Shear Correction Factor Research Articles

Assessment of inverse hyperbolic zigzag theory for buckling analysis of laminated composite and sandwich plates using finite element method

In the present study, a recently developed non-polynomial zigzag theory by the authors is assessed for the buckling analysis of the laminated composite and sandwich plates. This theory assumes a nonlinear distribution for the transverse shear stresses. The model satisfies the transverse shear stress free boundary conditions at top and bottom surfaces of the laminate obviating the need for a shear correction factor. An efficient C0 continuous isoparametric serendipity element is employed for the discretization of plate structure for the assessment of buckling responses. The Gauss quadrature rule with reduced integration scheme for thin laminate and full integration rule for thick laminate are implemented in the analysis. Some representative results are obtained on buckling analysis covering different features of laminated composite and sandwich structures such as modular ratios, aspect ratios, loading conditions, boundary conditions and span to thickness ratios. The evaluated results are compared with results available in the literature to ensure the efficacy of the present theory. The excellent agreement of the evaluated results with the three-dimensional elasticity results concludes that the presented models are not only accurate but also efficient.

Static Stability Analysis of Mass Sensors Consisting of Hygro-Thermally Activated Graphene Sheets Using a Nonlocal Strain Gradient Theory

This paper develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygro-thermal environments with mass sensors. For a more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. The graphene sheet is modeled via a two-variable shear deformation plate theory that does not need shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on the elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors, such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, nanoparticle mass and geometrical parameters, on buckling characteristics of graphene sheets are examined and presented as dispersion graphs.

A refined inverse hyperbolic shear deformation theory for bending analysis of functionally graded porous plates

The modern engineering structures require the advanced engineering materials to resist the high temperatures and to provide high stiffness. In particular the functionally graded porous materials (FGPMs) introduced are expected to have these desired properties, consequently eliminating local stress concentration and de-lamination. In the present paper, a new shear strains shape function is chosen to research the bending analysis of functionally graded plates (FGPs) with uneven symmetrical, uneven asymmetrical and even distributions of porosity. The material properties of uneven porosity distributions along the thickness of the FGPs vary with cosine function. The present theory includes the influence of thickness stretching. This theory also fulfills the nullity of the shear stresses in the transverse direction on the top and bottom of the plate, thus avoids the use of a shear correction factor. The virtual displacement principle is employed to develop the equilibrium equations for porous FGPs. The Navier’s method is used to obtain the solutions of porous FGPs for simply supported (SS) conditions. The accuracy of the developed theory is established with numerical results of perfect and porous FGPs available in the open source. The transverse displacements and stress results have been reported and studied for evenly, unevenly symmetrical and unevenly asymmetrical distributions with different porosity volume fraction (PVF), thickness ratios and aspect ratios. From the numerical results it is concluded that the type of porosity distribution needs to be considered as a key factor in the optimal design of the porous FGPs.

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.

Magneto-electro-elastic vibration analysis of modified couple stress-based three-layered micro rectangular plates exposed to multi-physical fields considering the flexoelectricity effects

In this paper, based on the CPT, motion equations for a sandwich plate containing a core and two integrated face-sheets have derived. The structure rests on the Visco-Pasternak foundation, which includes normal and shear modules. The piezo-magnetic core is made of CoFe2O4 and also is subjected to 3D magnetic potential. Two face sheets at top and bottom of the core are under electrical fields. Also, in order to obtain more accuracy, the effect of flexoelectricity has took into account at face sheets' relations in this work. Flexoelectricity is a property of all insulators whereby they polarize when subject to an inhomogeneous deformation. This property plays a crucial role in small-scale rather than macro scale. Employing CPT, Hamilton's principle, flexoelectricity considerations, the governing equations are derived and then solved analytically. By present work a detailed numerical study is obtained based on Piezoelectricity, Flexoelectricity and modified couple stress theories to indicate the significant effect of length scale parameter, shear correction factor, aspect and thickness ratios and boundary conditions on natural frequency of sandwich plates. Also, the figures show that there is an excellent agreement between present study and previous researches. These finding can be used for automotive industries, aircrafts, marine vessels and building industries.

Locking-free robust finite element approximation of thin shell-like structures

In the structural analysis of thin elastic structures using standard FEMs within the poor approximation space, the numerical results may suffer from the quality deterioration called the locking phenomenon. This situation becomes more crucial for the structures with non-vanishing curvatures, such as arch- and shell-like bodies, owing to the additional membrane constraints. In addition, the choice of unsuitable shear correction factor (SCF), in dimen-sionally-reduced first-order analysis models, may become an additional source for the quality deterioration. In this context, we here propose a locking-free robust finite element approximation for the robust structural analysis of shell-like structures, by improving the degenerated 8-node shell element. In order to completely overcome the quality deterioration according to the thickness reduction, we combine the reduced integration (RI) and the modified shear correction factor.

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories

Composite laminated plates have found several applications in engineering field. For instance, aerospace and automotive industry use these structures in several components of their products due to their exceptional specific properties along the direction of the fibres. There are several two-dimensional plate models that aim to more accurately and more efficiently predict the kinematics of those structures. Unlike the Classical Plate Theory (CLPT), which can only be applied to thin laminates, the First-Order Shear Deformation Theory (FSDT), developed by Reissner and Mindlin, already accounts shear effects and, due to that, it can describe in a satisfactory way the kinematic of a generic thick laminated plate. Nevertheless, the FSDT considers linear shape functions to describe the in-plane displacements through the plate thickness, which results in constant shear strains. Thus, the traction boundary conditions on the top and bottom surface of plate are violated. Additionally, the FSDT needs shear correction factors, which are not easily obtained for a generic problem. Equivalent single layer (ESL) theories following high-order shear deformation theories (HSDTs) have been proposed and more efficiently applied to composite laminated plates. These theories allow to represent the nonlinear variation of transverse shear stresses through thickness of plate because they possess nonlinear shape functions interpolating the in-plane displacements. Thus, the kinematics of these structural elements can be better described and more accurate stress fields can be obtained for problems that cannot be handled with the FSDT.These structures have been analysed for several years using the Finite Element Method (FEM), which is the most popular numerical tool in structural analysis. However, several other advanced numerical techniques, such as meshless methods, can also handle this kind of problems and, in some cases, in a more efficient way. This work makes use of a recently developed meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) (Belinha, 2014) – to study the bending of antisymmetric angle-ply composite laminates using distinct HSDTs.

A new C0 layerwise wavelet finite element formulation for the static and free vibration analysis of composite plates

In this paper, a new C0 layerwise wavelet finite element is proposed for the static and free vibration analysis of composite plates. The refined zigzag theory is adopted to introduce the zigzag effects in multilayered plate structures by using piecewise linear C0 continuous functions. Then the layerwise wavelet-based BSWI element is derived based on the higher-order plate theory by means of two-dimensional BSWI scaling functions. The proposed model satisfies the conditions of transverse shear stress continuity at the layer interfaces as well as yields to the stress-free boundary conditions on the surface of plate without a shear correction factor. What’s more, the layerwise wavelet-based BSWI element also possesses the advantages of high convergence, high accuracy and reliability with fewer degrees of freedom on account of the excellent approximation property of BSWI. The accuracy and effectiveness of proposed layerwise wavelet-based BSWI element is assessed for static and free vibration analysis of laminated composite and sandwich plates with available 3D elasticity solutions, finite element solutions and other referential solutions in published literatures.

Development of an equivalent shell finite element for modelling damped multi-layered composite structures

A new equivalent shell finite element (FE) for modelling damped multi-layered structures is presented in this study. The method used for developing the new FE for such structures is based on the idea that the strain energy of the equivalent single-layer FE must be equal to the sum of the strain energies of individual layers. The so-called energy coefficients are defined for this purpose for the extensional, bending and shear deformations of the composite structure. These coefficients are then determined and used as correction multipliers during stacking the elemental matrices of individual layers. Two approaches, based on second-order strain or stress distribution assumption through the composite thickness, are investigated for deriving the shear energy coefficients. The damping capability of the FE developed here originates from using complex Young’s modulus to define the material properties of individual layers. The resulting equivalent single-layer shell element with four nodes has six degrees-of-freedom per node. The accuracy, advantages and limitations of the composite FE developed in this work are investigated using experimental as well as theoretical results. In the light of the finding of these investigations, further enhancement in the formulation is made by also utilising a new shear correction factor for the individual layers in the equivalent shell element. Final results for free- and constrained-layered structures confirm that the equivalent shell FE developed here can be used effectively for the prediction of the modal properties of damped multi-layered structures.

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.

A finite element model for static analysis of curved thin-walled beams based on the concept of equivalent layered composite cross section

In this study, an accurate 1D finite element model with low degrees of freedom (DOFs) is presented for the static extensional-shearing-bending analysis of curved thin-walled beams. In order to incorporate the nonclassical effects like transverse shear flexibility in the formulation, the concept of equivalent layered composite cross section (ELCS) is employed. Based on this new concept, the cross section of a thin-walled beam is replaced with an equivalent layered composite one. A global–local layered beam theory is employed to describe the displacement fields of the curved beams. A curved beam finite element with three nodes and 13 DOFs is developed for the approximation of the unknown variables of the displacement fields. For validation of the proposed model, the results of 1 D models of other researchers as well as the results of the 2D/3D finite element simulations (ABAQUS) are employed. Needlessness to the shear correction factor, satisfaction of zero-conditions of shear stresses on the upper and lower planes of the curved beam, high accuracy in the prediction of structural responses, and significant reduction in the time of computations are some of distinct features of the proposed finite element model.

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.

Effect of Fluid–Structure Interaction on Vibration of Moving Sandwich Plate With Balsa Wood Core and Nanocomposite Face Sheets

This research presents theoretical investigation to analyze vibration of axially moving sandwich plate floating on fluid. This plate is composed of balsa wood core and two nanocomposite face sheets where the three layers vibrated as an integrated sandwich. The fluid–structure interaction (FSI) effects on the stability of moving plate are considered for both ideal and viscous fluid. Halpin–Tsai model is utilized to determine the material properties of two-phase composite consist of uniformly distributed and randomly oriented carbon nanotubes (CNTs) through the PmPV (poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) matrix. The governing equations are derived based on sinusoidal shear deformation plate theory (SSDT) which is more accurate than the conventional theories, and significantly, it does not require a shear correction factor. Employing Hamilton’s principle, the equations of motion are obtained and solved by the semi-analytical method. Results indicated that the dimensionless frequencies of moving sandwich plate decrease rapidly with increasing the water level and they are almost independent of fluid level when it is higher than 50% of the plate length. The results of this investigation can be used in design and manufacturing of marine vessels and aircrafts.

Free Vibration of Thick Quadrilateral Laminates Using Third-Order Shear-Normal Deformation Theory

Free vibrations of thick and skew quadrilateral laminates have been analyzed by using the Ritz method and a third-order shear and normal deformable plate theory (TSNDT) that does require a shear-correction factor. The weighted orthogonal Jacobi polynomials are employed as basis functions that exactly satisfy essential boundary conditions at the plate edges. An in-house-developed software is first verified by comparing computed frequencies with those either available in the literature or found by using the three-dimensional linear elasticity-theory-based finite-element-based commercial software ABAQUS. The method is applied first to find the lowest few (six in most examples) frequencies of isotropic cantilever plates of different skew angles and thickness-to-side length ratios. Subsequently, natural frequencies of laminated composite plates for various skew angles, thickness-to-side ratios, numbers of layers, and stacking sequences are compared with those obtained from converged solutions using either shell or brick elements in ABAQUS. It is demonstrated that the TSNDT/Ritz method provides highly accurate frequencies. Three-dimensional mode shapes are presented for a better understanding of the dynamic behavior of skew quadrilateral laminates.

A thermodynamically consistent non-linear mathematical model for thermoviscoelastic plates/shells with finite deformation and finite strain based on classical continuum mechanics

This paper presents a kinematic assumption free and thermodynamically consistent non-linear formulation incorporating finite strain and finite deformation for thermoviscoelastic plates/shells based on the conservation and balance laws of the classical continuum mechanics (CCM) in R3 (see Surana and Mathi, (2020) for linear theory). The conservation and balance laws in Lagrangian description with finite strain measure and the conjugate stress measure in R3 are considered. The conjugate pairs in the second law of thermodynamics (SLT), the consideration of additional physics and the principle of equipresence are utilized to determine the constitutive variables and their argument tensors. The constitutive theory for the contravariant second Piola–Kirchhoff stress tensor is derived using Green’s strain tensor and its convected time derivatives of up to order n as its argument tensors using representation theorem with complete basis (i.e. using integrity). The convected time derivatives of the Green’s strain tensor up to order n provide ordered rate constitutive theory for the dissipation mechanism that is naturally non-linear due to Green’s strain measure. The constitutive theory for heat vector derived using representation theorem and integrity is also a non-linear constitutive theory. Simplified linear forms of these constitutive theories are also presented. The solution methods for the mathematical model for the BVPs as well as the IVPs using p-version hierarchical higher degree and higher order finite element method are presented. Due to dissipation, the energy equation is integral part of the mathematical model that accounts for rate of mechanical work resulting in rate of entropy production, hence heat.The plate/shell geometry (flat or curved) is described by its middle surface containing the nodal vectors (at each of the nine nodes), the ends of which define bottom and top surfaces of the plate/shell (Surana and Mathi, 2020). The geometry is mapped into ξηζ natural coordinate space in a two unit cube. The p-version hierarchical local approximations in ξηζ are constructed to describe the deformation of the plate/shell middle surface as well as its faces that is controlled by p-levels in ξ,η and ζ directions (element local approximation). The formulation presented here is accurate for very thin as well as very thick plate/shells and for small as well as finite deformation and finite strains. Non-linear constitutive theories based on integrity allow more complex constitutive behavior in term of strains as well as strain rates, hence dissipation mechanism. Dissipation mechanism is described by an ordered rate theory based on SLT and additional physics. The formulation presented here always ensures thermodynamic equilibrium during the evolution as it is derived using the conservation and balance laws of CCM in R3. The formulation always preserves three dimensional nature of the deformation physics regardless of the plate/shell thickness and is free of locking problems and shear correction factors that plague most of the currently used plate/shell formulations.

A Non-Intrusive Stochastic Isogeometric Analysis of Functionally Graded Plates with Material Uncertainty

A non-intrusive approach coupled with non-uniform rational B-splines based isogeometric finite element method is proposed here. The developed methodology was employed to study the stochastic static bending and free vibration characteristics of functionally graded material plates with inhered material randomness. A first order shear deformation theory with an artificial shear correction factor was used for spatial discretization. The output randomness is represented by polynomial chaos expansion. The robustness and accuracy of the framework were demonstrated by comparing the results with Monte Carlo simulations. A systematic parametric study was carried out to bring out the sensitivity of the input randomness on the stochastic output response using Sobol’ indices. Functionally graded plates made up of Aluminium (Al) and Zirconium Oxide (ZrO2) were considered in all the numerical examples.

AVA Biochem teams with Michelin Group

In this paper, the buckling of cross-ply and angle-ply composite beams under hygro-thermo-mechanical loading is presented by the analytical approach. This approach is based on theory of hyperbolic refined shear deformation. This approach which has strong resemblance with beam theory of Euler Bernoulli in various aspects, accounts for a second-degree variation of the transverse shear strains through the thickness and verified the zero traction of boundary conditions on the bottom and top beam surfaces sans employing shear correction factors. The effects of temperature and moisture concentration, thus the effect of the ratios of modulus, length to thickness and coefficient of thermal expansion respectively on hygro-thermo-mechanical buckling of cross-ply and angle-ply laminated beams are studied.

Dynamic Behavior of a Bidirectional Functionally Graded Sandwich Beam under Nonuniform Motion of a Moving Load

A bidirectional functionally graded Sandwich (BFGSW) beam model made from three distinct materials is proposed and its dynamic behavior due to nonuniform motion of a moving point load is investigated for the first time. The beam consists of three layers, a homogeneous core, and two functionally graded face sheets with material properties varying in both the thickness and longitudinal directions by power gradation laws. Based on the first-order shear deformation beam theory, a finite beam element is derived and employed in computing dynamic response of the beam. The element which used the shear correction factor is simple with the stiffness and mass matrices evaluated analytically. The numerical result reveals that the material distribution plays an important role in the dynamic response of the beam, and the beam can be designed to meet the desired dynamic magnification factor by appropriately choosing the material grading indexes. A parametric study is carried out to highlight the effects of the material distribution, the beam layer thickness and aspect ratios, and the moving load speed on the dynamic characteristics. The influence of acceleration and deceleration of the moving load on the dynamic behavior of the beam is also examined and highlighted.

Hygro-thermo-mechanical buckling of laminated beam using hyperbolic refined shear deformation theory

In this paper, the buckling of cross-ply and angle-ply composite beams under hygro-thermo-mechanical loading is presented by the analytical approach. This approach is based on theory of hyperbolic refined shear deformation. This approach which has strong resemblance with beam theory of Euler Bernoulli in various aspects, accounts for a second-degree variation of the transverse shear strains through the thickness and verified the zero traction of boundary conditions on the bottom and top beam surfaces sans employing shear correction factors. The effects of temperature and moisture concentration, thus the effect of the ratios of modulus, length to thickness and coefficient of thermal expansion respectively on hygro-thermo-mechanical buckling of cross-ply and angle-ply laminated beams are studied.

Simple Two Variable Refined Theory for Shear Deformable Isotropic Rectangular Beams

In this paper, a displacement-based, variationally consistent, two variable refined theory for shear deformable beams is presented. The beam is assumed to be of linearly elastic, homogeneous, isotropic material and has a uniform rectangular cross-section. In this theory, the beam axial displacement and beam transverse displacement consist of bending components and shearing components. The assumed displacement field of this theory is such that, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. This theory utilizes linear strain-displacement relations. The displacement functions give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. Hence the shear correction factor is not required. Hamilton’s principle is utilized to derive governing differential equations and variationally consistent boundary conditions. This theory involves only two governing differential equations of fourth-order. These governing equations are only inertially coupled for the case of dynamics and are decoupled for the case of statics. This theory is simple and has a strong resemblance with the Bernoulli-Euler beam theory. To demonstrate the efficacy of the present theory, illustrative examples pertain to the static bending and free vibrations of shear deformable isotropic rectangular beams are presented.