Nine-node Isoparametric Element Research Articles

Grey wolf optimizer based-natural frequency optimization of laminated composite shells carrying distributed mass

In this study, a grey wolf optimizer is proposed to find the optimum stacking sequence to maximize the fundamental frequency of shells. The vibration analysis of the shells is done using first-order shear deformation theory and a nine-node isoparametric element with five degrees of freedom per node. A mass lumping scheme with rotary inertia is considered in the finite element formulation to include the rotary inertia effect. The design variable considered is the fiber angle orientation of the shells. The present approaches are compared with previously published results using layerwise optimization theory, the Ritz method, and classical lamination theory. A comparison study is also performed, comparing the GWO-based finite element model against six different metaheuristic approaches, and it demonstrates that the Grey Wolf Optimizer consistently outperforms these competing algorithms. The optimum fiber angle orientation of shells carrying distributed mass for various boundary conditions and shell types is analyzed for the first time, to the best of the author’s knowledge. This novel discovery will benefit future studies.

Static Analysis of Skew Functionally Graded Plate Using Novel Shear Deformation Theory.

In this article, the static response of a functionally graded material (FGM) plate is studied via hybrid higher-order shear deformation theory which uses hyperbolic and polynomial shape functions and includes the effect of thickness stretching. The composition of the plate comprises metallic and ceramic phases. The ceramic volume fraction varies gradually along with the thickness following the power law. The mechanical properties of the FGM plate are determined by the rule of mixtures and the Mori–Tanaka homogenization scheme. The displacement fields are defined to satisfy the requirement of traction-free boundary conditions at the bottom and top surfaces of the plate surface removing the need for determination of shear correction factor. A C0 continuity FE model is developed for the present mathematical model. Nine-node isoparametric elements with eight nodal unknowns at each node are developed. The present model comparison with existing literature is completed and found to be coherent. Inhouse MATLAB code is developed for the present work. Sinusoidal and uniformly distributed loading is analyzed in the present work. The parametric study is undertaken to explore the effect of the side-to-thickness ratio, aspect ratio, thickness, and volume fraction index on stresses and transverse displacements.

Dynamic Analysis of FGM Plate Under Moving Load Considering Effect of Temperature

The paper used the moving element method (MEM) to analyze the dynamics of the functionally graded material (FGM) plate under the moving load considering the effect of temperature. To calculate the displacement of the plate, the study applied the Mindlin plate theory. A nine-node isoparametric element, each with five degrees of freedom, is used to model the plate element. According to the MEM, the equation of motion of the FGM plate is established based on the principle of virtual work and on a coordinate system that moves along with the moving load. The temperature field is assumed to be constant in the plane and varies across the plate thickness. By solving the governing equation of temperature transfer, it is possible to obtain a temperature distribution function. Both mechanical strain and temperature-induced strain are considered to determine plate strain. Numerical results were surveyed with different parameters and compared with published results to verify the reliability of the model. It is found that temperature significantly affects the dynamic response of the FGM plate. This study shows that the displacement of the plate increases when the temperature increase.

Geometrically nonlinear dynamic analysis of variable stiffness composite laminated and sandwich shell panels

The nonlinear free and forced vibration analyses of variable stiffness composite laminated (VSCL) and sandwich shell panels having curvilinear fibres, not yet reported in literature, are conducted based on a higher-order theory. The present structural model considers a third-order shear deformation theory incorporating Murakami zig-zag effects. The analyses are performed using C0 nine-noded isoparametric element with thirteen degrees of freedom and geometrical nonlinearity is included through von Kármán nonlinear strain–displacement relations. The free and forced vibrational responses of the VSCL are established using the eigenvalue solutions and Newmark’s time integration technique, respectively. The results obtained using the present finite element formulation are compared with the available three-dimensional exact and two-dimensional analytical and numerical solutions for establishing the validity of the model. The fundamental frequencies and nonlinear transient deflections of VSCL and sandwich spherical and cylindrical shell panels are evaluated considering the effects of curvilinear fibre path, curvature ratio, number of layers and lamination configuration.

Geometrically nonlinear bending analysis of variable stiffness composite laminated shell panels with a higher-order theory

A finite element (FE) formulation is developed by extending the third-order shear deformation theory, including the zig-zag effects using the Murakami zig-zag function, for the linear and nonlinear static analyses of variable stiffness composite laminated (VSCL) shell panels. In this work, a nine-noded isoparametric element with thirteen degrees of freedom is employed to model the VSCL shell panels. Geometrical nonlinearity according to von Kármán nonlinear strain-displacement relations is considered for the evaluation of the deflections and stresses. The assumed displacement field considers up to third-order terms for in-plane displacements and second-order terms for the transverse displacement through-the-thickness. The accuracy of the formulation is established by comparing the obtained results with the three-dimensional elasticity and two-dimensional numerical/analytical results available in the literature for static analysis of constant stiffness composite laminates (CSCL) and VSCL. New results are presented for the nonlinear deflection and stresses of VSCL shell panels considering the effect of the fibre path angles, defined by the curvilinear fibre orientation for the VSCL spherical, cylindrical and hyperboloid shells. Further, the effect of curvature ratios, number of layers, lamination configurations and boundary conditions on the nonlinear static behaviour of VSCL shells is presented.

The displacement view of a multilayered HSDT plate

This paper presents the state of displacement of a multilayered composite laminate subjected to transverse static load with varying balance, symmetric and anti-symmetric angle-ply and cross-ply staking sequences. Higher-order shear deformation theory (HSDT) is considered in the finite element formulation of nine-noded isoparametric element with seven degrees of freedom at each node. The finite element formulation is transformed into computer codes. A convergence study is carried out first to obtain the optimal mesh size for minimizing the computational time. The maximum deflection at the center of plate for both fixed and simply supported edges is verified with reported literature and a good conformity is found. An attempt has been made to observe the minimum value of maximum deflection in the laminate for attaining the maximum strength of laminate with a suitable combination of stacking sequences with a constant volume of material.

Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates with Initial Geometric Imperfections

The aim of the present study is to investigate thermo-mechanical buckling response of skew functionally graded laminated plates (FGLP) with initial geometric imperfections. The formulation has been performed using Reddy’s higher order shear deformation theory (HSDT) with the [Formula: see text] continuous displacement field. A nine-noded isoparametric element has been employed to discretize the domain of the plate. Variational principle has been used to derive the governing differential equation of the problem. Several examples with various comparison and parametric studies have been shown to prove the efficiency and effectiveness of the present formulation. The numerical results have been highlighted with different system parameters and boundary conditions.

Nonlinear transient analysis of FG pipe subjected to internal pressure and unsteady temperature in a natural gas facility

This study investigates the response of functionally graded (FG) gas pipe under unsteady internal pressure and temperature. The pipe is proposed to be manufactured from FGMs rather than custom carbon steel, to reduce the erosion, corrosion, pressure surge and temperature variation effects caused by conveying of gases. The distribution of material graduations are obeying power and sigmoidal functions varying with the pipe thickness. The sigmoidal distribution is proposed for the 1st time in analysis of FG pipe structure. A Two-dimensional (2D) plane strain problem is proposed to model the pipe cross-section. The Fourier law is applied to describe the heat flux and temperature variation through the pipe thickness. The time variation of internal pressure is described by using exponential-harmonic function. The proposed problem is solved numerically by a two-dimensional (2D) plane strain finite element ABAQUS software. Nine-node isoparametric element is selected. The proposed model is verified with published results. The effects of material graduation, material function, temperature and internal pressures on the response of FG gas pipe are investigated. The coupled temperature and displacement FEM solution is used to find a solution for the stress displacement and temperature fields simultaneously because the thermal and mechanical solutions affected greatly by each other. The obtained results present the applicability of alternative FGM materials rather than classical A106Gr.B steel. According to proposed model and numerical results, the FGM pipe is more effective in natural gas application, especially in eliminating the corrosion, erosion and reduction of stresses.

Investigation of Coating Layer to Reduce Thermal Stresses in Steel Fiber Reinforced Aluminum Metal Matrix Composite

In this study, by using coating layers to reduce thermal stresses in the metal matrix composites with a mismatch in coefficients of thermal expansions of fiber and matrix is investigated. The thermoelastic solutions based on a three cylinder model are deformed. It is shown that the effectiveness of the layer can be defined by the product of its cofficients of thermal expansions and thickness and that a compensating layer with a sufficiently high coefficient of thermal expansions can reduce the thermal stresses in the metal matrix. . In order to verify the results were compared with the finite element method. In this solution, 224 nodes and 44 nine-node isoparametric elements are used. The study is based on a three cylinder model isolating one steel fiber with a coating layer and a aluminum matrix layer. Only monotonic cooling is studied and the variation of the material properties with temperatures is not considered. The results have been presented in graphics.

Numerical-experimental investigation on vibration and bending failure of stiffened composite plates

In this paper, free vibration and bending of laminated stiffened glass fiber/polyester composite plates with laminated open section (rectangular or T shaped) and closed section (hat shaped) of stiffeners have been studied by finite element method and experiment. A nine-noded isoparametric element with nine degrees of freedom per node is developed for the plates. The stiffener element is a three-noded isoparametric beam element with five degrees of freedom per node and the stiffeners can be positioned anywhere within the plate element. The natural frequencies of the laminated stiffened plates are determined experimentally by Dewebook device and DasyLab 7.0 software. The results calculated by computational model for above plates under different boundary conditions are in good agreement with experiments. The failure problems of these stiffened glass fiber/polyester composite plates are also investigated.

A new element for analyzing large deformation of thin Naghdi shell model. Part 1: Elastic

One of the best approaches for modeling large deformation of shells is the Cosserat surface. However, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. The basic assumption of this theory is that the mid-surface of the shell is regarded as a Cosserat surface with one inextensible director. In this paper, it is shown that by constraining the director vector normal to the mid-surface, besides very good and accurate results, shear locking is also eliminated. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirichhoff’s hypothesis is enforced. Numerical solution is performed using nine-node isoparametric element. The principal of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.

Finite element study of multi-size particulate flow in horizontal pipes

Fully developed pseudo-homogeneous horizontal pipe flow of slurry with multi-size particles is modelled via Galerkin finite element method using nine-noded isoparametric elements. The momentum equation for the slurry mixture is based on a pseudo-fluid model. For each particle size (termed as specie), a convection-diffusion equation is solved to obtain the volumetric concentration distribution for that specie. A mixing length model, with a concentration-dependent eddy viscosity, is used to account for turbulence. Wall functions are used to prescribe mixture velocity at the wall. For any specified axial pressure gradient, the finite element code predicts the volumetric concentration distribution of each size fraction and the mixture axial velocity distribution. These are found to agree favourably with results published in the open literature. Overall mass conservation for each size fraction is satisfied to within 1%. The variations of concentration and mixture velocity along the pipe circumference are presented and possible insight into the sliding wear distribution along the pipe wall is deduced.

Optimum design of laminated composite plates undergoing large amplitude oscillations

The optimum design of composite laminated plates under going large amplitude free vibration is discussed. Von Karman's nonlinear strain displacement relations are considered to account for large amplitude. A higher order shear deformation theory with parabolic variation of transverse shear stresses through thickness is used in the finite element formulation. A nine-noded isoparametric element with 7 dof per node is adopted. Ritz formulation for nonlinear finite element analysis is implemented and the direct iteration method is used to solve the governing nonlinear equation. Optimization is carried out using genetic algorithm (GA) with tournament selection scheme.

Impact Response Analysis of Partially Delaminated Composite Laminates.

This paper concerns the dynamic response of partially delaminated composite laminates impacted by a rigid ball. Nine-node isoparametric elements including the influence of shear deformation were used in the finite element analysis. In order to evaluate the contact force between the impactor and the laminate, the Hertzian contact law was adopted. The Newmark algorithm was employed to solve the motion of the impactor. To deal with the dynamic contact problem between two delamlnated layers effectively, a Lagrange multiplier technique incorporated with a special time integration algorithm was presented to overcome the deficiency of the classical Lagrange multiplier method. The numerical results give much information of impact phenomena of delaminated composite laminates.

Bending and vibration analysis of composite sandwich plates

A finite element analysis of composite sandwich plates is studied in this paper. The face plates of composite sandwich structures are modeled based on the Mindlin's plate theory. The displacement fields of the sandwich core material are linearly interpolated in terms of the displacements of two face plates. According to this assumption, the transverse shear strains are linearly varied while the transverse normal strain is constant through the core thickness. Finite element analysis based on the Mindlin formulation is adopted using a nine-node isoparametric element. Static and free vibrational problems have been solved to investigate the effect of considering the transverse normal deformation of the core. It was found that when the sandwich plate is subjected to a concentrated load, the aforementioned normal deformation of the core should not be neglected. For the case of free vibration, natural frequencies decrease when the core is considered to be flexible in the transverse normal direction while the natural mode shapes do not have prominent change.

Moisture diffusion in variously shaped fibre reinforced composites

The versatility of composite material lies in its unique characteristics of high strength and stiffness combined with light weight. Like all other materials this is also susceptible to moisture coupled with temperature. Moisture susceptibility is directly associated with the diffusion of moisture through polymeric composites. Moisture diffusion coefficients of polymer composites are found to depend on a number of factors, such as moisture concentration and environmental temperature, fibre volume fraction, and the diffusivity of the fibre (if permeable) and polymer matrix. Moisture diffusion coefficients of polymer composites also depend on the shape of the reinforcing fibre. In the present investigation single fibre composite models reinforced with variously shaped reinforcing fibres are used for the evaluation of the diffusion coefficients. The finite element analysis approach is developed based on the Galerkin formulation using nine-node isoparametric elements.

Impact Behavior of Thick Laminated Composite Beams

In the present investigation a higher-order shear deformation theory (HST) and the conventional first-order theory (FST) are used to develop a finite element method to analyze the impact behavior of laminated composite beams. The higher-order theory assumes all the displacement components u, v, w that contain variation up to cubic power of z. The effects of various parameters, such as span to thickness ratios, support conditions and stacking sequence on the impact behavior of laminated composite beams are studied. Both eight-noded and nine-noded isoparametric elements are employed to compare the computational efficiency. However, to save computational time, numerical results are generated using only eight-noded isoparametric elements. The present results compare well with those of W. Goldsmith. It is observed that stresses computed using HST and FST exhibit wide variations.

Thermally induced vibration of laminated circular cylindrical shell panels

This work is concerned with a two-dimensional finite element study of the transient response of laminated thin circular cylindrical shell panels subjected to thermal impact under the framework of linear coupled thermoelasticity. A nine-node isoparametric element is applied to the FEM formulation. Numerous examples are given, in which the shell panels are assumed to be thermally isolated at four edges and the bottom surfaces and a heat flux is then imposed suddenly on their top surfaces. The effects of boundary conditions, ply-angles and radii of curvature are discussed. In addition, the differences between uncoupled and coupled problems are especially addressed.

Non-linear static and dynamic analysis of heated laminated plates: A finite element approach

The static and dynamic thermoelastic responses of laminated plates have been investigated by the use of a shear flexible finite element model. The model accounts for large deflections of the plate and non-uniform distributions of temperature. A nine-noded isoparametric element is implemented to obtain the numerical solutions. The validity of the model is demonstrated by comparison with solutions available in the literature. A wide variety of results are presented for the non-linear response of rectangular and circular plates under thermal and thermo-mechanical loads. The influences of anisotropy, the boundary conditions, aspect ratio, rotary inertia and stacking sequence on the thermally induced response are studied.

Bending and free vibration analysis of shear deformable laminated composite beams by finite element method

In the present investigation a higher-order shear deformation theory and the conventional first-order theory are used to develop a finite element method to analyse accurately the bending and free vibration behaviour of laminated composite beams, using nine-noded isoparametric elements. The higher-order theory assumes all the displacement components, u, v and w, which contain variation up to a cubic power of z. The effects of various parameters such as fibre orientation, stacking sequence, span-to-thickness ratio and support condition on the non-dimensionalised deflections, stresses and fundamental frequencies are investigated. Cases where only the higher-order theory is likely to yield accurate results are highlighted.