and the buckling parameters are predicted employing the eigenvalue analysis. The curved laminated beam element efficacy is examined for the known analytical solutions in the literature. Extensive numerical analysis by opting for different structural parameters like curved beam angle, thickness, ply-angle, curvilinear fiber angle variation from center-to-edge, layup, and beam end condition are made to visualize the buckling characteristics of general layup composite curved beams.

Critical electro-magneto-mechanical loads of a higher-order shear deformable cylindrical nano panel

This work studies electro-magneto-mechanical critical loads of a shear deformable cylindrical panel subjected to a combination of electric, magnetic, and mechanical loads. A higher-order shear deformation model is used to derive kinematic relations. The critical load equations are extracted using the virtual work’s principle. The buckling loads are obtained with calculation of external works due to electro-magneto-mechanical loads. Zero out of plane shear strains at the top and the bottom surfaces are satisfied using application of third-order shear deformation theory with some modifications. The analytical solution is presented to investigate impact of influencing parameters such as nano-scale length, length to radius ratio, span angle, and Pasternak’s foundation parameters on the electro-magneto-mechanical buckling loads. A large parametric analysis is presented to discuss the impact of influencing parameters such as nano-scale parameter, span angle, Pasternak’s foundation parameters, and length to radius ratio on the critical loads.

On frequency response of FG-CNT reinforced composite pipes in thermally pre/post buckled configurations

This paper presents a theoretical analysis for the frequency response of thermally pre/post buckled composite pipes reinforced with carbon nanotubes (CNTs). The vibrational behavior of the uniformly heated nanocomposite pipe surrounded by a nonlinear elastic foundation is analytically studied. The nanocomposite media is introduced using a refined rule of mixtures that is calibrated with the available data on the basis of molecular dynamics simulations. Distribution of CNTs across the radius of composite pipes can be uniform or functionally graded (FG) and properties are considered to be temperature-dependent. A higher-order shear deformation model and the von Kármán assumption are employed to extract the nonlinear equations of motion. The system of partial differential equations is established by means of Hamilton’s principle. A two-step perturbation method is utilized to carry out closed-form solutions for these nonlinear equations. The natural frequencies are obtained for the immovable simply-supported boundary conditions. A comparison study is provided for the case of isotropic homogeneous pipes in which good agreement is obtained. The novel numerical results are presented to show the influences of geometric characteristics, CNT distribution pattern, foundation stiffness, and the CNT volume fraction upon the frequency response of the nanocomposite pipe subjected to uniform temperature rise loading.

A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates

A novel unified framework of higher-order shear deformation theories is developed for modeling and analysis of laminated and functionally graded plates. The major objective of this study is to (1) unify all higher-order shear deformation plate theories in a unique formulation; (2) propose a new higher-order shear deformation model based on the unified plate model. In this study, the existing displacement fields in the equivalent single layer theories are classified into three categories based on the present unified method. Some of the existing equivalent single-layer plate theories are unified according to different types of transverse displacements and transverse shear strains based on the present method and several new plate theories are proposed and assessed. The Reddy shear strain function and a changeable shear function proposed by Li et al. are applied to represent the distribution of transverse shear strains through the thickness of composite laminated and FGM plates, respectively. One of the new higher-order shear deformation theory is then ascertained by comparing the obtained bending, free vibration and buckling results of composite laminated and FGM plates with other published ones. The results show that the unified plate model can theoretically cover all existing equivalent single-layer models and is able to encourage interested readers to search for more feasible and accurate plate theories.

Time-Dependent Deflection Responses of Porous FGM Structure Including Pattern and Porosity

The transient deflections of the functionally graded structure considering various types of patterns (power-law, sigmoid and exponential) are computed in this paper numerically using a higher-order shear deformation model. Also, the model includes variable distribution of porosity, i.e., the even and the uneven types, through the thickness direction ([Formula: see text]-axis) of the graded panel. The transient deflection data are obtained computationally via a customized computer code prepared in MATLAB in association with Newmark’s constant acceleration-type time-integration technique. The model accuracy is checked by comparing the present time-dependent data with the published transient deflection values and the simulated results (modeled through a commercial package, ANSYS). Further, the effects of several design parameters (aspect ratio, thickness ratio, power exponent, porosity index, type of porosity, geometry and end-support conditions) on the transient deflection responses of the graded structure are computed through the derived numerical model.

Comparing Different Theories for Dynamic Behavior of a Functionally Graded Beam

Theoretical formulation, Navier’s solutions of rectangular plates based on a new higher order shear deformation model are presented for the static and dynamic analysis of functionally graded plates (FGPs). This theory enforces traction free boundary conditions at plate surfaces. Shear correction factors are not required because a correct representation of transverse shearing strain is given. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concern flexural behavior of FG plates with Metal–Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. Results are verified with available results in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static and dynamic behavior of functionally graded plates. This paper presents a theoretical investigation in free vibration of simply supported FG beam. Young’s modulus of beam varies in the thickness direction according to power law. Governing equations were found by applying Hamilton’s principle. Navier type solution method was used to obtain frequencies. Different higher order shear deformation theories and classical beam theories were used in the analysis. A free vibration frequency is given for different material properties.

Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model

In this paper, a new higher order shear deformation model is developed for static analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. The model account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG beam. Based on the present higher-order shear deformation model, the equations of motion are derived by the principle of virtual works. Navier type solution method was used to obtain displacement and stresses, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, and geometry on the bending of imperfect FG beams. It can be concluded that the proposed model is simple and precise for the resolution of the behavior of flexural FGM beams while taking into account the shape of distribution of the porosity.

An analytical solution for bending and free vibration responses of functionally graded beams with porosities Effect of the micromechanical models

In this paper, a new higher order shear deformation model is developed for static and free vibration analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. Different patterns of porosity distributions (including even and uneven distribution patterns, and the logarithmic-uneven pattern) are considered. In addition, the effect of different micromechanical models on the bending and free vibration response of these beams is studied. Various micromechanical models are used to evaluate the mechanical characteristics of the FG beams whose properties vary continuously across the thickness according to a simple power law. Based on the present higher-order shear deformation model, the equations of motion are derived from Hamilton\'s principle. Navier type solution method was used to obtain displacement, stresses and frequencies, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, micromechanical models, mode numbers, and geometry on the bending and natural frequencies of imperfect FG beams.

On the Nonlinear Vibrations of Polymer Nanocomposite Rectangular Plates Reinforced by Graphene Nanoplatelets: A Unified Higher-Order Shear Deformable Model

This paper presents a unified higher-order shear deformable plate model to numerically examine the nonlinear vibration behavior of thick and moderately thick polymer nanocomposite rectangular plates reinforced by graphene platelets (GPLs). Four distribution patterns of graphene nanoplatelet nanofillers across the plate thickness are considered. The effective material properties of graphene platelet-reinforced polymer (GPL-RP) nanocomposite plate are approximately calculated by employing the modified Halpin–Tsai model and rule of mixture. Using a generalized displacement field, a unified mathematical formulation is derived based on Hamilton’s principle in conjunction with von Karman geometrical nonlinearity. By selecting appropriate shape functions, the proposed unified nonlinear plate model can be reduced to that on the basis of Mindlin, Reddy, parabolic, trigonometric and exponential shear deformation plate theories. The investigation of nonlinear vibration behavior is performed by employing a multistep numerical solution approach. In this regard, the discretization process is done through the generalized differential quadrature method. Then, the discretized governing equations are solved by employing the numerical-based Galerkin technique, periodic time differential operators and pseudo-arc length continuation algorithm. A detailed parametric study is carried out to examine the effect of GPL distribution pattern, weight fraction, geometry of GPL nanofillers and boundary constraints on the nonlinear vibration characteristics of the GPL-RP nanocomposite rectangular plates.

A higher order shear deformation model of a periodically sectioned plate

This talk develops a higher order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton’s principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the...

A Higher-Order Shear Deformation Model of a Periodically Sectioned Plate

This paper develops a higher-order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton's principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the system. This new model is validated with finite-element theory and Kirchhoff plate theory for a thin plate simulation and verified with comparison to experimental results for a 0.0191 m thick sectional plate.

A new higher order shear deformation model for functionally graded beams

In this paper, a new higher order shear deformation model is developed for static and free vibration analysis of functionally graded beams. The theory account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Based on the present higher-order shear deformation beam theory, the equations of motion are derived from Hamilton's principle. Navier type solution method was used to obtain frequencies. A static and free vibration frequency is given for different material properties. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions.

Numerical free vibration analysis of higher-order shear deformable beams resting on two-parameter elastic foundation

Free vibration analysis of higher-order shear deformation beam resting on one- and two-parameter elasticfoundation is studied using differential transform method (DTM) as a part of a calculation procedure. First,the governing differential equations of beam are derived in a general form considering the shear-freeboundary conditions (zero shear stress conditions at the top and bottom of a beam). Using DTM the derivedequations governing beams, followed by higher-order shear deformation model, Timoshenko model andBernoulli-Euler model are transformed to algebraic forms and a set of recurrence formulae is then derived.Upon imposing the boundary conditions of the beam at hand, a set of algebraic equations are obtained andexpressed in matrix form. Finally, the transverse natural frequencies of the beam are calculated through aniterative procedure. Several numerical examples have been carried out to demonstrate the competency ofthe present method and the results obtained by DTM are in good agreement with those in the literature.Afterward, the free vibration of beams followed up by different models (i.e. Bernoulli-Euler, Timoshenkoand different higher-order models) are taken into consideration.

A New Higher Order Shear Deformation Model of Functionally Graded Beams Based on Neutral Surface Position

This paper uses a new refined beam theory based on neutral surface position for the static and free vibration analysis of functionally graded material beams. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The neutral surface position for a functionally graded beam with its material properties varying in the thickness direction is determined. Based on the present refined shear deformation beam theory and the neutral surface concept, the equations of motion are derived from Hamilton’s principle. A static and free vibration frequency is given for different material properties. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions.

A Higher Order Shear Deformation Model for Bending Analysis of Functionally Graded Plates

In this paper, the static response of simply supported functionally graded plates subjected to a transverse uniform load and resting on an elastic foundation is examined by using a new higher order displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the new higher order shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.

A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates

AbstractIn this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.

Analytical solution for bending analysis of functionally graded plates

A theoretical formulation, of Navier solutions of rectangular plates based on a new higher order shear deformation model is presented for the static response of functionally graded plates (FGPs). This theory enforces traction free boundary conditions at plate surfaces. Shear correction factors are not required because a correct representation of transverse shearing strain is given. Unlike any other theory, the number of unknown functions involved is only four, as against five in cases of other shear deformation theories. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concern the flexural behavior of FG plates with metal–ceramic composition. Parametric studies are performed for varying ceramic volume fractions, volume fraction profiles, aspect ratios and length to thickness ratios. Results are verified with available results in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static bending behavior of functionally graded plates.