Noded Isoparametric Element Research Articles

Free Vibration Analysis of Composite Hat-stiffened Panels By Method of Finite Elements

Free vibration analysis of laminated composite plates/shells stiffened with ‘open’ and ‘closed’ type sections as stiffeners have been carried out in the present investigation. The plate/shell has been analyzed using eight noded isoparametric shell elements and the stiffeners have been modeled using three noded isoparametric curved beam elements. There is no restriction on the positioning of stiffener elements within a shell element. The stiffness and mass matrices of the stiffener and plate/shell elements are calculated irrespective of the stiffener position within the shell element. A unified formulation describing the rigidity matrix of laminated stiffeners having open (rectangular, ‘T’, ‘L’, and ‘I’ shaped) and closed type (hat or box shaped) sections has been presented. Free vibration analysis of laminated plates/shells with open and closed section stiffeners have been analysed using the method of finite elements. Numerical results for the free vibration analysis solved by the present formulation are presented and compared with published results.

Dynamic response of multidirectional composites in hygrothermal environments

The present paper deals the effects of temperature and moisture on the free vibration and transient response of multidirectional composites. A three-dimensional finite element analysis procedure is developed using 20 noded isoparametric quadratic elements. The analysis accounts for the degradation of composite properties due to both temperature and moisture concentration. A typical multidirectional unit cell is assumed to consist of several unidirectional composite blocks. The transformation based relationships are used to generate the stiffness properties of multidirectional composite plates. The numerical results for the natural frequencies and transient response of multidirectional composites under the action of both temperature and moisture concentration are presented and discussed.

The large rotations theory of elasto‐viscoplastic shells subjected to the dynamic and thermal loads

In this paper, the problem of the elasto‐viscoplastic dynamic and thermal behaviour of geometrically non‐linear plates and shells is studied under the assumption of small strains and large rotations. The first‐order shear deformation shell theory and the Chaboche constitutive viscoplastic model taking the temperature fields into account are used for computations. An effective procedure using the central difference method of solving the equations of motion is applied. The trapezoidal method is used to integrate the constitutive viscoplastic law. A nine node isoparametric shell element has been utilised for the finite element algorithm. Finally, some examples are presented and compared with the results obtained by moderate rotation theory.

Progressive Failure Analysis of Laminated Composite Plates by Finite Element Method

The progressive failure analysis of laminated composite plates under transverse static loading has been carried out in linear and elastic range. The laminated composite plate has been modeled using eight noded isoparametric plate bending elements. The first order shear deformation theory has been applied and a shear correction factor is used. After the failure of the weakest ply, the stiffness is reduced by either fiber failure or matrix failure. The stiffness of failed lamina has been totally discarded and other existing laminae are considered to remain unchanged after the weakest ply failure. The strength of the laminate at the same point is evaluated again to see if the laminate can carry additional load. This ply-by-ply analysis progresses until the ultimate strength of the laminate is reached. The results in terms of first-ply failure load obtained in the present investigation have been compared with those available in published literature. A parametric study has been done on the progressive failure analysis.

Finite element buckling analysis of laminated composite stiffened shells

The paper presents finite element buckling analysis of laminated stiffened plates and stiffened cylindrical shells using modified approach of shell and stiffener modelling. An eight noded isoparametric quadratic element for the shell and a three noded curved stiffener element for the stiffeners on the concept of equal displacements at the shell- stiffener junction is used in the analysis. In the present formulation, stiffener can be placed anywhere within the shell element and obviates the constraints of aligning the mesh lines along the stiffeners. The buckling analysis of stiffened and unstiffened laminated structures subjected to various loading cases has been studied and the results are compared with the published ones.

First Ply Failure Analysis of Laminated Panels Under Transverse Loading

The present investigation deals with the failure analysis of the weakest ply in a laminated composite plate or shell using the method of finite elements. The objective of the study is to detect the stress level at the stage of initiation of damage in the composite panels. An eight noded isoparametric quadratic shell element has been used applying first order shear deformation theory. The elegance of the present formulation lies in the capability of analysing plate as well as shallow and deep shell using a single formulation. Various failure criteria have been studied to predict the load when the weakest lamina fails under transverse static load. The investigation has been made on various plates and shells having varying lamination schemes. Published results on first ply failure analysis of laminated plates are scanty, however the results on laminated shells are not available in the literature. A number of problems on laminated plates have been solved and the results are compared with those available in the relevant literature. A parametric study has been carried out in the present investigation to predict the failure load of laminated shells.

Modeling microarchitecture and mechanical behavior of nacre using 3D finite element techniques Part I Elastic properties

Three dimensional finite element models of nacre were constructed based on reported microstructural studies on the 'brick and mortar' micro-architecture of nacre. 3D eight noded isoparametric brick elements were used to design the microarchitecture of nacre. Tensile tests were simulated using this model. The tests were conducted at low stresses of 2 MPa which occur well within the elastic regime of nacre and thus effects related to locus and extent of damage were ignored. Our simulations show that using the reported values of elastic moduli of organic (0.005 GPa) and aragonitic platelets (205 GPa), the displacements observed in nacre are extremely large and correspond to a very low modulus of 0.011 GPa. The reported elastic modulus of nacre is of the order of 50 GPa. The reason for this inconsistency may arise from two possibilities. Firstly, the organic layer due to its multilayered structure is possibly composed of distinct layers of different elastic moduli. The continuously changing elastic modulus within the organic layer may approach modulus of aragonite near the organic-inorganic interface. Simulations using variable elastic moduli for the organic phase suggest that an elastic modulus of 15 GPa is consistent with the observed elastic behavior of nacre. Another explanation for the observed higher elastic modulus may arise from localized platelet-platelet contact. Since the observed modulus of nacre lies within the above two extremes (i.e. 15 GPa and 205 GPa) it is suggested that a combination of the two possibilities, i.e. a higher modulus of the organic phase near the organic-inorganic interface and localized platelet-platelet contact can result in the observed elastic properties of nacre.

Three-dimensional finite element for thick shells of general shape

When the thickness of a shell increases in comparison with other dimensions as seen in arch dams, pressure vessels and shelter roofs, the effect of the unit deformations and shear deformations along the thickness have to be taken into account for achieving an acceptable approximation. This however, may entail the use of three dimensional elasticity solutions in developing suitable finite elements. In the present study, a twenty node isoparametric finite element is developed for analysis of thick shells having a general shape by using the three-dimensional theory of elasticity in orthogonal curvilinear coordinates. Shell geometric properties and their derivatives, which are necessary for obtaining a displacement-strain matrix, are determined by means of the shape functions. The accuracy of the presented formulation is assessed by comparing the numerical results with those given in the three dimensional elasticity solutions as well as with those obtained by the use of three dimensional finite elements.

First ply failure analysis of stiffened panels – a finite element approach

A first-ply failure analysis method has been developed for predicting the failure load on the laminated composite stiffened panels under various loading conditions. The present finite element formulation is generalised to analyse both laminated bare/stiffened plate and shell panels elegantly. Eight noded isoparametric quadratic elements for shell and three noded curved beam elements for the stiffener are used for the analysis of stiffened panel. The stiffness matrix of the stiffener is computed independently and then transferred to that of the shell. The formulation eliminates the restriction on the location and orientation of the stiffener within the shell element. Several prominent failure theories have been considered in the present work for the prediction of failure load. The results obtained by the present formulation have been compared with those available in literature and some parametric study has also been performed on the basis of variety in shell geometry.

TRANSIENT RESPONSE OF ROTATING LAMINATED PLATES WITH INTERFACIAL FRICTION UNDER ACCELERATING CONDITIONS

This paper is concerned with laminated plates mounted on a rotating disk under accelerating conditions. Interfacial friction is included at specified locations on the plate. The effects of in-plane loads and non-linear Coriolis forces are included. An eight noded isoparametric element is derived for the analysis. Under the action of combined accelerating conditions and Coriolis forces, it is shown that the blade is subjected to shock forces. Employing the Newmark direct integration method, the results obtained are presented.

A large displacement formulation for anisotropic constitutive laws

This work has been motivated by configurations designed in our laboratory for which wood samples undergo large displacements that result in more accurate measurements. From a theoretical point of view, such configurations require a formulation able to take into account the geometrical non-linearities in the case of a highly anisotropic material. This paper proposes a solution to this specific problem in the case of large displacements, where the rotations may be large but the pure strain field is assumed to remain small. We propose to show how it is possible to extend existing rate-form small displacement constitutive laws in order to use them in a large displacement formulation. The strategy adopted in order to account for the anisotropy of the material and, at the same time, to respect the principle of tensorial objectivity, is to express the constitutive law in a co-ordinate system, which rotates during the deformation. Therefore we use tensors whose eigenvectors have a Lagrangian orientation while the eigenvalues have a sense in the Eulerian configuration. In order to exhibit the possibilities of the resulting formulation, two different configurations are tested: • The curvature due to shrinkage of a board section submitted to a linear moisture content profile, • The deflection of a cantilever beam. Both examples are compared to an analytical calculation obtained using the material resistance results. The numerical calculation uses a viscoelastic orthotropic constitutive law, which is integrated using an implicit time scheme and eight node isoparametric finite elements.

Laminated Stiffened Plate-A First Ply Failure Analysis

A first ply failure analysis of composite stiffened plates has been presented using the finite element method. A new scheme has been formulated to model stiffened composite plates. The plate element has been formulated using an eight noded isoparametric quadratic plate bending element. A three noded isoparametric beam element has been used to formulate the stiffener element. The stiffness matrix of the stiffener element has been developed independently and then the stiffness matrix is transferred to that of the plate element. This technique eliminates the restriction on the orientation of the stiffener within the plate element. The formulation has been validated by analysing a number of problems. The results obtained are compared with those available in published literature.

Computer Simulation of the Inflation Process in Blow Molding

A finite element numerical analysis of parison inflation associated with the extrusion blow molding process has been developed using a novel overlay viscoelastic model. The model uses 2-D six noded overlay isoparametric elements. The analysis is capable of predicting final wall thickness distribution for axisymmetric mold geometries. Experimental studies were carried out on an extrusion blowing molding machine with two axisymmetric molds. High density polyethylene was used in the experiment. The predicted wall thickness distribution is in reasonable agreement with the experimental data. Non-uniformity in the initial thickness and the non-isothermal temperature profile of the parison was found to have the most significant impact on the inflation behavior and the resulting wall thickness. Numerical simulation of a blow molding process can be used to optimize the molds and the molding parameters to achieve a product with ideal thickness distribution.

Three Dimentional Finite Element Modeling of Microstructural Development of Nacre in Seashells and Implication on Mineralization of CaCO3

AbstractThree dimensional finite element models of nacre were constructed based on reported microstructural studies on the ‘brick and mortar’ micro-architecture of nacre. 3D eight noded isoparametric brick elements were used to design the microarchitecture of nacre. Tensile tests were simulated using this model at stresses of 2 MPa which occur well within the elastic regime of nacre and thus effects related to locus and extent of damage were ignored. The reported values of elastic moduli of organic (0.005 GPa) and aragonitic platelets (205 GPa) were used in our simulations and the resulting displacements were found to be extremely large and corresponding to a very low modulus of 0.011 GPa. The reported elastic modulus of nacre is of the order of 50 GPa. The large displacements can possibly result from two possibilities. Firstly. the organic layer due to its multilayered structure is possibly composed of distinct layers of different elastic moduli. A significantly higher modulus of the organic phase may be possible near the organic-inorganic interface. Simulations using variable elastic moduli for the organic phase suggest that an elastic modulus of 15 GPa is consistent with the observed elastic behavior of nacre. A second possibility for the observed higher elastic modulus may arise from localized platelet-platelet contact. Since the observed modulus of nacre lies within the above two extremes (i.e. 15 GPa and 205 GPa) it is suggested that a combination of the two, i.e. a higher modulus of the organic phase near the organic-inorganic interface and localized platelet-platelet contact can result in the observed elastic properties of nacre.

Geometric Nonlinear Static Analysis of Laminated Hat Stiffened Plates

Composite plates stiffened with close-section stiffeners have been analysed for large deflection using the method of finite elements.The laminated plate has been modelled using eight noded isoparametric quadratic plate bending elements which include transverse shear deformation. Three noded isoparametric beam elements have been used to formulate the stiffener elements.The stiffener can be incorporated at any arbitrary direction within the plate element. The nonlinear equations of the static equilibrium assuming small rotation have been solved by the Newton-Raphson iteration technique. The composite stiffened plates with both the open and the closed type stiffeners have been analysed in the present investigation and the results have been compared with the published results to validate the present formulation.

Buckling of moderately thick rectangular composite plates subjected to partial edge compression

The objective of the present paper is to investigate the influence of partial edge compression on the critical loads of moderately thick laminated plates. Towards this, an eight node isoparametric plate element is developed. The element has five degrees of freedom per node. The computer code developed accepts two sets of boundary conditions, one for pre-buckling stress analysis and the second for stability analysis. This flexibility is proposed to exploit the mid-line symmetry conditions. Two types of partial edge compression, viz., (I) uniform partial edge compression near the corners and (II) uniform partial edge compression at the middle of edges are considered. The effect of percentage of loaded edge length on the critical load of thin and thick composite plates with simply supported and clamped edge conditions is studied in detail.

Finite element formulation of the Ogden material model with application to rubber-like shells

The present contribution proposes a variational procedure for the numerical implementation of the Ogden material model. For this purpose the strain energy density originally formulated in terms of the principal stretches is transformed as variational quantities into the invariants of the right Cauchy–Green tensor. This formulation holds for arbitrary three-dimensional deformations and requires neither solving eigenvalue problems nor co-ordinate system transformations. Particular attention is given to the consideration of special cases with coinciding eigenvalues. For the analysis of rubber-like shells this material model is then coupled with a six parametric shells kinematics able to deal with large strains and finite rotations. The incompressibility condition is considered in the strain energy, but it is additionally used as 2-D constraint for the elimination of the stretching parameter at the element level. A four node isoparametric finite element is developed by interpolating the transverse shear strains according to assumed strain concept. Finally, examples are given permitting to discuss the capability of the finite element model developed concerning various aspects. © 1998 John Wiley & Sons, Ltd.

Finite Element Thermostructural Analysis of Laminated Composite Shells of Revolution Under Asymmetric Thermal Loading

This paper presents a finite element thermostructural analysis procedure for the generally laminated composite shells of revolution subjected to an asymmetric thermal loading. Governing finite element equations are obtained from the conduction and equilibrium equations of the shell. A general thermal load, both symmetric and asymmetric, that constitutes heat fluxes and convective fluxes, are expressed in terms of trigonometric Fourier series. Two noded and three noded isoparametric axisymmetric elements are used in the analysis. For thermal analysis, the nodal degree of freedom is temperature and for structural analysis, the nodal degrees of freedom are three translational displacements and two rotations. The present analysis also includes the shear flexibility of the composite laminates. The governing finite element equations are solved for the individual Fourier modes and these solutions are superposed to obtain the final solution. Numerical examples are chosen to study the effects of geometry, lamination and structural boundary conditions on both the thermal and the structural response. An interesting observation is that anisotropy in thermal conductivity of the composite influences the maximum temperature rise. Circumferential displacement prevails under the asymmetric loads and/or the antisymmetric laminations of a composite.

Finite Element Free Vibration Analysis of Smart Laminated Composite Beams and Plates

The present paper deals with the development of a finite element method for the free vibration analysis of smart laminated composite beams and plates. The structure consists of a laminated composite substrate to which are surface bonded or embedded a set of piezoelectric layers, one of which is a sensor while the others are actuators. The analysis considers the linear piezoelectric constitutive relations and the first order shear deformation behaviour. An eight noded quadratic isoparametric piezoelectric element is developed to discretize the analysis domain, and the same shape functions are used to describe the electrical field, the charge density, the structural geometry as well as the structural displacements. Numerical results are obtained to study the effects of (1) stacking sequences of the composite laminae and piezoelectric layers, (2) geometric boundary conditions and (3) application of electrical voltages on the free vibration frequencies.

A Direct Caustics Simulation Method Based on FEM.

In this paper, for the sake of prediction of caustic experiments, explanation and verification of experimental results, a new numerical method based on the finite element method, which can directly simulate the caustic patterns, is presented. Based on this new idea, the program for directly determining the gradient solution from the 2 dimensional displacement solution is presented. It is also proven to be necessary for only 8 or 12 node isoparametric element to apply this new method to the 2 dimensional problem. For verifying the accuracy of this method, the simulations of 4 point bending problem and concentrated load problem are performed. The simulated results have good agreement with the theoretical values. As the applied examples, the simulation is carried out for the plate specimen with a single edge crack, or double edge crack, or a short single edge crack under mode I condition. The changes of caustic image under the influence of the interferometry between cracks and the influence of the boundary are given.