Degree Of Orthotropy Research Articles

Analytical study on bending behaviour of plates made of functionally graded materials using higher-order shear

Functionally Graded Materials (FGM) are the alternate replacement of laminated composites and metallic substances in structural members from daily use to aerospace applications due to their superior characteristics like ability to with stand large temperature gradients, competence to endure to heavy forces and superior sturdy properties, lesser residual and thermal stresses, high capacity to resist surface wear and tear, and an overall increment in strength to weight ratio. As this advanced class of materials are being used as the structural components their ability to with stand various loads considering different boundary conditions under various analytical approaches is very essential even before put into use. The key objective of the current study is to apply higher–order displacement model to determine bending behaviour of plate made of functionally graded material with distinct geometrical constraints, forces and degree of orthotropy under simply supported boundary conditions. An analytical investigation on the behaviour stresses developed in plates made of functionally graded material by applying higher order shear displacement model approach is carried-out by considering the effective material properties according to power law distribution of the volume fraction of the constituents and the equations obtained for simply supported beam made functionally graded materials from closed form using Navier’s empirical equations. A Computer code has been developed using MATHEMATICA 7.0 Software for displacement model to predict deflections and stresses. The accurateness of the current higher-order shear deformation theory has been tested for static bending of functionally graded material plates. The findings are reported in non-dimensional form for various thickness ratios, aspect ratios, degrees of orthotropy, volume fraction index values, and loading circumstances for static bending issues involving mechanical loading of functionally graded material plates. All the solutions derived by the current higher order shear deformation theory agree well with the other plate theory solutions that are documented in the literature.

Buckling behavior of laminated composite stiffened cylindrical shell panels having cutout for different parametric variations

Buckling of composite stiffened cylindrical shell panels having cutout is analyzed using the finite element approach. Shell panel and stiffeners are modeled using first-order shear deformation theory and considering displacement compatibility at the stiffener-panel interface. Convergence and validation studies are conducted to establish the accuracy of the present approach. An attempt is made to maximize the uniaxial and biaxial buckling loads by varying the lamination, boundary conditions, load position, depth of stiffeners, and their arrangements. Parametric studies show that buckling load and mode shapes depend on the size of cutout, ratio of shell width to thickness, degree of orthotropy, and fiber orientation angle. The parametric variations considered here may help the practicing engineers to have an elaborate design aid for improving the buckling strength of such shell panels.

Buckling characteristics of cut-out borne composite stiffened hyperbolic paraboloid shell panel

The present study investigates buckling characteristics of cut-out borne stiffened hyperbolic paraboloid shell panel made of laminated composites using finite element analysis to evaluate the governing differential equations of global buckling of the structure. The finite element code is validated by solving benchmark problems from literature. Different parametric variations are studied to find the optimum panel buckling load. Laminations, boundary conditions, depth of stiffener and arrangement of stiffeners are found to influence the panel buckling load. Effect of different parameters like cut-out size, shell width to thickness ratio, degree of orthotropy and fiber orientation angle of the composite layers on buckling load are also studied. Parametric and comparative studies are conducted to analyze the buckling strength of composite hyperbolic paraboloid shell panel with cut-out.

On thermal-expansion properties of more-orthotropic prepreg laminates with and without interleaf layers

More-orthotropic laminates are increasingly used in today’s carbon-fiber-epoxy composite airframe structures. A recent study reported on a considerable discrepancy between measured and calculated coefficients of thermal expansion (CTEs) for M21E/IMA prepreg laminates, which was found to increase with the laminates’ degree of orthotropy ξ=Ex/Ey. In this paper it is analyzed whether the observed effect is related to more-orthotropic laminate stackings in general or to the particular inner laminate architecture of M21E/IMA laminates, which is characterized by discrete interleaf layers. Therefore, a quasi-isotropic and three more-orthotropic laminates, made from 8552/AS4 and M21E/IMA prepreg, were analyzed by means of thermo-mechanical analysis (TMA) and classical-laminate theory (CLT). Calculated and measured CTEs are compared and the effect of orthotropy is quantified. Potential affectations of TMA measurements and CTE determination due to edge effects and sample size are examined by means of finite-element analyses.

On abnormal thermal-expansion properties of more orthotropic M21E/IMA carbon-fiber-epoxy laminates

The present paper reports on an anomaly of laminates’ coefficients of thermal expansion (CTE), which has been observed for more-orthotropic laminates of a series-quality M21E/IMA prepreg-made composite frame. It was found that measured CTEs increasingly deviate from theoretical CTE values, derived from unidirectional ply data, when the degree of laminate orthotropy increases. The paper comprises experimental CTE measurements for the unidirectional M21E/IMA ply and six specific laminates of a full-scale frame. Analyses suggest that the observed discrepancy correlates with the laminates' degree of orthotropy, which is quantified by the in-plane Young's modulus ratio Ex/Ey in this paper. The observed effect is considered critical as inhomogeneous CTEs lead to process-induced distortions of curved composite structures. Thus, they play a key role when carbon-fiber-epoxy composite frames shall be made in tight dimensional tolerances. Wrong CTE calculation leads to erroneous shape predictions, inaccurate tool compensation and finally to assembly problems.

Orientable wrinkles in stretched orthotropic films

Tensional wrinkles are widely observed in elastic thin films, with mono-orientation of wrinkles being usually perpendicular to the stretching direction. Here, by changing material orthotropic direction, we present orientable wrinkles in uniaxially stretched orthotropic membranes. To quantitatively explore orthotropy-related wrinkles and their morphological evolution, we develop a mathematical model by introducing orthotropic, elastic constitution into the extended Föppl–von Kármán nonlinear plate theory that can describe large in-plane anisotropic deformations. We find that degree of orthotropy and shear modulus significantly affect the critical buckling strain, restabilization point and wrinkling amplitude. A 3D phase diagram on stability boundaries is drawn and we find that no wrinkles emerge when the orthotropic parameters are located in the stability region. Orientable oblique wrinkles, depending on the angle between orthotropy and stretching direction, are carefully examined, which could be used to guide effective designs of wrinkle-tunable membrane surfaces and structures.

Collapse of Orthotropic Spherical Shells.

We report on the buckling and subsequent collapse of orthotropic elastic spherical shells under volume and pressure control. Going far beyond what is known for isotropic shells, a rich morphological phase space with three distinct regimes emerges upon variation of shell slenderness and degree of orthotropy. Our extensive numerical simulations are in agreement with experiments using fabricated polymer shells. The shell buckling pathways and corresponding strain energy evolution are shown to depend strongly on material orthotropy. We find surprisingly robust orthotropic structures with strong similarities to stomatocytes and tricolpate pollen grains, suggesting that the shape of several of Nature's collapsed shells could be understood from the viewpoint of material orthotropy.

Mode frequency analysis of antisymmetric angle-ply laminated composite stiffened hypar shell with cutout

Abstract This article deals with finite element method for the analysis of antisymmetric angle-ply laminated composite hypar shells (hyperbolic paraboloid bounded by straight edges) that applies an eight-noded isoparametric shell element and a three-noded beam element to study the mode-frequency analysis of stiffened shell with cutout. Two-, 4-, and 10-layered antisymmetric angle-ply laminations with different lamination angles are considered. Among these, 10-layer antisymmetric angle-ply shells are considered for elaborate study. The shells have different boundary conditions along its four edges. The formulation is based on the first-order shear deformation theory. The reduced method of eigen value solution is chosen for the undamped free vibration analysis. The first five modes of natural frequency are presented. The numerical studies are conducted to determine the effects of width-to-thickness ratio (b/h), degree of orthotropy (E 11/E 22), and fiber orientation angle (θ) on the nondimensional natural frequency. The results reveal that free vibration behavior mainly depends on the number of boundary constraints rather than other parametric variations such as change in fiber orientation angle and increase in degree of orthotropy and width-to-thickness ratio.

Buckling analysis of cracked laminated plates by domain decomposition method

ABSTRACTBuckling of cracked laminated plates has been analysed by generalised differential quadrature (GDQ) method combined with domain decomposition technique. Crack was modelled as a discontinuity which divides the plate into some segments. Governing equations, boundary and continuity conditions were applied on each segment and the problem was converted to algebraic system of equations. Critical buckling load was determined by eigenvalue analysis. Effects of layers arrangements, crack lengths, plate aspect ratios and degree of orthotropy of each layer on the buckling are presented. The accuracy of the presented method is demonstrated using several numerical samples. Also, results illustrate that present approach by using a small number of nodes can consider the material discontinuity and achieve high accuracy. The buckling load of a cracked plate is smaller than the perfect one, especially when the load is normal to the crack edge. Strength ratio can alter the buckling mode of a cracked laminate.

Contact mechanics problem between an orthotropic graded coating and a rigid punch of an arbitrary profile

Singular integral equation (SIE) and finite element methods are developed for sliding contact analysis of a finite thickness orthotropic graded coating, which is perfectly bonded to an isotropic substrate. Orthotropic stiffness coefficients of the coating vary exponentially through the coating thickness. The coating is assumed to be loaded by a frictional rigid punch of an arbitrary profile. In the SIE formulation, governing partial differential equations are derived in accordance with the theory of plane elasticity. Applying Fourier transformation techniques, the problem is reduced to a singular integral equation of the second kind. Flat and triangular punch profiles are considered in the formulation. The integral equation is solved numerically through an expansion-collocation technique. Roots of Chebyshev polynomials of the first kind are employed as collocation points. The posed contact problem is also solved by means of the finite element method. The augmented Lagrange algorithm is applied in the iterative solution procedure. Comparisons of the contact stresses obtained by singular integral equation and finite element methods demonstrate that the findings of these two separate techniques are in excellent agreement. Further numerical results are generated to be able to assess the influences of coefficient of friction, non-homogeneity constants, relative coating thickness, and degree of orthotropy upon the contact behavior of the coating-substrate system.

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 Instability of Laminated-Composite and Sandwich Plates Using a New Inverse Trigonometric Zigzag Theory

A structure with periodic dynamic load may lead to dynamic instability due to parametric resonance. In the present work, the dynamic stability analysis of laminated composite and sandwich plate due to in-plane periodic loads is studied based on recently developed inverse trigonometric zigzag theory (ITZZT). Transverse shear stress continuity at layer interfaces along with traction-free boundary conditions on the plate surfaces is satisfied by the model obviating the need of shear correction factor. An efficient C0 continuous, eight noded isoparametric element with seven field variable is employed for the dynamic stability analysis of laminated composite and sandwich plates. The boundaries of instability regions are determined using Bolotin's approach and the first instability zone is presented either in the nondimensional load amplitude–excitation frequency plane or load amplitude–load frequency plane. The influences of various parameters such as degrees of orthotropy, span-thickness ratios, boundary conditions, static load factors, and thickness ratios on the dynamic instability regions (DIRs) are studied by solving a number of problems. The evaluated results are validated with the available results in the literature based on different deformation theories. The efficiency of the present model is ascertained by the improved accuracy of predicted results at the cost of less computational involvement.

Layerwise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations

Layerwise plate theory of Reddy, extended for the analysis of delaminations, has served as a basis for development of enriched finite elements. The proposed model assumes layerwise linear variation of in-plane displacements and constant transverse displacement through the plate thickness. Jump discontinuities in displacement field in three orthogonal directions are incorporated using Heaviside step functions, depending on delamination position through the plate thickness. Equations of motion are derived using Hamilton’s principle. Using the proposed model laminated composite and sandwich plates were analyzed. All numerical solutions are obtained using originally coded MATLAB programs. Proposed model is verified using existing results from the literature. Results for natural frequencies, mode shapes and critical buckling loads for intact and damaged plates are compared. Effects of plate geometry, lamination scheme, degree of orthotropy and delamination size or position on dynamic characteristics of the plate are presented. Excellent agreement is obtained and a family of new results is presented as a benchmark for future investigations.

Parametric instability of delaminated composite plates subjected to periodic in-plane loading

This paper deals with the study of the parametric instability characteristics of delaminated composite plates subjected to periodic in-plane load. A first order shear deformation theory based on finite element method is developed for studying the instability region of mid-plane delaminated composite plate. A detailed parametric investigation is carried out to study the influence of the delamination area, number of layers, degree of orthotropy, aspect ratio and static in-plane load on the dynamic stability characteristics of delaminated cross-ply plates. The results of vibration and buckling of delaminated composite plates are compared with previous numerical studies in the literature and the authors’ experimental investigations. The boundaries of instability regions are obtained by using Bolotin’s method and are represented in the non-dimensional load amplitude-excitation frequency plane. It is observed that the natural frequencies and the critical buckling load of the plates decrease with an increase in delamination size. The increase in delamination size also causes dynamic instability regions to be shifted to lower excitation frequencies.

One Explanation for Two-Times Running Speed Response Due to Misalignment in Rotors Connected by Flexible Couplings

Misalignment in turbomachinery is commonly thought to produce two-times running-speed (2N) response. The source of 2N vibration response was investigated, starting with the development of finite-element models for three flexible disk-pack couplings (four-bolt, six-bolt, and eight-bolt couplings). Parallel and angular misalignments were analyzed. The resultant lateral stiffness terms had 1N, 2N, and 3N harmonic components versus the shaft rotation angle. The four-bolt coupling had large 1N stiffness components under angular and parallel misalignment. The six-bolt coupling had only a 1N reaction component under angular misalignment, while parallel misalignment showed a strong 2N reaction component, larger than either the 1N or 3N components. Under angular misalignment, the eight-bolt model produced large 1N reaction components. Under parallel misalignment, it produced 1N, 2N, and 3N components that were similar in magnitude. All the couplings behaved linearly in the range studied. Some experts attribute observed 2N response to nonlinear bearing forces produced by bearings at high unit loads. Static tests for a five-pad tilting-pad journal bearing with unit loads up to 34.5 bars produced small 2N motion components that did not grow with increasing unit load. A Jeffcott-rotor model with shaft stiffness orthotropy and a fixed-direction side load predicts that 2N response depends on three related factors: (1) the degree of orthotropy (the 1N stiffness variation magnitude), (2) the magnitude of the side load, and (3) the relative ratio of running speed to rotor first natural frequency, (ω/ωn). The 2N response magnitude is largest when ω is close to ωn/2. The side load is required to create 2N response due to shaft stiffness orthotropy. Misaligned couplings create precisely the same (very old) physical model as a two-pole turbogenerator rotor with a gravity side load (gravity critical speed). The response of a two-rotor/coupling system with parallel and angular misalignment was simulated using a time-transient code. When the frequency ratio was 0.5, the system response with the four-bolt and six-bolt coupling had a synchronous 1N component as well as a significant 2N component. Parallel misalignment at a coupling produces stiffness orthotropy and a fixed-direction side load. For ranges of running speed near ωn/2, these two elements can combine to produce 2N response.

Parametric Resonance Characteristics of Laminated Composite Curved Shell Panels in a Hygrothermal Environment

The present study deals with the parametric resonance behaviour of laminated composite curved shell panels in a hygrothermal environment using Bolotin’s approach. A simple laminated model is developed using first order shear deformation theory (FSDT) for the vibration and dynamic stability analysis of laminated composite shells subjected to hygrothermal conditions. A computer program based on the finite element method (FEM) in a MATLAB environment is developed to perform all necessary computations. Quantitative results are presented to show the effects of curvature, ply-orientations, degree of orthotropy and geometry of laminates on the parametric instability of composite curved shell panels for different temperature and moisture concentrations. The excitation frequencies of laminated composite panels decrease with the increase of temperature and moisture due to reduction of stiffness for all laminates.

Local fibre reinforcement of holes in composite multilayered plates

For various technical reasons, cutouts such as holes in thin-walled structures are inevitable and are of significant technical relevance. Unfortunately holes lead to an undesired stress concentration at the hole vicinity and a reduced strength of the structure. Therefore in practice a local reinforcement in the form of a ring is usually applied around the hole. The increasing requirements for modern structures in terms of low weight and high strength lead to the question of an optimal reinforcement design. The present paper addresses the new but well-approved techniques of the use of curved fibre format to determine the aforementioned optimal design of the reinforcement. The optimization of cutouts in laminated composite plates under bi-axial tensile loading conditions has been investigated using two approaches: the finite element and the Rayleigh–Ritz method. The used methodology implemented successfully theoretical results based on the complex potential theory. For the considered problems, the proposed methods were shown to successfully produce a constant objective function around the hole boundary under biaxial loading. The optimal reinforcement of holes in laminated composite plates illustrated that the optimum depends on the degree of orthotropy. Significant reduction of stress concentrations were demonstrated. The results obtained illustrate the necessity and usefulness of the applied optimization procedure.

Parametric Instability of Laminated Composite Doubly Curved Shell Panels Subjected to Hygrothermal Environment

The dynamic stability behavior of laminated composite shells subjected to hygrothermal loadings are studied in the present investigation. A simple laminated model is developed for the vibration and stability analysis of laminated composite shells subjected to hygrothermal conditions. A computer program based on FEM in MATLAB environment is developed to perform all necessary computations. An eight-node isoparametric element is employed in the present The analysis with five degrees of freedom per node. Element elastic stiffness matrices, mass matrices, geometric stiffness matrix due to mechanical and hygrothermal loads and load vectors are derived using the principle of minimum potential energy. Quantitative results are presented to show the effects of curvature, ply-orientation, degrees of orthotropy and static load factors of laminate on dynamic stability of composite shells for different temperatures and moisture concentrations.

Simulation of slitting method for calculation of compliance functions of laminated composites

In this research, the process of slitting with an assumed residual stress distribution is simulated to calculate compliance functions of laminated composites. The primary aim of this article is to investigate different parameters influencing the calculation of compliance coefficients for laminated composites using a finite element model. First, using two- and three-dimensional finite element models, the process of slitting in isotropic materials for calculating the compliance coefficients is simulated. The results show a complete agreement between the two models. Also, the results of simulation show that in the modeling of slitting method for laminated composites, real slit width must be considered. That is because composite laminates are generally thin and slitting in these materials results in large slit width to thickness ratio. Finally, the developed simulation method is applied to calculate compliance coefficients for two carbon/epoxy- and glass/epoxy-laminated composites with quasi-isotropic and cross-ply lay-ups using a three-dimensional finite element modeling. For a better precision, a full three-dimensional model without symmetry assumption is considered. The presented method is capable of simulating slitting method in parts and specimens with complicated geometry as well as in orthotropic materials with any degree of orthotropy.

Effect of loading and lamination parameters on the optimum design of laminated plates

This paper illustrates the effect of loading conditions and lamination parameters on the optimum design of the laminated composite plates. Optimum design procedure based on the flexibility criterion is presented in this paper. The objective is to determine the optimum thickness of the laminate layers and its optimum orientations without exhibiting any failure. The finite element method based on Mindlin plate theory is used in conjunction with an optimization method in order to determine the optimum design. Newmark implicit time integration scheme is used to discretize the time domain and calculate the transient response of the laminated plate. Exterior penalty method is exploited as a constrained minimization technique. For this purpose, four numerical examples are presented to figure out the effects of dynamic loading profile, boundary conditions, number of layers, and degree of orthotropy on the optimum weight design.