Simple Higher-order Shear Deformation Research Articles

Examination of the high-frequency behavior of functionally graded porous nanobeams using nonlocal simple higher-order shear deformation theory

Examination of the high-frequency behavior of functionally graded porous nanobeams using nonlocal simple higher-order shear deformation theory

An efficient model for wave propagation of temperature-dependent E-FGM plates resting on viscoelastic foundation

The present work shows the plate's wave propagation behavior made of exponent-law-based functionally graded materials (E-FGM) supported by viscoelastic foundations. In extreme thermal environments, where the E-FGMs are considered highly effective, the temperature influences the composite plate's properties. Therefore, this present study aims to investigate the wave propagation of the E-FGM plates by considering the temperature-dependent effective properties as functions of the position across the thickness and the thermal increase. The governing equations of the E-FGM plate are obtained by employing a simple higher-order shear deformation theory (HSDT) and utilizing the Hamilton principle. An eigenvalue problem is formulated to determine the principal wave propagation frequency. The effects of viscoelastic parameters on E-FGM plates' phase velocities under uniform thermal changes are investigated in detail. The results indicate that introducing metal-ceramic mixtures in the plate would lower the phase velocity in the plate. Furthermore, increasing a uniform temperature in the plate would deteriorate the foundation and plates and, therefore, result in lower velocity and less stiffness.

An exponential-trigonometric higher order shear deformation theory (HSDT) for bending, free vibration, and buckling analysis of functionally graded materials (FGMs) plates

Two assumptions have been made based on by this proposed theory, which come from recently developed exponential–trigonometric shape function for transverse shear deformation effect and a simple higher order shear deformation theory for plate, based on a constraint between two rotational displacements of axis parallel to the plate midplane, about the axes x, y Cartesian coordinates system, which caused fewer unknown number. For the application of this method, a displacement field extended as only bending membrane for transverse displacement is used, a governing equations of motion as a result are determined according to Hamilton’s principle, and simplified using Navier analytical solutions, as well as the transverse shear stresses effect that satisfied the stress-free boundary conditions on the simply supported plate free faces as a parabolic variation along the thickness are taken into account. A functionally graded materials plates are chosen for the parametric study, where the plates are functionally graded continuously in materials through the plate thickness as a function of power law or exponential form. The aim of this study is to analyze the bending, free vibration as well as the buckling mechanical behaviors, where the results are more focused on the investigation of different parameters such as the volume fraction index, geometric ratios, frequency modes, in-plane compressive load parameters and material properties effects on the deflection, stresses, natural frequencies, and critical buckling load, which are validated in terms of accuracy and efficiency with other plate theories results found in the literature.

Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory

Free vibration and mechanical buckling of porous functionally graded (FG) sandwich plates is studied in this paper by using a new and simple higher-order shear deformation theory (SHSDT). The present theory accounts for the distribution of transvers shear stresses that satisfy the free transverse shear stress conditions on the upper and lower surfaces of the sandwich plate. The FG sandwich plate is composed of three layers, an isotropic core and two FG face layers. Four porosities distribution and new model of FG sandwich plates based on a sigmoid function are presented. The boundary conditions for the FG sandwich plate are assumed to be simply-supported. Navier’s solution is used to obtain the closed-form solutions of FG sandwich plates. The effect of porosity, sandwich plate geometry and inhomogeneity parameter on the buckling and free vibration of FG sandwich plate is investigated.

A new refined simple TSDT-based effective meshfree method for analysis of through-thickness FG plates

In this paper a novel numerical method based on the Moving Kriging (MK) interpolation meshfree method, integrated with a simple higher-order shear deformation plate theory for analysis of static bending, free vibration and buckling of functionally graded (FG) plates is presented. In the proposed technique, the shape functions are built by the Kriging technique which possesses the property of Kronecker delta function which makes it easy to enforce essential boundary conditions. The present formulation is based on a refined simple third-order shear deformation theory (R-STSDT), which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the plate present in the original simple third-order shear deformation theory (STSDT). In this theory, instead of assuming a specific distribution for the displacement field, the theory of elasticity is used for obtaining the kinematics of the plate deformation. We first propose the formulation, and then several numerical examples are provided to show the merits of the proposed approach.

Creep and recovery of viscoelastic laminated composite plates

In this study, the dynamic relaxation method is employed for higher-order bending analysis of isotropic and viscoelastic laminated composite plates. Using Prony series, a linear viscoelastic model, is considered for material behavior. The displacement field is expressed according to two-variable refined plate theory. This simple higher-order shear deformation theory, which involves only two unknown functions, predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies the zero traction condition on the plate surfaces. Virtual work principle is employed to derive the governing equations. Time integrals are approximated by trapezoidal rules and the dynamic relaxation method combined with finite difference technique is used in order to solve governing equations. Creep and recovery of isotropic and laminated plates are studied under various loadings. Efficiency and accuracy of present formulation is proved by solving several benchmark problems. The effects of different parameters such as side-to-thickness ratio, types of loadings and boundary conditions, and stacking sequence are investigated on the behavior of viscoelastic laminated composite plates.

A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate

In this work, a new higher shear deformation theory (HSDT) is developed for the free vibration and buckling of functionally graded (FG) sandwich plates. The proposed theory presents a new displacement field by using undetermined integral terms. Only four unknowns are employed in this theory, which is less than the classical first shear deformation theory (FSDT) and others HSDTs. Equations of motion are obtained via Hamilton\'s principle. The analytical solutions of FG sandwich plates are determined by employing the Navier method. A good agreement between the computed results and the available solutions of existing HSDTs is found to prove the accuracy of the developed theory.

Free vibrations of laminated composite plates using a novel four variable refined plate theory

In this research, the free vibration response of laminated composite plates is investigated using a novel and simple higher order shear deformation plate theory. The model considers a non-linear distribution of the transverse shear strains, and verifies the zero traction boundary conditions on the surfaces of the plate without introducing shear correction coefficient. The developed kinematic uses undetermined integral terms with only four unknowns. Equations of motion are obtained from the Hamilton's principle and the Navier method is used to determine the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical examples studied using the present formulation is compared with three-dimensional elasticity solutions and those calculated using the first-order and the other higher-order theories. It can be concluded that the present model is not only accurate but also efficient and simple in studying the free vibration response of laminated composite plates.

A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams

In this article, a novel simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded (FG) beams is proposed. The beauty of this theory relies on its 2-unknowns displacement field as the Euler- Bernoulli beam theory, which is even less than the Timoshenko beam theory. A shear correction factor is, therefore, not needed. Equations of motion are obtained via Hamilton‟s principle. Analytical solutions for the bending and free vibration analysis are given for simply supported beams. Efficacy of the proposed model is shown through illustrative examples for bending and dynamic of FG beams. The numerical results obtained are compared with those of other higher-order shear deformation beam theory results. The results obtained are found to be accurate.

A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates

In this paper, a new simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded (FG) plates is developed. The significant feature of this formulation is that, in addition to including a sinusoidal variation of transverse shear strains through the thickness of the plate, it deals with only three unknowns as the classical plate theory (CPT), instead of five as in the well-known first shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Equations of motion are derived from Hamilton's principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. Verification studies show that the proposed theory is not only accurate and simple in solving the bending and free vibration behaviours of FG plates, but also comparable with the other higher-order shear deformation theories which contain more number of unknowns.

An efficient and simple higher order shear deformation theory for bending analysis of composite plates under various boundary conditions

In this study, the bending and dynamic behaviors of laminated composite plates is examined by using a refined shear deformation theory and developed for a bending analysis of orthotropic laminated composite plates under various boundary conditions. The displacement field of the present theory is chosen based on nonlinear variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theory is reduced and hence makes them simple to use. In the analysis, the equation of motion for simply supported thick laminated rectangular plates is obtained through the use of Hamilton´s principle. Numerical results for the bending and dynamic behaviors of antisymmetric cross-ply laminated plate under various boundary conditions are presented. The validity of the present solution is demonstrated by comparison with solutions available in the literature. Numerical results show that the present theory can archive accuracy comparable to the existing higher order shear deformation theories that contain more number of unknowns.

A simple higher order shear deformation theory for mechanical behavior of laminated composite plates

In the present study, the static, buckling, and free vibration of laminated composite plates is examined using a refined shear deformation theory and developed for a bending analysis of orthotropic laminated composite plates. These models take into account the parabolic distribution of transverse shear stresses and satisfy the condition of zero shear stresses on the top and bottom surfaces of the plates. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the equa- tion of motion for simply supported thick laminated rect- angular plates is obtained through the use of Hamilton's principle. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher order models and with data found in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static, the buckling, and free vibration behaviors of laminated composite plates.

Buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory

In this paper, the buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory are examined by using a refined shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In this analysis, the equations of motion for simply supported thick laminated rectangular plates are derived and obtained through the use of Hamilton’s principle. The closed-form solutions of anti-symmetric cross-ply and angle- ply laminates are obtained using Navier solution. Numerical results of the present study are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling and free vibration behaviors of laminated composite plates.

Stress analysis of laminated composite and soft core sandwich beams using a simple higher order shear deformation theory

In this paper, the refined beam theory (RBT) is examined for the bending of simply supported isotropic, laminated composite and sandwich beams. The axial displacement field uses parabolic function in terms of thickness ordinate to include the effect of transverse shear deformation. The transverse displacement consists of bending and shear components. The present theory satisfies the traction free conditions on the upper and lower surfaces of the beam without using problem dependent shear correction factors of Timoshenko. Governing differential equations and boundary conditions associated with the assumed displacement field are obtained by using the principle of virtual work. To prove the credibility of the present theory, we applied it to the bending analysis of beams. A simply supported isotropic, laminated composite and sandwich beams are analyzed using Navier approach. The numerical results of non-dimensional displacements and stresses obtained by using the present theory are presented and compared with those of other refined theories available in the literature along with the elasticity solution.

Modeling and Forced Vibration of Smart Piezoelectric Beams Using Simple Higher Order Shear Deformation Theory

In the current work, a finite element formulation is developed for modeling and forced vibration of isotropic as well as orthotropic composite beams with distributed piezoelectric actuators subjected to both mechanical and electrical loads. The proposed model is developed based on a simple higher order shear deformation theory where the displacements field equations in the model accounts for a parabolic distribution of the shear strain and the non-linearity of in-plane displacements across the thickness and subsequences the shear correction factor is not involved. The virtual displacement method is used to formulate the equations of motion of the structure system. The model is valid for both segmented and continuous piezoelectric elements which can be either surface bonded or embedded in the laminated beams. A two-node element with four mechanical degrees of freedom in addition to one electrical degree of freedom for each node is used in the finite element formulation. The electric potential is considered as function of the thickness and the length of the beam element. The steady state responses for damped and undamprd beams are formulated. A MATLAB code is developed to compute the static deformation, free vibration parameters, and the beams responses due to harmonic excitation of mechanical and electrical loads. The predictions from the proposed model are in good shape.

Modeling and analysis of smart piezoelectric beams using simple higher order shear deformation theory

In the current work, a finite element formulation is developed for modeling and analysis of isotropic as well as orthotropic composite beams with distributed piezoelectric actuators subjected to both mechanical and electrical loads. The proposed model is developed based on a simple higher order shear deformation theory where the displacement field equations in the model account for a parabolic distribution of the shear strain and the nonlinearity of in-plane displacements across the thickness and subsequently the shear correction factor is not involved. The virtual displacement method is used to formulate the equations of motion of the structure system. The model is valid for both segmented and continuous piezoelectric elements, which can be either surface bonded or embedded in the laminated beams. A two-node element with four mechanical degrees of freedom in addition to one electrical degree of freedom for each node is used in the finite element formulation. The electric potential is considered as a function of the thickness and the length of the beam element. A MATLAB code is developed to compute the static deformation and free vibration parameters of the beams with distributed piezoelectric actuators. The obtained results from the proposed model are compared with the available analytical results and the finite element results of other researchers.

Analysis of Deflections and Stresses for Laminated Composite Plates Based on a New Higher-Order Shear Deformation Theory

Based on the new simple third-order shear deformation theory, the deflections and stresses of the simply surported symmetrical laminated composite plates are obtained by using the principle of virtual work .The solutions are compared with the solutions of three-dimensional elasticity theory, the first-order shear deformation theory and the Reddy’s higher order shear deformation theory . Results show that the presented new theory is more reliable, accurate, and cost-effective in computation than the first-order shear deformation theories and other simple higher-order shear deformation theories.

A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates

In this paper, a new higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates is developed. The present theory has only four unknowns, but it accounts for a parabolic variation of transverse shear strains through the thickness of the plate. A shear correction factor is, therefore, not required. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3D and quasi-3D solutions and those predicted by other plate theories. Results show that the present theory can achieve the same accuracy of the existing higher-order shear deformation theories which have more number of unknowns, but its accuracy is not comparable with those of 3D and quasi-3D models which include the thickness stretching effect.

Optimum design of laminated composite under axial compressive load

In the present study optimal design of composite laminates, with and without rectangular cut-out, is carried out for maximizing the buckling load. Optimization study is carried out for obtaining the maximum buckling load with design variables as ply thickness, cut-out size and orientation of cut-out with respect to laminate. Buckling load is evaluated using a ‘simple higher order shear deformation theory’ based on four unknown displacements u, v, w b and w s . A C1 continuous shear flexible finite element based on HSDT model is developed using Hermite cubic polynomial. It is observed that for thick anti-symmetric laminates, the non-dimensional buckling load decreases with increase in aspect ratio and increase in fibre orientation angle. There is a decrease in the non-dimensional buckling load of symmetric laminate in the presence of cut-out.

Optimal design of composite laminates with and without cutout undergoing free vibration

In this article, the optimal designs of freely vibrating composite laminates with and without cutout undergoing are investigated. A simple higher order shear deformation theory (HSDT) with four unknown displacements (u 0, v 0, wb, ws ) is employed to obtain the vibration response. A C 1 continuity shear flexible element based on HSDT using the Hermite cubic polynomial is used for the rectangular element. The optimisation exercise is performed using genetic algorithms to maximise the fundamental frequency of vibration. The aspect ratio of the laminate, fibre orientation, thickness of plies, modulus are treated as design variables. On the basis of the investigation, it is observed that the non-dimensional frequency of the laminate with cutout can be higher or lower than those of the laminate without cutout depending on the size of the cutout. The presence of cutout significantly affects the frequency.