Application of the quadrilateral area coordinate method:a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate （QAC） method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF （5-DOF per node） quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-rocessing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Hamiltonian parametric element and semi-analytical solution for smart laminated plates

Based on the Hellinger-Reissner （H-R） mixed variational principle for piezoelectric material, a unified 4-node Hamiltonian isoparametric element of anisotropy piezoelectric material is established. A new semi-analytical solution for the natural vibration of smart laminated plates and the transient response of the laminated cantilever with piezoelectric patch is presented. The major steps of mathematical model are as follows： the piezoelectric layer and host layer of laminated plate are considered as unattached three-dimensional bodies and discretized by the Hamiltonian isoparametric elements. The control equation of whole structure is derived by considering the compatibility of generalized displacements and generalized stresses on the interface between layers. There is no restriction for the side-face geometrical boundaries, the thickness and the number of layers of plate by the use of the present isoparametric element. Present method has wide application area.

A 3-Node Co-Rotational Triangular Finite Element for Non-Smooth,Folded and Multi-Shell Laminated Composite Structures

Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite structures.The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system.In the global coordinate system,two smaller components of one vector,together with the smallest or second smallest component of another vector,of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables,whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell(away from non-smooth intersections)are defined as rotational variables.All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure,and thus improve the computational efficiency in the nonlinear solution of these composite shell structures.Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables,and the second derivatives of the strain energy functional with respect to local nodal variables,symmetric tangent stiffness matrices in local and global coordinate systems are obtained.To overcome shear locking,the assumed transverse shear strains obtained from the line-integration approach are employed.The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests,several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.

ON THE STABILITY ANALYSIS OF PLATES AND SHELLS USING A QUADRILATERAL,16-DEGREES OF FREEDOM PLAT SHELL ELEMENT DKQ16

The linear buckling problems of plates and shells were analysed using a recently developped quadrilateral,16-degrees of freedom flat shell element (called DKQ16).The geometrical stiffness matrix was established.Comparison of the numerical results for several typical problems shows that the DKQ16 element has a very good precision for the linear buckling problems of plates and shells.

ITERATIVE METHOD FOR LARGE DEFLECTION NONLINEAR PROBLEM OF LAMINATED COMPOSITE SHALLOW SHELLS AND PLATES

Based on the results by Wang,in this paper, the iterative method is presented for the study of large deflection nonlinear problem of laminated composite shallow shells and plates. The rectangular laminated composite shallow shells have been analyzed. The results have been compared with the small deflection linear analytical solution and finite element nonlinear solution. The results proved that the solution coincide with small deflection linear analytical solution in the condition of the low loads and finite element nonlinear solution in the condition of the high loads.

The Hamilton System and Hamilton Type Generalized Variational Principle for the Laminated Composite Plates and Shells

By introducing the Hamilton theory and algorithms into the problems of laminated composite plates andshells, the Hamiltion type generalized variational principle is established, and the Hamilton canonical equations andthe boundary conditions for the static and elastoplastic analysis of composite plates are presented. With thetransformation of phase variables, the Hamilton canonical equations and their boundary conditions for thecylindrical shells and doubly curved shells in the curvilinear coordinate are given.

Stochastic Finite Element Analysis of Plate and Shell Construction

The response of random plate and shell construction is analyzed with the stochastic finite element method (SFEM). Random material properties and geometric dimensions of construction are involved in this paper. A simplified isoparametric local average model is used to describe the random field. Numerical results of the examples indicate that the approach presented herein is an economical and efficient solution for such an analysis compared with Monte Carlo simulation (MCS).

A Novel Rectangular Element for Piezoelectric Laminated Plates

Based on the classical laminated plate theory, a novel finite element formulation is presented for modeling the static response of laminated composites containing distributed piezoelectric ceramic subjected to electric loadings. A four-node rectangular composite element with an additional voltage freedom per piezoelectric layer is implemented for the analysis. The element can predict more accurately the bending response of the structure because of its new displacement radixes. Numerical examples ere performed and the calculated data compare very well with existing results in the literatures.

Optimization of buckling load for laminated composite plates using adaptive Kriging-improved PSO:A novel hybrid intelligent method

An effective hybrid optimization method is proposed by integrating an adaptive Kriging(A-Kriging)into an improved partial swarm optimization algorithm(IPSO)to give a so-called A-Kriging-IPSO for maximizing the buckling load of laminated composite plates(LCPs)under uniaxial and biaxial compressions.In this method,a novel iterative adaptive Kriging model,which is structured using two training sample sets as active and adaptive points,is utilized to directly predict the buckling load of the LCPs and to improve the efficiency of the optimization process.The active points are selected from the initial data set while the adaptive points are generated using the radial random-based convex samples.The cell-based smoothed discrete shear gap method(CS-DSG3)is employed to analyze the buckling behavior of the LCPs to provide the response of adaptive and input data sets.The buckling load of the LCPs is maximized by utilizing the IPSO algorithm.To demonstrate the efficiency and accuracy of the proposed methodology,the LCPs with different layers(2,3,4,and 10 layers),boundary conditions,aspect ratios and load patterns(biaxial and uniaxial loads)are investigated.The results obtained by proposed method are in good agreement with the literature results,but with less computational burden.By applying adaptive radial Kriging model,the accurate optimal resultsebased predictions of the buckling load are obtained for the studied LCPs.

THE LAYOUT OPTIMIZATION OF STIFFENERS FOR PLATE-SHELL STRUCTURES

The plate-shell structures with stiffeners are widely used in a broad range of engineering structures. This study presents the layout optimization of stiffeners. The minimum weight of stiffeners is taken as the objective function with the global stiffness constraint. In the layout optimization, the stiffeners should be placed at the locations with high strain energy/or stress. Conversely, elements of stiffeners with a small strain energy/or stress are considered to be used inefficiently and can be removed. Thus, to identify the element efficiency so that most inefficiently used elements of stiffeners can be removed, the element sensitivity of the strain energy of stiffeners is introduced, and a search criterion for locations of stiffeners is presented. The layout optimization approach is given for determining which elements of the stiffeners need to be kept or removed. In each iterative design, a high efficiency reanalysis approach is used to reduce the computational effort. The present approach is implemented for the layout optimization of stiffeners for a bunker loaded by the hydrostatic pressure. The numerical results show that the present approach is effective for dealing with layout optimization of stiffeners for plate-shell structures.

NUMERICAL ANALYSIS OF DELAMINATION GROWTH FOR STIFFENED COMPOSITE LAMINATED PLATES

A study of postbuckling and delamination propagation behavior in delaminated stiffened composite plates was presented. A methodology was proposed for simulating the multi-failure responses, such as initial and postbuckling, delamination onset and propagation, etc. A finite element analysis was conducted on the basis of the Mindlin first order shear effect theory and the von-Krmn nonlinear deformation assumption. The total energy release rate used as the criteria of delamination growth was estimated with virtual crack closure technique (VCCT). A self-adaptive grid moving technology was adopted to model the delamination growth process. Moreover, the contact effect along delamination front was also considered during the numerical simulation process. By some numerical examples, the influence of distribution and location of stiffener, configuration and size of the delamination, boundary condition and contact effect upon the delamination growth behavior of the stiffened composite plates were investigated. The method and numerical conclusion provided should be of great value to engineers dealing with composite structures.

Are “Higher-Order” and “Layer-wise Zig-Zag” Plate & Shell Theories Necessary for Functionally Graded Materials and Structures?

Similar to the very vast prior literature on analyzing laminated composite structures,“higher-order”and“layer-wise higher-order”plate and shell theories for functionally-graded(FG)materials and structures are also widely popularized in the literature of the past two decades.However,such higher-order theories involve(1)postulating very complex assumptions for plate/shell kinematics in the thickness direction,(2)defining generalized variables of displacements,strains,and stresses,and(3)developing very complex governing equilibrium,compatibility,and constitutive equations in terms of newly-defined generalized kinematic and generalized kinetic variables.Their industrial applications are thus hindered by their inherent complexity,and the fact that it is difficult for end-users(front-line structural engineers)to completely understand all the newly-defined generalized DOFs for FEM in the higher-order and layer-wise theories.In an entirely different way,very simple 20-node and 27-node 3-D continuum solid-shell elements are developed in this paper,based on the simple theory of 3D solid mechanics,for static and dynamic analyses of functionally-graded plates and shells.A simple Over-Integration(a 4-point Gauss integration in the thickness direction)is used to evaluate the stiffness matrices of each element,while only a single element is used in the thickness direction without increasing the number of degrees of freedom.A stress-recovery approach is used to compute the distribution of transverse stresses by considering the equations of 3D elasticity in Cartesian as well as cylindrical polar coordinates.Comprehensive numerical results are presented for static and dynamic analyses of FG plates and shells,which agree well,either with the existing solutions in the published literature,or with the computationally very expensive solutions obtained by using simple 3D isoparametric elements(with standard Gauss Quadrature)available in NASTRAN(wherein many 3D elements are used in the thickness direction to capture the varying material properties).The effects of the material gradient index,the span-to-thickness ratio,the aspect ratio and the boundary conditions are also studied in the solutions of FG structures.Because the proposed methodology merely involves:(2)standard displacement DOFs at each node,(2)involves a simple 4-point Gaussian over-integration in the thickness direction,(3)relies only on the simple theory of solid mechanics,and(4)is capable of accurately and efficiently predicting the static and dynamical behavior of FG structures in a very simple and cost-effective manner,it is thus believed by the authors that the painstaking and cumbersome development of“higher-order”or“layer-wise higher-order”theories is not entirely necessary for the analyses of FG plates and shells.

Vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers under blast load

In this article,vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers subjected to blast load are studied.Higher-order ES-MITC3 element based on higher-order shear deformation theory(HSDT)to achieve the governing equations.The sandwich plates with the ultra-light feature of the auxetic honeycomb core layer(negative Poisson’s ratio)and reinforced by two laminated three-phase skin layers.The obtained results in our work are compared with other previously published to confirm accuracy and reliability.In addition,the effects of parameters such as geometrical and material parameters on the vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers are fully investigated.

Prediction of Natural Frequency of Laminated Composite Plates Using Artificial Neural Networks

The paper is focused on the application of artificial neural networks (ANN) in predicting the natural frequency of laminated composite plates under clamped boundary condition. For training and testing of the ANN model, a number of finite element analyses have been carried out using D-optimal design in the design of experiments (DOE) by varying the fibre orientations, –45?, 0?, 45? and 90?. The composite plate is modeled using linear layered structural shell element. The natural frequencies were found by analyses which were done by finite element (FE) analysis software. The ANN model has been developed using multilayer perceptron (MLP) back propagation algorithm. The adequacy of the developed model is verified by coefficient of determination (R). It was found that the R2 (R: coefficient of determination) values are 1 and 0.998 for train and test data respectively. The results showed that, the training algorithm of back propagation was sufficient enough in predicting the natural frequency of laminated composite plates. To judge the ability and efficiency of the developed ANN model, absolute relative error has been used. The results predicted by ANN are in very good agreement with the finite element (FE) results. Consequently, the D-optimal design and ANN are shown to be effective in predicting the natural frequency of laminated composite plates.

A FINITE ELEMENT ANALYSIS OF THE FLANGE EARRINGS OF STRONG ANISOTROPIC SHEET METALS IN DEEP-DRAWING PROCESSES

Flange earrings of strong anisotropic sheet metals in deep-drawing process are numerically analyzed by the elastic-plastic large deformation finite element formulation based on a discrete Kirchhoff triangle plate shell element model. A Barlat-Lian anisotropic yield function and a quasi-flow corner theory are used in the present formulation. The numerical results are compared with the experimental ones of cylindrical cup drawing process. The focus of the present researches is on the numerical analysis and the constraining scheme of the flange earring of circular sheets with strong anisotropy in square cup drawing process.

Three Dimensional Coupling between Elastic and Thermal Fields in the Static Analysis of Multilayered Composite Shells

This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model.This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic,composite and sandwich structures.Cylindrical and spherical panels,cylinders and plates are analyzed in orthogonal mixed curved reference coordinates.The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels.The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate.The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature.The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses.As a consequence,the sovra-temperature description is assumed to be an unknown variable of themodel and it is calculated in the sameway as the three displacements.The final systemis based on a set of coupled homogeneous differential relations of second order in the thickness coordinate.This system is reduced in a first order differential relation system by redoubling the number of unknowns.Therefore,the exponential matrix methodology is applied to calculate the solution.The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components.After an appropriate preliminary validation,new benchmarks are discussed for several thickness ratios,geometrical data,lamination sequences,materials and sovra-temperature values imposed at the outer faces.Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation.Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.

Numerically Simulating the Sandwich Plate System Structures

Sandwich plate systems (SPS) are advanced materials that have begun to receive extensive attention in naval architecture and ocean engineering.At present, according to the rules of classification societies, a mixture of shell and solid elements are required to simulate an SPS.Based on the principle of stiffness decomposition, a new numerical simulation method for shell elements was proposed.In accordance with the principle of stiffness decomposition, the total stiffness can be decomposed into the bending stiffness and shear stiffness.Displacement and stress response related to bending stiffness was calculated with the laminated shell element.Displacement and stress response due to shear was calculated by use of a computational code write by FORTRAN language.Then the total displacement and stress response for the SPS was obtained by adding together these two parts of total displacement and stress.Finally, a rectangular SPS plate and a double-bottom structure were used for a simulation.The results show that the deflection simulated by the elements proposed in the paper is larger than the same simulated by solid elements and the analytical solution according to Hoff theory and approximate to the same simulated by the mixture of shell-solid elements, and the stress simulated by the elements proposed in the paper is approximate to the other simulating methods.So compared with calculations based on a mixture of shell and solid elements, the numerical simulation method given in the paper is more efficient and easier to do.

The Interlaminar Stress of Laminated Composite under Uniform Axial Deformation

The interlaminar stresses are analyzed by combining the first shear theory with the layerwise theory method. And the plate subjected to a uniform axial strain is studied by the simplified displacement field. Using the simplified displacement field, the equations of finite element method is developed by the principle of virtual work. And the amount of calculation is reduced by using the linear element. Then, some numerical examples are given to verify the accuracy of the method and analyze the distribution of interlaminar stresses along y-axis and z-axis. The shapes of the stresses’ curves in the vicinity of the free edge are very different from the interior area. Moreover, the influence of the ply angle on the interlaminar stresses is analyzed for the plate [θ/-θ]s. It can be found that the shapes of the stresses along z-axis are similar when the angle is different, while the values of the interlaminar stresses are changed apparently with the ply angle.

Postbuckling of Marine Stifened Composite Plates with Initial Geometric Imperfections Using Progressive Failure Analysis

This work explores the postbuckling behavior of a marine stifened composite plate in the presence of initial imperfections.The imperfection shapes are derived from buckling mode shapes and their combinations.Thereafter,these imperfection shapes are applied to the model,and nonlinear large defection fnite element and progressive failure analyses are performed in ANSYS 18.2 software.The Hashin failure criterion is employed to model the progressive failure in the stifened composite plate.The efect of the initial geometric imperfection on the stifened composite plate is investigated by considering various imperfection patterns and magnitudes.Results show that when the magnitude of the imperfection is 20 mm,the ultimate strength of the stifened composite plate decreases by 31%.Moreover,global imperfection shapes are found to be fundamental in determining the ultimate strength of stifened composite plates and their postbuckling.

考虑局部屈曲加劲钢箱形构件极限承载力的纤维梁柱模型

大型桥梁加劲钢箱形构件的极限承载力会受到板件局部稳定性能的影响,而目前尚缺乏成熟的可直接计入局部屈曲的高效计算模型。归纳总结国内外钢构件计算方法对局部屈曲的处理方式,明确以有效应力法即采用板件(或加劲板)平均受压应力-应变曲线简化局部屈曲效应的研究思路。通过有限元程序ANSYS二次开发钢板拉压非对称本构模型,提出一种可直接计入局部屈曲的纤维梁柱模型。材料本构模型中受压应力-应变曲线通过小型的加劲板板壳模型得到,且全面考虑了焊接残余应力与几何缺陷。搜集已开展的钢箱梁、钢桥塔及钢拱肋的稳定性试验,采用提出的纤维梁柱模型、板壳模型与传统弹塑性梁柱模型进行分析,展示截面纤维划分策略,验证文中纤维梁柱模型的可靠性与高效性;通过设置不同的缺陷参数,探讨缺陷水平与局部屈曲对各个构件极限承载力的影响。研究表明,随着缺陷水平的严重,局部屈曲会显著降低极限承载力,提出的纤维梁柱单元模型能够有效地预测加劲钢箱形构件的极限承载力,具有较广泛的应用前景。