Present Finite Element Formulation Research Articles

Stochastic impact responses analysis of functionally graded plates

The present paper portrays the mapping for stochasticity in low-velocity impact responses of functionally graded material (FGM) plates by employing multivariate adaptive regression splines (MARS) surrogate model in conjunction with finite element (FE) approach. The unavoidable stochastic variabilities (caused due to numerous errors involved in manufacturing processes) in material properties of FGM plates are considered in order to map the effect of elemental variabilities on global response of the structure. The material properties of FGM plates are considered to follow the rule of mixture in conjunction to power law. The Newmark’s time integration scheme and modified Hertzian contact law are employed to solve the time-dependent equation. The present FE formulation is based on an eight noded isoparametric element in which each element has five degrees of freedom. The effects of variability in temperature and power-law exponent on stochastic low-velocity impact responses are also portrayed. The maximum contact force, plate and impactor displacement are considered as the response parameters. The present MARS model is coupled with the finite element to achieve the higher efficiency with adequate accuracy as compared to the FE-based full-scale Monte Carlo simulation. The statistical results illustrate that the stochasticity in material properties significantly influences the low-velocity impact responses of FGM plates.

The Cosserat Point Element as an Accurate and Robust Finite Element Formulation for Implicit Dynamic Simulations

The reliability of numerical simulations manifested the need for an accurate and robust finite element formulation. Therefore, in the present study, an eight node brick Cosserat point element ( CPE ) for the nonlinear dynamic analysis of three-dimensional (3D) solids including both thick and thin structures is developed. Within the present finite element formulation, a strain energy function is proposed and additively decoupled into two parts. One part is characterized by any 3D strain energy function, while the other part controls the response to inhomogeneous deformations. Several example problems are presented, which demonstrate the accuracy and the robustness of the developed CPE in modeling the dynamic response of elastic structures.

Parametric Study on the First Ply Failure Load of Delaminated Shallow Pretwisted Conical Shells – A Finite Element Approach

The present study deals with the effect of offset length on the first ply failure load of undelaminated and mid plane delaminated composite shallow conical shells. Numerical results were obtained using finite element codes for the failure loads of undelaminated and mid plane delaminated composite conical shell for different offset length considering twisted and untwisted instance. From Lagrange’s equation generalized dynamic equations have been derived for moderate rotational speed ignoring the Coriolis Effect. Based on Mindlin’s theory the theoretical formulation has been carried out and it has been implemented by using an isoparametric plate bending element having eight nodes. Multi-point constraint algorithm has been used to ensure the equilibrium of resultant forces, moments and compatibility of deformation at delamination crack front. The standard Eigen value problem has been solved based on standard QR iteration algorithms. Convergence studies are carried out considering mesh sizes, and results are validated with open literature to verify the correctness of the present finite element formulation. The first ply failure load considering different failure criteria namely Max stress (independent), Tsai Hill, Tsai Wu Hahn and Tsai Hill Hoffman are evaluated. The variation of the failure loads for different offset length and twist angle are illustrated and presented.

Vibration Analysis of Woven Fiber Metal Laminated Plates — Experimental and Numerical Studies

The present study deals with numerical and experimental investigations on the vibration behavior of fiber-metal-laminated (FML) plates, a new aircraft material. A finite element (FE)-based formulation is established for the plate using the first-order Reissner–Mindlin theory, including both fibers and metals of different material properties in alternate layers. A four-node isoparametric quadratic element with five degrees of freedom per node is adopted in the analysis. Convergence studies and comparison with previous studies are made to validate the present FE formulation. A set of experiments was conducted to get natural frequencies of vibration for glass FML (GFML) plates using Bruel and Kjaer (B&K) Fast Fourier Transform (FFT) analyzer with PULSE platform. The effects of different parameters such as aspect ratio, side-to-thickness ratio, ply orientation, and boundary conditions on the dynamic behavior of the FMLs are studied. Good agreement is achieved between the numerical and experimental results. Both results indicate that increasing the aspect ratio can increase the natural frequency of the FML plate, while the increase in the side-to-thickness ratio decreases the natural frequency of vibration. The boundary conditions can significantly affect the natural frequency of the FML plates due to the restraint effect at the edges.

Analysis of functionally graded sandwich plates using a higher-order layerwise theory

A higher-order layerwise finite element formulation is presented for static and dynamic analyses of functionally graded material (FGM) sandwich plates. A higher-order displacement field is assumed for core and first-order displacement field is assumed for top and bottom facesheets maintaining a continuity of displacement at layer interface. An eight noded isoparametric element using a C0 based finite element formulation with thirteen degrees of freedom per node has been considered in the present work. Two configurations of FGM sandwich plates, one with FGM core and homogenous facesheets and second having top and bottom layers made of FGM and homogenous core are considered. Effective material properties of the FGM are computed using rule of mixture (ROM). In order to establish the correctness of the present finite element formulation for wide range of problems for two configurations of FGM sandwich plates, comparison studies are presented. Next, parametric studies are taken up to investigate the effects of volume fraction index, span to thickness ratio and boundary conditions on static and dynamic behavior of FGM sandwich plate. It is shown here that present formulation is simple, straightforward and accurate for static and dynamic analyses of functionally graded material (FGM) sandwich plates.

Dynamic characterization of the thickness-tapered laminated plant fiber-reinforced polymer composite plates

Evolution of the laminated woven natural fiber fabric-reinforced polymer composite structures makes a way to the development of the non-uniform laminated composite structures in order to achieve the stiffness variation throughout the structure. An attempt is made in this work to carry out the experimental and numerical investigations on the dynamic characteristics of the thickness-tapered laminated woven jute/epoxy and woven aloe/epoxy composite plates. The governing differential equations of motion for the thickness-tapered laminated composite plate are developed using the h-p version FEM based on higher order shear deformation theory. The validation of the present finite element formulation is carried out by comparing the natural frequencies obtained using the finite element formulation with those natural frequencies determined experimentally. The developed model is further validated with the available literature works on tapered composite plate to confirm the efficiency of h-p version FEM. This work also explores the study of the vibrational characteristics of composite plates under the influence of plant fiber’s transverse isotropic material characteristics and porosity associated with plant fiber composites through the elastic constants evaluated in the author’s previous work. Also the influences of aspect ratios, ply orientations, and taper angles under various end conditions on the natural frequencies of the woven jute/epoxy composite plate are studied using the present finite element formulation. The forced vibration response of the thickness-tapered laminated woven jute/epoxy composite plate under the harmonic force excitation is carried out considering CFCF and CFFF end conditions.

Numerical Analysis of Stress Concentration in Isotropic and Laminated Plates with Inclined Elliptical Holes

The design of high-performance composite structures frequently includes discontinuities to reduce the weight and fastener holes for joining. Understanding the behavior of perforated laminates is necessary for structural design. In the current work, stress concentrations taking place in laminated and isotropic plates subjected to tensile load are investigated. The stress concentrations are obtained using a recent quadrilateral finite element of four nodes with 32 DOFs. The present finite element (PE) is a combination of two finite elements. The first finite element is a linear isoparametric membrane element and the second is a high precision Hermitian element. One of the essential objectives of the current investigation is to confirm the capability and efficiency of the PE for stress determination in perforated laminates. Different geometric parameters, such as the cutout form, sizes and cutout orientations, which have a considerable effect on the stress values, are studied. Using the present finite element formulation, the obtained results are found to be in good agreement with the analytical findings, which validates the capability and the efficiency of the proposed formulation. Finally, to understand the material parameters effect such as the orientation of fibers and degree of orthotropy ratio on the stress values, many figures are presented using different ellipse major to minor axis ratio. The stress concentration values are considerably affected by increasing the orientation angle of the fibers and degree of orthotropy.

A layerwise finite element for geometrically nonlinear analysis of composite shells

This work aims to develop a nonlinear layerwise shell element formulation for shear-deformable laminated composite plate and shell structures. The element is formulated based on a zigzag theory in presence of individual local coordinates in the thickness direction for separate layers. In order to properly employ the zigzag theory, the considered local coordinates have different ranges of variation for middle, upper and lower layers. Using Mindlin-Reissner theory a convenient displacement field is derived for each layer and an ordered algorithm is adapted to calculate increments in the director vector of each layer due to relative finite rotations of its adjacent layers. Employing this shear deformable displacement field in the principle of virtual displacement leads to integral governing equations with various through-the-thickness parameter ranges. To overcome this challenge, the stress and strain tensors are rewritten in terms of through-the-thickness parameters and an explicit integration is performed in thickness direction. In this way, the nonlinear formulation is derived using the updated Lagrangian approach in accompany with a particular linearization scheme. To assess the performance of the present finite element formulation, a proprietary nonlinear finite element program is developed. Some illustrative problems are solved and comparisons with available solutions are presented.

Spatial vulnerability analysis for the first ply failure strength of composite laminates including effect of delamination

The present investigation deals with the first ply failure strength of laminated composite plates for spatial variation of loading position, which can render a clear idea about the locational sensitivity of the loading positions in a two dimensional space. In this context, the effect of delamination is investigated on the failure strengths considering angle ply and cross ply laminates. A finite element model is developed based on different failure criteria of composites, such as maximum stain, maximum stress, Tsai-Hill, Tsai-Wu-Hahn and Tsai-Hill-Hoffman. An eight noded isoparametric quadratic element is considered in the present finite element formulation incorporating transverse shear and rotary inertia. Results are presented in deterministic as well as stochastic regime. For obtaining the probabilistic descriptions of failure strengths following different failure criteria, Monte Carlo simulation is carried out in conjunction with the finite element model following a non-intrusive approach. The variation of failure strength is portrayed considering the effect of stacking sequence, ply orientation, number of layers, degree of orthotropy and ply thickness. In this article, consideration of stochastic material and structural attributes along with critical service-life characteristics such as delamination for spatially varying loading positions provides a comprehensive understanding about the failure strength of composite laminates for practical applications.

Dynamic Stability of Woven Fiber Laminated Composite Shallow Shells in Hygrothermal Environment

The dynamic stability of bidirectional woven fiber laminated glass/epoxy composite shallow shells subjected to harmonic in-plane loading in hygrothermal environment is considered. An eight-noded isoparametric shell element with five degrees of freedom is used in the analysis. In the present finite element formulation, a composite doubly curved shell model based on first-order shear deformation theory (FSDT) is used for the dynamic stability analysis of shell panels subjected to hygrothermal loading. A program is developed using MATLAB for the parametric study on the dynamic stability of shell panels under the hygrothermal field. The effects of various parameters like static load factor, curvature, shallowness, temperature, moisture, stacking sequence and boundary conditions on the dynamic instability regions of woven fiber glass/epoxy shell panels are investigated. The location of dynamic instability regions is shown to affect significantly due to presence of the hygrothermal field.

First-ply Failure Analysis of Delaminated Rotating Composite Conical Shells: A Finite Element Approach

The present analysis evaluates the first-ply failure load of delaminated composite conical shells. Single and multiple delamination is considered across the thickness of the conical shell. The dynamic equilibrium equation considers only moderate rotational speeds for which Coriolis effect is neglected. Convergence studies are performed in respect of mesh sizes, and comparisons of the present solutions and those reported in open literature are provided to verify the accuracy of the present finite element formulation. Computer codes are developed to obtain the numerical results for the failure loads of delaminated composite conical shells. The first-ply failure load considering different failure criteria namely Maximum stress (independent), Tsai-Hill, Tsai-Wu-Hahn and Tsai-Hill-Hoffman are evaluated. The variation of the failure loads with the position of delamination across the thickness as well as along longitudinal direction are illustrated and presented.

Influence of porosity on the flexural and vibration response of gradient plate using nonpolynomial higher-order shear and normal deformation theory

The present article deals with flexural and vibration response of functionally graded plates with porosity. The basic formulation is based on the recently developed non-polynomial higher-order shear and normal deformation theory by the authors’. The present theory contains only four unknowns and also accommodate the thickness stretching effect. The effective material properties at each point are determined by two micromechanics models (Voigt and Mori–Tanaka scheme). The governing equations for FGM plates are derived using variational approach. Results have been obtained by employing a C0 continuous isoparametric Lagrangian finite element with eight degrees of freedom per node. Convergence and comparative study with the reported results in the literature, confirm the accuracy and efficiency of the present model and finite element formulation. The influence of the porosity, various boundary conditions, geometrical configuration and micromechanics models on the flexural and vibration behavior of FGM plates is examined.

Hygrothermal effects on buckling of composite shell-experimental and FEM results

The effects of moisture and temperature on buckling of laminated composite cylindrical shell panels are investigated both numerically and experimentally. A quadratic isoparametric eight-noded shell element is used in the present analysis. First order shear deformation theory is used in the present finite element formulation for buckling analysis of shell panels subjected to hygrothermal loading. A program is developed using MATLAB for parametric study on the buckling of shell panels under hygrothermal field. Benchmark results on the critical loads of hygrothermally treated woven fiber glass/epoxy laminated composite cylindrical shell panels are obtained experimentally by using universal testing machine INSTRON 8862. The effects of curvature, lamination sequences, number of layers and aspect ratios on buckling of laminated composite cylindrical curved panels subjected to hygrothermal loading are considered. The results are presented showing the reduction in buckling load of laminated composite shells with the increase in temperature and moisture concentrations.

A Nonclassical Finite Element Approach for the Nonlinear Analysis of Micropolar Plates

Based on the micropolar elasticity theory, a size-dependent rectangular element is proposed in this article to investigate the nonlinear mechanical behavior of plates. To this end, a novel three-dimensional formulation for the micropolar theory with the capability of being used easily in the finite element approach is developed first. Afterward, in order to study the micropolar plates, the obtained general formulation is reduced to that based on the Mindlin plate theory. Accordingly, a rectangular plate element is developed in which the displacements and microrotations are estimated by quadratic shape functions. To show the efficiency of the developed element, it is utilized to address the nonlinear bending problem of micropolar plates with different types of boundary conditions. It is revealed that the present finite element formulation can be efficiently employed for the nonlinear modeling of small-scale plates by considering the micropolar effects.

Piecewise shear deformation theory and finite element formulation for vibration analysis of laminated composite and sandwich plates in thermal environments

In this paper, a piecewise shear deformation theory for laminated composite and sandwich plates is presented by integrating the advantages of the layerwise theory and equivalent single layer theory. A C0 continuous four-noded quadrilateral isoparametric plate element based on this theory is developed for free and forced vibration analysis of laminated composite and sandwich plates in thermal environments. The accuracy and effectiveness of piecewise shear deformation theory and finite element formulation are validated by comparing the numerical results obtained by the present finite element formulation with the analytical and exact results available in literatures as well as the numerical results computed by MSC.Nastran software. It is demonstrated that the piecewise shear deformation theory and finite element formulation are suitable for the vibration analysis of thin and thick laminated composite and sandwich plates. As compared with the layerwise theory, high-order zigzag theory and high-order equivalent single layer theory, the piecewise shear deformation theory is able to produce a sufficiently accurate result at a very low computational cost. The work reported in this paper provides an efficient modeling approach for thermal vibration analysis of laminated composite and sandwich plates in practical engineering.

Electro-Magneto-Elastic Response of Laminated Composite Plate: A Finite Element Approach

The flexural behaviour of the laminated composite plate embedded with two different smart materials (piezoelectric and magnetostrictive) and subsequent deflection suppression have been investigated in this article. The mathematical model of the laminated composite plate embedded with and without smart materials are developed using the higher-order shear deformation kinematics in conjunction with finite element steps. The plate is assumed to be subjected to the combined effect of the mechanical load, electrical potential and the magnetic field induction. The desired responses are computed numerically with the help of a homemade computer code developed in MATLAB environment in association with the present finite element formulation. The convergence and the validity of the presently computed numerical responses have been established by comparing the responses with those available numerical and analytical results. Further, the desired responses of the laminated composite structure bonded with and without functional materials are computed using the commercial finite element package (ANSYS) and compared with the present numerical results. Finally, the present higher-order model is extended to examine the static responses of the laminated composite plate embedded with piezo and magnetostrictive material by solving the wide variety of numerical examples for different design parameters. The degree of deflection suppression capability due to the smart layers has been underlined and discussed in details.

Geometric nonlinear finite element and genetic algorithm based optimal vibration energy harvesting from nonprismatic axially functionally graded piezolaminated beam

The present article deals with optimal vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated beam using genetic algorithm (GA). Geometric nonlinear based finite element (FE) formulation using Newmark method in conjunction with Newton-Raphson method is formulated to solve the obtained governing equation. A two noded beam element with two degrees of freedom (DOF) at each node is used in the present FE formulation. The FG material (i.e. non-homogeneity) in the axial direction is considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) are modelled using the Euler-Bernoulli beam theory. The effects of nonlinearity on the responses (such as displacement, voltage and output power) are deliberated with arbitrary power gradient index. The effects of tapers (both width and height in length directions) on the output power and voltage are analysed. A real-coded genetic algorithm based constrained optimization technique is also proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of beam and breakdown voltage of PZT sensor.

Stochastic Damped Free Vibration Analysis of Composite Sandwich Plates

In this paper, stochastic damped free vibration analysis of composite sandwich plates has been carried out using a nine node Heterosis plate bending element based on a first order shear deformation theory with a-priori shear correction factors. The plate bending element contains one transverse displacement and two rotations of the normals about the plate's mid-plane. Selective reduced integration scheme is adopted to integrate terms associated with the stiffness matrix formulations. Both lumped and consistent mass matrices are considered in the analysis. The accuracy and reliability of the present finite element formulation is verified with previously published results in the literature. New results are presented which are beneficial for designers of composite structures.

Finite element formulations and algorithms for coupled multiphysics analysis of 0-1 and 0-3 polarized PLZT actuators

The present paper derives a 3-D finite element formulation and an iterative algorithm for multiphysics analysis of coupled anomalous photovoltaic, photo-thermal and pyroelectric effects in PbLaZrTi (PLZT) ceramics. The distinction in the computation aspects of electrical potential, electrical field strength and electrical charge for 0-1 and 0-3 polarized PLZT wafers are discussed thoroughly. The element performance is improved by using an assumed natural strain approach and an enhanced assumed strain method. The transverse displacements of a bimorph 0-1 polarized PLZT actuator were predicted using the present finite element formulation and then verified with the existing analytical solution and experimental results. The static analysis of a cantilevered beam bonded with a 0-1 or 0-3 polarized PLZT actuators was also presented and compared with the theoretical solutions. The numerical simulations demonstrate that the present formulations and algorithm possess superior performance.

Micromorphic first-order shear deformable plate element

Based on the micromorphic theory (MMT), a size-dependent plate element is developed for the finite element analysis of materials with considering the microstructure effect. To this end, the strain energy and constitutive relations of MMT are generally written first. The relations are represented in the matrix form so as to obtain the finite element matrices. Then, based on the first-order shear deformation theory, the matricized formulation is reduced for the case of a plate model, and the micromorphic plate element is formulated accordingly. In order to show the efficiency of the developed element, it is utilized to address the bending problem of micromorphic plates subject to various kinds of boundary conditions. In the numerical results, the effects of small scale and other parameters on the bending behavior of micromorphic plates are studied. It is revealed that the present finite element formulation can be efficiently used in the analysis of small-scale structures owing to considering micro-deformation and micro-rotation degrees of freedom of material particles.