Quadratic Isoparametric Elements Research Articles

Static and free vibration analysis of delaminated composite plates using secant function based shear deformation theory

In this work, the static and free vibration analysis of delaminated composite plates are studied. The secant function based shear deformation theory (SFSDT), is used for the analysis of multiple delaminated models. This theory exhibits a non-linear shear stress distribution and fulfills the traction-free boundary conditions at the top and bottom surfaces of the plates, hence there is no need for a shear correction factor. The finite element method, a numerical solution-based approach is used here considering an eight-noded isoparametric quadratic element. The investigation explores the effects of delamination size, position and stacking sequence on the flexural and dynamic characteristics of the composite plates. Additionally, other geometric factors such as delamination thickness and shape are investigated to understand their impact. The study aims to identify the optimal configurations that minimize the detrimental effects of delaminations. The results provide valuable insights into the static and free vibration behaviour of delaminated composite plates, contributing to design guidelines and optimization strategies for structurally sound and efficient composite structures. Overall, this research offers a comprehensive investigation of the static and free vibration analysis of delaminated composite plates, advancing knowledge in the field and facilitating improved design practices.

An isoparametric quadratic boundary element for coupled stretching-bending analysis of thick laminated composite plates with transverse shear deformation

Recently, we derived the Green's function for an infinite thick laminated composite plate subjected to concentrated forces and moments. By serving this Green's function as the fundamental solutions, in this paper for the first time we develop the associated boundary element method (BEM), which is applicable to general problems of thick laminated plates with various types of deformations. For example, the in-plane stretching and out-of-plane bending deformations may be coupled together, which occurs for the unsymmetric laminated composite plates. Besides, the transverse shear deformations are also incorporated based upon the first order plate theory. To complete the entire development of the proposed BEM, all the involved issues, such as the boundary integral equations, the fundamental solutions and their derivatives, the mathematical formulation for the selected isoparametric quadratic boundary element, and the treatments of singular integrals are elaborated in this paper. The calculations of complete solutions at boundary or internal points are provided as well. The accuracy and efficiency of the proposed BEM are demonstrated through numerical examples, which show that it outperforms the conventional finite element method and is applicable for general thick laminated plates with unrestricted coupling effects.

Two-dimensional radial integration displacement discontinuity method with high-order isoparametric elements

To improve the accuracy of Displacement Discontinuity Method and enhance its adaptivity, a general-purpose two-dimensional displacement discontinuity method with both linear and quadratic isoparametric elements has been developed to model engineering discontinuity problems where cracks or cavities are involved. Linear and quadratic isoparametric elements have linear and quadratic distributions of displacement discontinuity, respectively. Both of them belong to the discontinuous element type, in which the geometry shape functions are different from the interpolation shape functions. A new general formulation, based on the boundary integral functions, is given for displacement discontinuity problems with arbitrary boundary shapes. This formulation contains singular and strongly singular integrals which can be evaluated in the sense of Cauchy principal value and Hadamard principal value, respectively. The radial integration technique is applied to perform these singular integrals with sufficiently high accuracy. Five numerical examples, most of which have analytic solutions, are given to illustrate the improved accuracy of the proposed approach. Compared with the constant displacement discontinuity element, the numerical results by using the present isoparametric displacement discontinuity elements show better accuracy.

Effect of Orientation of Laminates on Bending Behaviour of Delaminated Composite Conoidal Shell Roofs

Literature review shows that research works on the bending behaviour of delaminated composite conoidal shells on rectangular planform are very few in number. In order to fill up this gap a study on static analysis of delaminated composite conoidal shells is done using finite element method. An eight noded curved quadratic isoparametric shell element having five degrees of freedom per node is used for the present work. To ensure the compatibility of deformation and equilibrium of forces and moments at the delamination crack tip a multiple constraint algorithm is developed and incorporated. Results of the present model are validated with help of some benchmark problems available in literature. The study is done to characterize delaminated composite conoids with different types of conventional boundary conditions. Parametric studies are done for angle of orientation, number of layers of laminates and degree of fixity of shells under uniformly distributed loading. Finally after a careful and thorough investigation some pin point conclusions is made as outcome of present study.

Immersed boundary Mindlin-Reissner 3D shell element for modeling isotropic and laminated composite shells

An immersed boundary 3D shell element is presented here that is based on Mindlin-Reissner shell theory assumptions and uses quadratic B-spline approximation for the solution. It enables mesh independent analysis wherein the surface representing the shell geometry is defined independently and not by the mesh. The shell geometry is immersed in a uniform background mesh whose elements are 3D B-spline shell elements. By making the geometric model independent of the mesh, the typical difficulties associated with mesh generation can be avoided. The quadratic 3D B-spline shell elements in the mesh can represent the displacement field as a tangent continuous piece-wise polynomial approximation. The nodes of the element have three translational and no rotational degrees of freedom. The element is formulated under small-deformation and plane stress assumptions. Constructing the element based on shell theory enables the use of effective properties to model thin fiber reinforced laminated composite shell-like structures. The element formulation and the mesh independent approach are validated against a series of common shell element benchmark problems. • Immersed boundary 3D B-spline element based on Mindlin-Reissner shell theory assumptions. • Mesh independent analysis where geometry is embedded in a uniform background mesh. • The iso-parametric quadratic B-spline shell elements are used for C 1 continuous approximation. • Step boundary method (SBM) is used for applying boundary conditions. • Immersed boundary FEM for shells is validated for isotropic and composite shells.

Isoparametric numerical integration on enriched 4D simplicial elements

The enriched finite element method is a natural improvement of the classical finite element method for solving complicated engineering problems. It is an essential tool for successful treating difficulties, singularities, non-Lipschitz internal boundaries, curved boundaries with complex geometry, crack propagations, etc. This paper deals with an enrichment of the four-dimensional isoparametric quadratic Lagrangian finite element. The paper discusses a new approach in which bubble functions are replaced with specific cubic ones in the enrichment process. The new method is applied to simplicial elements in four-dimensional Euclidean space, where lumping of the mass matrix is obtained. Error estimates in numerical integration and the interpolation by the new elements are proved. Applications of the quadrature rules for solving elliptic eigenvalue problems are demonstrated. A new operator that maps tesseract onto a four-dimensional ball is defined. The new quadrature rules are tested by integrating Bessel functions and by solving a second-order elliptic eigenvalue problem with Dirichlet boundary conditions.

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

This paper emphasized the size-dependent thermo-elastic vibration characteristics of uncertain functionally graded (FG) porous nano-beams by employing the first-order perturbation theory (FOPT) with the finite element method. An isoparametric quadratic beam element having three nodes is used for the finite element analysis of the nano-beams with variation in degree of uncertainty in material properties, geometric configuration, and thermal environment. The power-law model is employed to obtain the effective material properties of the considered beams. The governing equations are presented according to Eringen’s elasticity theory and Reddy’s beam theory using the minimum potential energy. The computed FOPT results for the vibration statistics are assessed with the help of Monte Carlo simulation (MCS) and a well agreement is found. Further, results are presented by analyzing the effects of degree of source uncertainty, size-dependent parameter, volume fraction indices, porosity distribution, porosity index, thermal environment, and aspect ratios on the fundamental frequency of the FGM nano-beams. The numerical results revealed the significant influence of thermal environment and porosity on the vibration statistics of the nano-beams and provided guidance for the precise and reliable design of nano-devices for the application in thermal environment.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.

A modified higher order zigzag theory for response analysis of doubly curved cross-ply laminated composite shells

The present theory deals with static and free vibration analyses based on higher order zigzag theory for laminated composite shells. The finite element shell model is developed considering the thickness coordinate to radius ratio (z/R) using nine noded isoparametric quadratic element. The proposed formulation satisfies the interlaminar stress continuity at the interface of laminae and zero shear boundary conditions at free surfaces. The response results obtained by the proposed formulations are compared with the published results. Moreover, new responses results of hyperbolic paraboloidal laminated shells are obtained using the present laminated shell theory and illustrated in the present study.

Stochastic low-velocity impact analysis of sandwich plates including the effects of obliqueness and twist

Abstract This paper quantifies the influence of uncertainty in the low-velocity impact responses of sandwich plates with composite face sheets considering the effects of obliqueness in impact angle and twist in the plate geometry. The stochastic impact analysis is conducted by using finite element (FE) modelling based on an eight nodded isoparametric quadratic plate bending element coupled with multivariate adaptive regression spline (MARS) in order to achieve computational efficiency. The modified Hertzian contact law is employed to model contact force and other impact parameters. Newmark's time integration scheme is used to solve the time-dependent equations. Comprehensive deterministic as well as probabilistic results are presented by considering the effects of location of impact, ply orientation angle, impactor velocity, impact angle, face-sheet material property, twist angle, plate thickness and mass of impactor. The relative importance of various input parameters is determined by conducting a sensitivity analysis. The results presented in this paper reveal that the impact responses of sandwich plates are significantly affected by the effect of source-uncertainty that in turn establishes the importance of adopting an inclusive stochastic design approach for impact modelling in sandwich plates.

Effect of hybridisation in laminated composites on the first ply failure behaviour: Experimental and numerical studies

First ply failure analysis of hybrid laminated plate panels subjected to transversely applied static load is carried out and presented in this article. Hybrid laminates are fabricated in the laboratory as a combination of carbon and glass fabrics along with epoxy resin for this purpose. Strength and elastic properties of epoxy based laminates with carbon fibre (CFRP) and glass fibre (GFRP) are determined experimentally according to ASTM standard. Various multilayered hybrid laminates (HFRP) and GFRP laminates having all edges fixed boundary condition are tested under transverse static load. An improved eight noded quadratic isoparametric plate bending element based on refined higher order zigzag theory (HRZT) has been developed in the present study to evaluate the interlaminar stresses of multilayered composite laminates. The warping function has been considered in the displacement field to formulate a C° continuous plate bending element. The transverse shear stresses are computed in a simplified manner using the three dimensional differential equations of stress equilibrium. The first ply failure load is determined within linear and elastic range. The correctness and competence of the present finite element model is verified by comparing the solutions with relevant published data and also experimental outcome. The comparison between HFRP and GFRP laminates in terms of failure load has been made to identify the effectiveness of hybridisation.

Comparative study on transient response analysis of hybrid laminated composite plates with experimental verification

This paper attempts in analysing the transient dynamic response behaviour of multi layered hybrid composite plates by a modified higher order refined zigzag theory (HRZT). An improved C0 continuous model is developed based on the refined higher order zigzag theories (HRZT) with warping effect for multilayered plates using eight noded isoparametric quadratic plate bending element to get the stress distribution across the thickness under dynamic load in a realistic manner. The continuity condition of transverse shear stresses is ensured with the help of 3D equilibrium equation at the interfaces of two adjacent layers along with the shear stress free condition at the outermost faces of a laminate. A computer code is developed with the help of MATLAB software package. The hybrid laminate is modeled by placing carbon fibres in the outermost layers and glass fibres in the rest and thus the flexural behaviour is improved. The experimental study on the transient dynamic response of hybrid laminates is also performed. Transient dynamic response obtained from the finite element based numerical model is verified by comparing the time history data obtained numerically and experimentally. The interlaminar stresses are also extracted and recorded under dynamic loading condition. The displacement and the stress components of laminated plates under static load are also computed in the present study and the results are compared with the theoretical outcome obtained in the relevant literature for validation of the present FE formulation. An exhaustive study on glass and hybrid fibre reinforced laminates is carried out to investigate the effectiveness of the hybrid laminate in terms of the response data of laminates with mono fibre as well as multiple (hybrid) fibres.

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.

Isogeometric boundary element method for calculating effective property of steady state thermal conduction in 2D heterogeneities with a homogeneous interphase

Based on the generalized self-consistent scheme (GSCS), the isogeometric boundary element method (IGABEM) is used to calculate the effective thermal conductivity of two dimensional (2D) steady state heat conduction heterogeneities with a homogeneous interphase. The heat energy formulation and the boundary integral equations adopted in this paper only contain the temperatures from the inclusion–interphase and interphase–matrix interfaces, respectively, so that the effective thermal conductivity from the generalized self-consistent model with a homogeneous interphase can be conveniently calculated. In numerical implementation, the Non-Uniform Rational B-Splines (NURBS) are employed not only to construct the exact interface shapes but also to approximate the interface temperatures. The results show that the conductivity and the thickness of the homogeneous interphase have great influence on the effective thermal conductivity. Higher interphase thermal conductivity enhances the effective thermal conductivity, whereas thinner thickness of the interphase results in the reduction of the effective thermal conductivity. Numerical examples show that the proposed method works well compared with the conventional quadratic isoparametric boundary element method.

Stochastic dynamic analysis of twisted functionally graded plates

Abstract This paper presents a stochastic dynamic analysis of functionally graded plates by following an efficient neural network based approach coupled with the finite element method. An isoparametric quadratic element having eight nodes is considered for the finite element analysis of pre-twisted functionally graded cantilever plates subjected to variation in geometric parameters, material properties and temperature. Both individual and compound effects of stochasticity in the uncertain input parameters are accounted to quantify their influence on the first three natural frequencies, mode shapes, and frequency response functions of functionally graded plates. A sensitivity analysis is conducted to ascertain the relative effects of various prospective sources of uncertainty. Latin hypercube sampling is utilised to train the efficient surrogate models, which are employed as a medium of uncertainty propagation. The comparative performance of artificial neural network and polynomial neural network is assessed in the stochastic dynamic analysis of the pre-twisted functionally graded plates from the viewpoint of accuracy and computational efficiency. The results are validated with respect to direct Monte Carlo simulation based on the finite element model of the functionally graded plates. It is observed that the artificial neural network based algorithm can achieve a significant level of computational efficiency without compromising the accuracy of results. The results presented in this article reveal that the source uncertainties of functionally graded plates have a significant effect on the dynamic responses of the structure.

Element differential method with the simplest quadrilateral and hexahedron quadratic elements for solving heat conduction problems

In this article, two types of new quadrilateral and hexahedron quadratic isoparametric elements are proposed for the element differential method (EDM) for solving heat conduction problems. These elements, called as the Ultra elements, have the minimum numbers of nodes comparing with the existing elements and have the feature that a central node is included inside them, which is necessary for the EDM analysis. The EDM is a strong-form method, which does not require control volumes and any integration. In the previous EDM for solving heat conduction problems, the Lagrange elements were used, which had many elemental nodes. The proposed new types of elements can circumvent this deficiency, in which only a few nodes are required. The shape functions for these elements are constructed for the first time, and the first and the second order derivatives of the shape functions with respect to intrinsic and global coordinates are analytically derived. Several 2D and 3D numerical examples are given to demonstrate the effectiveness, the accuracy and the efficiency of the newly proposed elements for the EDM for solving heat conduction problems.

Novel shear deformation model for moderately thick and deep laminated composite conoidal shell

ABSTRACTThis article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C1 continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.

Dynamic Analysis of Sandwich Plate Structure by Finite Element Method

An 8-noded quadratic isoparametric plate bending finite element that incorporates first-order transverse shear deformation and rotary inertia is used to predict the free vibration response of sandwich plate structures. A programme has been developed using MATLAB. The finite element results presented here show good agreement with the available semi-analytical solutions and finite element results. Parametric studies have been conducted by incorporating variation in support conditions, fibre angles of the skins and overall thickness and detail interpretations are provided.

Simply supported composite hypar shells under free vibration–some observations

A general finite element procedure is presented to model the hypar shells using eight ndash noded curved quadratic isoparametric elements with five degrees of freedom per node including two inplane displacements and one transverse displacement and two rotations Problems of twisted cantilever plate which have structural resemblance with skewed hypar shell are solved using the present approach and the results are compared with published ones Having established the exactitude of the present formulation numerical experiments with simply supported skewed composite hypar shells are conducted for four different types of laminations including four layered symmetric and antisymmetric cross and angle ply laminates The first four natural frequencies are presented in tabular forms and are studied critically and a set of meaningful conclusions are derived nbsp

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.