Large reference displacement analysis of composite plates. Part II: Computer implementation

This investigation concerns itself with the computer implementation of the dynamic formulation of thin laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A finite element preprocessor computer program is developed to automatically generate the invariants of the laminae, which may have arbitrary orientations. The laminae invariants are then used to obtain the invariants of the elements and the composite laminated plate. The consistent and lumped mass formulations of the invariants of motion of composite plates are compared and it is concluded that the two methods are comparable, if a fine enough finite element mesh is used. The structure of the dynamic equations of motion, based on the formulation presented in Part I of this paper, is examined. Non-linear centrifugal and Coriolis forces arising as the result of the finite rotations of the laminae are defined, and the solution schemes of the resulting non-linear differential equations of motion are discussed. Numerical examples illustrating the differences between homogeneous isotropic and laminated composite plates are presented. An RSSR (Revolute-Spherical-Spherical-Revolute) mechanism is used in the numerical examples, with the coupler modelled as a laminated plate flexible body. It is found that the inertia of the plate contributed greatly to the transverse deformation. The effects of laminae orientation is also investigated.     

Finite element analysis of laminated composite plates using a higher-order displacement model

A C° continuous displacement finite element formulation of a higher-order theory for flexure of thick arbitrary laminated composite plates under transverse loads is presented. The displacement model accounts for non-linear and constant variation of in-plane and transverse displacement model eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with nine degrees-of-freedom per node. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form, the theory of elasticity and the finite element solutions with another higher-order displacement model by the same authors. A computer program has been developed which incorporates the realistic prediction of interlaminar stresses from equilibrium equations.     

Finite element analysis of laminated composite plates

A finite element analysis technique for an arbitrarily laminated anisotropic plate is described. A superparametric quadratic plate element with five degrees-of-freedom per node is used in the analysis. A stress-strain relation is derived from a three-dimensional approach to the problem. The volume integration of the stiffness matrix is evaluated by numerical integration using the Gauss quadrature formula with 2 × 2 × 2 sampling points. A variety of laminated plate problems is solved and the results are compared with the exact solutions, which demonstrate the validity of the method.     

Finite element formulation of smart piezoelectric composite plates coupled with acoustic fluid

In the context of noise and vibration reduction by passive piezoelectric devices, this work presents the theoretical formulation and the finite element (FE) implementation of vibroacoustic problems with piezoelectric composite structures connected to electric shunt circuits. The originalities of this work concern (i) the formulation of the electro-mechanical-acoustic coupled system, (ii) the implementation of an accurate and inexpensive laminated composite plate FE with embedded piezoelectric layers connected to resonant shunt circuits, and (iii) the development of an efficient fluid-structure interface element. Various results are presented in order to validate and illustrate the performance of the proposed fully coupled numerical approach.     

Finite element analysis of anisotropic plates

Two aspects of the finite element analysis of mid-plane symmetrically laminated anisotropic plates are considered in this paper. The first pertains to exploiting the symmetries exhibited by anisotropic plates in their analysis. The second aspect pertains to the effects of anisotropy and shear deformation on the accuracy and convergence of shear-flexible displacement finite element models. Numerical results are presented which show the effects of increasing the order of approximating polynomials and of using derivatives of generalized displacements as nodal parameters.     

Finite element analysis of laminated composite plates using zeroth-order shear deformation theory

In this paper a generalized finite element model is developed for static and dynamic analyses of laminated composite plates using zeroth-order shear deformation theory (ZSDT). The theory ensures the parabolic distribution of transverse shear stresses across the plate thickness. A four-noded plate element is considered in this model and the generalized nodal variables are expressed using Lagrangian linear interpolation functions and Hermitian cubic interpolation functions. The solutions of the finite element model have been compared with the existing solutions for symmetric and antisymmetric laminated composite plates. The comparison confirms that the ZSDT can be efficiently used for finite element analysis of both thin and thick plates with high accuracy.     

Equivalent homogeneous finite element for composite materials via reissner principle. Part I: Finite element for plates

The stiffness matrix in the finite element method for multi-layered materials is generally computed by expressing the strain energy in each layer and adding them together. In order to lower the computing time, which may be prohibitive if the number of layers is high, and to get accurate information on the stresses, especially on transverse shear stresses, we present a new finite element using the Reissner principle. In the first part the case of plates will be detailed: extensions to shell problems will be presented in the second part. The efficiency of the method is tested on a special analytic solution, and some examples are given.     

Finite element bending analysis of multilayer plates

The development of a general quadratic multilayer plate element is presented for the analysis of arbitrarily layered curved plates. In the formulation, each layer of the multilayer plate can have different orthotropic properties and can deform locally. Examples of bending problems are presented which demonstrate the applicability of the formulation.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite element analysis of laminated composite stiffened plates using FSDT and HSDT: A comparative perspective

Significance of using higher-order shear deformation theory (HSDT) over the first-order shear deformation theory (FSDT) for analyzing laminated composite stiffened plates is brought out using the finite element method (FEM). For this purpose, a C⁰ HSDT, is extended for application to stiffened configurations, for linearly elastic static and natural vibration analysis. The spatial displacement fields of both the plate and the stiffener are derived as functions of reference plane variables using Taylor series expansion. The developed computational tool is employed for analyzing systems having varying configurations using the FSDT and two different HSDTs, and their comparative effects are systematically studied, demonstrating the need for using HSDT instead of FSDT, for obtaining accurate structural response of such stiffened configurations.     

Finite element analysis of plates with curved edges

The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.     

模具对柱面复合材料构件固化变形影响的有限元分析

针对复合材料构件固化变形问题, 分析了复合材料热压罐成型固化过程的多场耦合关系, 考虑模具的作用, 建立了柱面复合材料构件固化过程的有限元分析模型。基于此模型, 研究了模具材料、 模具厚度和模具形式对柱面复合材料构件固化变形的影响。结果表明: 模具对柱面件固化变形的影响较大, 模具材料与构件材料热膨胀系数(CTE)不匹配程度影响构件回弹角的大小; 模具的厚度不同, 导致构件的不同回弹角; 采用阴模时, 构件回弹角小于阳模的, 且回弹方向相反。     

Total Lagrangian formulation for the large displacement analysis of rectangular plates

In this paper, a method for the non-linear dynamic analysis of rectangular plates that undergo large rigid body motions and small elastic deformations is presented. The large rigid body displacement of the plate is defined by the translation and rotation of a selected plate reference. The small elastic deformation of the midplane is defined in the plate co-ordinate system using the assumptions of the classical theories of plates. Non-linear terms that represent the dynamic coupling between the rigid body displacement and the elastic deformation are presented in a closed form in terms of a set of time-invariant scalars and matrices that depend on the assumed displacement field of the plate. In this paper, the case of simple two-parameter screw displacement, where the rigid body translation and rotation of the plate reference are, respectively, along and about an axis fixed in space, is first considered. The non-linear dynamic equations that govern the most general and arbitrary motion of the plate are also presented and both lumped and consistent mass formulations are discussed. The non-linear dynamic formulation presented in this paper can be used to develop a total Lagrangian finite element formulation for plates in multibody systems consisting of interconnected structural elements.     

Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT

The finite element dynamic stability analysis of laminated composite skew structures subjected to in-plane pulsating forces is carried out based on the higher-order shear deformation theory (HSDT). The two boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for square and skew plates with or without central cutout are in good agreement with those reported by other investigators. The new results for laminated skew plate structures containing cutout in this study mainly show the effect of the interactions between the skew angle and other various parameters, for example, cutout size, the fiber angle of layer and thickness-to-length ratio. The effect of the magnitude of the periodic in-plane load on the dynamic instability index is also investigated.     

Finite element analysis of substation composite insulators

Composite insulators are rapidly replacing their porcelain counterparts in electrical substation applications. These insulators consist of a glass-reinforced polymer (GRP) rod, with two metal end fittings radially crimped onto the ends of the rod during assembly. In this paper, axisymmetric finite element models are developed to evaluate the mechanical performance of composite insulators under externally applied axial compression. The analyses are performed by assuming both a perfectly bonded interface between the composite rod and the end fittings, and an imperfect interface which permits large relative sliding with Coulomb friction. Results indicate that the perfect interface model is unrealistic since it predicts singular stresses at the interface comer and an overall linear structural response. On the other hand, the imperfect interface model is found to simulate accurately the structural non-linearity caused by relative sliding of the GRP rod within the end fittings. The imperfect interface model has therefore been used to evaluate the effects of interface friction, and the extent of crimping, on the maximum load-bearing capacity of substation composite insulators.     

Finite element buckling analysis of homogeneous and sandwich plates

Finite element buckling analyses of plates are performed by means of quardrilateral elements having 12 degrees-of-freedom. It is found that if the plate is capable of appreciable transverse shear deformation, initial-stress stiffness matrices that represent the lateral dispalcement by cubic polynomials do not yield convergence to correct results. An initial-stress stiffness that works satisfactorily for both homogeneous and sandwich plates is formulated. A Fortran listing of the associated subroutine is provided.     

Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory

A higher-order shear deformable C° continuous finite element is developed and employed to investigate the transient response of isotropic, orthotropic and layered anisotropic composite plates. The governing ordinary linear differential equations are integrated using the central difference explicit time integration scheme. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. Numerical results for deflections and stresses are presented for rectangular plates under various boundary conditions and loadings. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. The numerical results are compared with those available in the literature, and with the results obtained by solving the same problems using the Mindlin plate element.     

Finite element analysis of textile composite preform stamping

The forming or draping of a textile composite preform may result in large changes in the fibrous microstructure of the preform. This change in the local fiber orientation leads to significant changes in the fabric permeability as well as the mechanical properties of the ensuing composite structure. Therefore, this change in orientation of the tows of the preform needs to be known accurately to calculate the various effective properties of the composite. A new finite element approach for stamping analysis of a plain-weave textile composite preform has been developed. This model is simple, efficient and can be used in the existing finite element codes. The model represents the preform as a mesh of 3-D truss elements and 3-D shell elements. The truss elements model the tows, which are allowed to both scissor and slide relative to one another. The shell elements represent a fictitious material that accounts for inter-tow friction and fiber angle jamming. The model takes into account large strains and large deformations. In-plane uniaxial tension tests have been performed on plain-weave specimens for determining the constitutive law of the transforming medium and to show the inter-tow sliding. Application of the model is demonstrated by simulating the stamping of a preform by a spherical punch. The results from the simulation show good correlation with results from the experiments.     

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

A new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case. Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.