Conventional First-order Shear Deformation Theory Research Articles

Transient Vibroacoustic Analysis of Functionally Graded Plates

Abstract This paper presents the vibroacoustic response of pure functionally graded (FG) plates under transient loading of mechanical nature. The functionally graded plate is modeled using the conventional first-order shear deformation theory (FSDT) to incorporate the effects of transverse shear and rotary inertia. The mid-surface variables are determined using the finite element method. Transient structural response is determined using Newmark Beta time marching scheme and the acoustic pressure in the free field is obtained using the time-domain Rayleigh integral. The effective material properties of the FG plate and the transient response of both the structural and acoustic fields have been computed in matlab. The influence of the volume fraction index, thickness ratio, and boundary conditions of pure FG plate on its transient vibroacoustic response is investigated by a detailed parametric study.

Global perspectives on Higher Education: A social constructivist case for diversification of Complementary Medicine curricula

In the present investigation, a new first-order shear deformation theory (OVFSDT) on the basis of the in-plane stability of the piezo-magnetoelectric composite nanoplate (PMEN) has been developed, and its precision has been evaluated. The OVFSDT has many advantages compared to the conventional first-order shear deformation theory (FSDT) such as needless of shear correction factors, containing less number of unknowns than the existing FSDT and strong similarities with the classical plate theory (CPT). The composite nanoplate consisted of BaTiO3-CoFe2O4, a kind of material by which coupling between piezoelectric and piezomagnetic in nanosize was established. The plate is surrounded by a motionless and stationary matrix that is embedded in a hygrothermal surround in order to keep it more stable, and to take into consideration the influences of the moisture and temperature on the plate's mechanical behavior. The governing equilibrium equations for the smart composite plate have been formulated using the higher-order nonlocal strain gradient theory within which both stress nonlocality and second strain gradient size-dependent terms are taken into account by using three independent length scale parameters. The extracted equations are solved by utilizing the analytical approaches by which numerical results are obtained with various boundary conditions. In order to evaluate the proposed theory and methods of solution, the outcomes in terms of critical buckling loads are compared with those from several available well-known references. Finally, after determining the accuracy of the results of the new plate theory, several parameters are investigated to show the influences of material properties of the ceramic composite nanoplate on the critical buckling loads.

A Four-Variable First-Order Shear Deformation Theory Considering the Variation of In-plane Rotation of Functionally Graded Plates

This paper presents a four-variable first-order shear deformation theory considering in-plane rotation of functionally graded plates. In recent studies, a simple first-order shear deformation theory was developed and extended to functionally graded plates. It has only four variables, separating the deflection into bending and shear parts, while the conventional first-order shear deformation theory has five variables. However, this simple first-order shear deformation theory only provides good predictions for simply supported plates since it does not consider in-plane rotation varying through the thickness of the plates. The present theory also has four variables, but considers the variation of in-plane rotation such that it is able to correctly predict the responses of the plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with various boundary conditions. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the responses of functionally graded plates.

Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory

In the present investigation, a new first-order shear deformation theory (OVFSDT) on the basis of the in-plane stability of the piezo-magnetoelectric composite nanoplate (PMEN) has been developed, and its precision has been evaluated. The OVFSDT has many advantages compared to the conventional first-order shear deformation theory (FSDT) such as needless of shear correction factors, containing less number of unknowns than the existing FSDT and strong similarities with the classical plate theory (CPT). The composite nanoplate consisted of BaTiO3-CoFe2O4, a kind of material by which coupling between piezoelectric and piezomagnetic in nanosize was established. The plate is surrounded by a motionless and stationary matrix that is embedded in a hygrothermal surround in order to keep it more stable, and to take into consideration the influences of the moisture and temperature on the plate's mechanical behavior. The governing equilibrium equations for the smart composite plate have been formulated using the higher-order nonlocal strain gradient theory within which both stress nonlocality and second strain gradient size-dependent terms are taken into account by using three independent length scale parameters. The extracted equations are solved by utilizing the analytical approaches by which numerical results are obtained with various boundary conditions. In order to evaluate the proposed theory and methods of solution, the outcomes in terms of critical buckling loads are compared with those from several available well-known references. Finally, after determining the accuracy of the results of the new plate theory, several parameters are investigated to show the influences of material properties of the ceramic composite nanoplate on the critical buckling loads.

A two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates

• A two-variable first-order plate theory considering in-plane rotation is presented. • Analytical solutions for various boundary conditions are obtained in bending, buckling and free vibration analyses. • The in-plane rotation occurs close to the edges, so it should be considered to correctly predict the responses of plates. This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.

A Simplified First-order Shear Deformation Theory for Bending, Buckling and Free Vibration Analyses of Isotropic Plates on Elastic Foundations

This paper presents analytical solutions for bending, buckling and free vibration analyses of isotropic plates on elastic foundations using a simplified first-order shear deformation theory. Unlike the conventional first-order shear deformation theory, the present theory contains only two variables and has many similarities to the classical plate theory. For the elastic foundations, the Pasternak model which has two parameters is used. Equations of motion are derived from Hamilton’s principle. Analytical solutions of deflections, moments, shear forces, buckling loads and natural frequencies are obtained for rectangular plates with various boundary conditions. Numerical examples for various aspect ratios, side-to-thickness ratios and foundation parameters are presented to verify the validity of the present theory. Comparative study shows that the present theory is accurate and efficient in predicting bending, buckling and free vibration responses of isotropic plates on elastic foundations. Parametric study shows the effect of the elastic foundations on the behavior of the plates.

Size effects of functionally graded moderately thick microplates: A novel non-classical simple-FSDT isogeometric analysis

Abstract This paper presents an effective plate formulation coupling the merits of isogeometric analysis (IGA) and a new non-classical simple first-order shear deformation theory (SFSDT) for static bending, free vibration, and buckling of functionally graded (FG) moderately thick microplates. In contrast to the conventional first-order shear deformation theory (FSDT), the new SFSDT adopted here inherently owns several advantages such as free from shear-locking, capturing the shear-deformation effect, and fewer unknowns. In order to capture the small scale effects, we thus introduce a non-classical SFSDT based on a modified couple stress theory. The requirement for C 2 -continuity in terms of the non-classical SFSDT is straightforwardly treated with the aid of inherent high-order continuity of non-uniform rational B-spline (NURBS), which serves as basis functions in our IGA framework. Numerical examples are presented and the obtained numerical results reveal that the deflection decreases while the frequency and buckling load increase with decreasing the plate thickness. Results also show that the small size effect can lead to an increase of microplate stiffness.

A simple FSDT-based meshfree method for analysis of functionally graded plates

Modeling of mechanical behavior of plates has been accomplished in the past decades, with different numerical strategies including the finite element and meshfree methods, and with a range of plate theories including the first-order shear deformation theory (FSDT). In this paper, we propose an efficient numerical meshfree approach to analyze static bending and free vibration of functionally graded (FG) plates. The kinematics of plates is based on a novel simple FSDT, termed as S-FSDT, which is an effective four-variable refined plate theory. The S-FSDT requires C1-continuity that is satisfied with the basis functions based on moving Kriging interpolation. Some major features of the approach can be summarized: (a) it is less computationally expensive due to having fewer unknowns; (b) it is naturally free from shear-locking; (c) it captures the physics of shear-deformation effect present in the conventional FSDT; (d) the essential boundary conditions can straightforwardly be treated, the same as the FEM; and (e) it can deal with both thin and thick plates. All these features will be demonstrated through numerical examples, which are to confirm the accuracy and effectiveness of the proposed method. Additionally, a discussion on other possible choices of correlation functions used in the model is given.

New enhanced first-order shear deformation theory for thermo-mechanical analysis of laminated composite and sandwich plates

Abstract This paper proposes an efficient and accurate thermal stress analysis method based on the conventional first-order shear deformation theory (FSDT). The transverse shear strain energy is modified through the mixed variational theorem. The independent transverse shear stresses are obtained from a third-order zigzag model to improve the accuracy, whereas the FSDT displacements are employed to improve the numerical efficiency. The transverse displacement field includes higher-order terms without introducing additional unknown variables by superimposing a prescribed thermal field. This allows one to consider the thermal transverse normal strain effect under the plane stress condition. The resulting strain energy expressions are collectively referred to as the enhanced first-order shear deformation theory including the transverse normal effect based on the mixed variational theorem (EFSDTM_TN). The EFSDTM_TN offers the same computational advantages as the conventional FSDT while allowing for improved local variations of stresses and displacements via a post-processing procedure. In this procedure, in-plane correction factors are introduced and determined by applying equivalent stress resultants so that the improved stresses satisfy the plate equilibrium. Finally, the obtained displacements and stress distributions are compared with the data available in the literature.

Thermal stability of functionally graded sandwich plates using a simple shear deformation theory

In the present work, a simple first-order shear deformation theory is developed and validated for a variety of numerical examples of the thermal buckling response of functionally graded sandwich plates with various boundary conditions. Contrary to the conventional first-order shear deformation theory, the present first-order shear deformation theory involves only four unknowns and has strong similarities with the classical plate theory in many aspects such as governing equations of motion, and stress resultant expressions. Material properties and thermal expansion coefficient of the sandwich plate faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The thermal loads are considered as uniform, linear and non-linear temperature rises within the thickness direction. The results reveal that the volume fraction index, loading type and functionally graded layers thickness have significant influence on the thermal buckling of functionally graded sandwich plates. Moreover, numerical results prove that the present simple first-order shear deformation theory can achieve the same accuracy of the existing conventional first-order shear deformation theory which has more number of unknowns.

A Timoshenko Piezoelectric Beam Finite Element with Consistent Performance Irrespective of Geometric and Material Configurations

The conventional Timoshenko piezoelectric beam finite elements based on First-order Shear Deformation Theory (FSDT) do not maintain the accuracy and convergence consistently over the applicable range of material and geometric properties. In these elements, the inaccuracy arises due to the induced potential effects in the transverse direction and inefficiency arises due to the use of independently assumed linear polynomial interpolation of the field variables in the longitudinal direction. In this work, a novel FSDT-based piezoelectric beam finite element is proposed which is devoid of these deficiencies. A variational formulation with consistent through-thickness potential is developed. The governing equilibrium equations are used to derive the coupled field relations. These relations are used to develop a polynomial interpolation scheme which properly accommodates the bending-extension, bending-shear and induced potential couplings to produce accurate results in an efficient manner. It is noteworthy that this consistently accurate and efficient beam finite element uses the same nodal variables as of conventional FSDT formulations available in the literature. Comparison of numerical results proves the consistent accuracy and efficiency of the proposed formulation irrespective of geometric and material configurations, unlike the conventional formulations.

Improved Viscoelastic Analysis of Laminated Composite and Sandwich Plates With an Enhanced First-Order Shear Deformation Theory

An enhanced first-order shear deformation theory (EFSDT) is developed for linear viscoelastic analysis of laminated composite and sandwich plates. Improved strain energy expression of the conventional Reissner/Mindlin first-order shear deformation theory (FSDT) through strain energy transformation is derived in the Laplace domain by minimizing the strain energy difference between FSDT and an efficient higher-order zigzag theory (EHOPT). The convolution theorem of Laplace transformation is applied to circumvent the complexity of dealing with linear viscoelastic materials. The present EFSDT with the Laplace domain approach has the same computational advantage of the conventional FSDT while improving upon the accuracy of the viscoelastic response by utilizing the postprocess recovery procedure. The accuracy and efficiency of the proposed theory are demonstrated through the numerical results obtained herein by comparing to those available in the open literature.

A New First-Order Shear Deformation Theory for Free Vibrations of Rectangular Plate

A new first-order shear deformation theory (FSDT) with pure bending deflection and shearing deflection as two independent variables is presented in this paper for free vibrations of rectangular plate. In this two-variable theory, the shearing deflection is regarded as the only fundamental variable by which the total deflection and bending deflection can be expressed explicitly. In contrast with the conventional three-variable first-order shear plate theory, present variationally consistent theory derived by using Hamiltonian variational principle can uniquely define the bending and the shearing deflections, and give two rotations by the differentiations of bending deflection. Due to more restrictive geometrical constraints on rotations and boundary conditions, the obtained natural frequencies are equal to or higher than those by conventional FSDT for the rectangular plate with at least one pair of opposite edges simply supported. This new theory is of considerable significance in theoretical sense for giving a simple two-variable FSDT which is variational consistent and involve rotary inertia and shear deformation. The relation and differences of present theory with conventional FSDT and other relative formulations are discussed in detail.

Study on an alternative deformation concept for the Timoshenko beam and Mindlin plate models

Abstract An overall historical survey on the bending and shearing deformation concepts in first-order shear deformation theories is carried out in relation to the corrected classical models from the viewpoint of distinguishability of both the deflections in beams and plates, including the alternative deformation concept for the Timoshenko beam and Mindlin plate models proposed by the author and his co-worker. The survey shows that in the corrected classical models, the bending and shearing deflections are recognized as distinguishable physical entities, and which has gained widespread acceptance in the various fields of continuum and structure mechanics, though both deflections cannot be obtained concurrently using a deductive approach. With regard to the conventional first-order shear deformation theory, both deflections are not to be recognized as distinguishable physical entities, and it is shown that these two deflections cannot be uniquely determined for a beam and cannot be specifically defined for a plate. In contrast to this, the alternative deformation concept is shown to be effective in resolving all the above inconsistencies. To demonstrate its effectiveness and practical advantages, both static and dynamic analyses are carried out. The static analysis involves a simply supported beam subjected to a uniform or concentrated static load, for which both the bending and shearing deflections are obtained concurrently and uniquely in addition to the total deflection using a deductive approach, and a thought experiment that distinguishes both deflections as physical entities is also presented. The dynamic analysis involves the application of an extension of the Southwell–Dunkerley methods for synthesizing frequencies. Based on a series-type synthetic-frequency method established by the author and his co-worker, a frequency analysis with two deflections coupled in series is carried out for isotropic plates and carbon fiber reinforced plastic laminated composite beams. It is also shown that the alternative deformation concept is a model that can contribute to the development of a practical finite element formulation that is free from shear locking.

An accurate novel coupled field Timoshenko piezoelectric beam finite element with induced potential effects

An accurate coupled field piezoelectric beam finite element formulation is presented. The formulation is based on First-order Shear Deformation Theory (FSDT) with layerwise electric potential. An appropriate through-thickness electric potential distribution is derived using electrostatic equilibrium equations, unlike conventional FSDT based formulations which use assumed independent layerwise linear potential distribution. The derived quadratic potential consists of a coupled term which takes care of induced potential and the associated change in stiffness, without bringing in any additional electrical degrees of freedom. It is shown that the effects of induced potential are significant when piezoelectric material dominates the structure configuration. The accurate results as predicted by a refined 2D simulation are achieved with only single layer modeling of piezolayer by present formulation. It is shown that the conventional formulations require sublayers in modeling, to reproduce the results of similar accuracy. Sublayers add additional degrees of freedom in the conventional formulations and hence increase computational cost. The accuracy of the present formulation has been verified by comparing results obtained from numerical simulation of test problems with those obtained by conventional formulations with sublayers and ANSYS 2D simulations.

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

This paper presents a simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Unlike the conventional first-order shear deformation theory, the present first-order shear deformation theory contains only four unknowns and has strong similarities with the classical plate theory in many aspects such as governing equations of motion, boundary conditions, and stress resultant expressions. Equations of motion and boundary conditions are derived from Hamilton’s principle. Closed-form solutions of simply supported plates are obtained and the results are compared with the exact 3D and quasi-3D solutions and those predicted by other plate theories. Comparison studies show that the present theory can achieve the same accuracy of the conventional first-order shear deformation theory which has more number of unknowns.