Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory

The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite element solutions.     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

An accurate higher-order theory and C finite element for free vibration analysis of laminated composite and sandwich plates

At present, it is difficult to accurately predict natural frequencies of sandwich plates with soft core by using the C⁰ plate bending elements. Thus, the C¹ plate bending elements have to be employed to predict accurately dynamic response of such structures. This paper proposes an accurate higher-order C⁰ theory which is very different from other published higher-order theory satisfying the interlaminar stress continuity, as the first derivative of transverse displacement has been taken out from the in-plane displacement fields of the present theory. Therefore, the C⁰ interpolation functions is only required during its finite element implementation. Based on the Hamilton’s principle and Navier’s technique, analytical solutions to the natural frequency analysis of simply-supported laminated plates have been presented. To further extend the ranges of application of the proposed theory, an eight-node C⁰ continuous isoparametric element is used to model the proposed theory. Numerical results show the present C⁰ finite element can accurately predict the natural frequencies of sandwich plate with soft core, whereas other global higher-order theories are unsuitable for free vibration analysis of such soft-core structures.     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.     

A four‐noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory

A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on the refined zigzag theory is presented. The element has seven kinematic variables per node. Shear locking is avoided by introducing an assumed linear shear strain field. The performance of the element is studied in several examples where the reference solution is the 3D finite element analysis using 20‐noded hexahedral elements. Copyright © 2013 John Wiley & Sons, Ltd.     

复合材料夹层板屈曲强度的分散性分析

将面板单层及芯层厚度、 面板纤维方向角和材料参数作为随机变量, 通过编制基于高阶剪切理论的随机有限元程序, 对复合材料夹层板的屈曲强度分散性进行了分析, 得到了不同铺层方式下夹层板屈曲强度分散系数随纤维方向角及夹层板长厚比的变化规律。并通过对屈曲强度的分散系数进行灵敏度分析, 确定了影响分散系数大小的主要因素, 从而为材料设计、 制造工艺改进和夹层板结构可靠度的提高提供了理论依据。      

Modal validation of sandwich shell finite elements based on a p‐order shear deformation theory including zigzag terms

This paper describes a set of improved C⁰‐compatible composite shell finite elements for evaluating the global dynamic response (natural frequencies and mode shapes) of sandwich structures. Combining a through‐the‐thickness displacement approximation of variable high order with a first‐order zigzag function, the proposed finite elements are suited for modelling sandwich plates and doubly curved shells with a non‐uniform thickness and are more accurate than conventional models based on the first‐ and third‐order shear deformation theories, especially in sandwich panels with highly heterogeneous properties. The new finite element model is then validated by a comparison with the standard shell and 3D solid models. From these investigations, it can be concluded that adding a zigzag function even to high‐order polynomial approximations of the through‐the‐thickness displacement is a useful tool for refining the modelling of sandwich structures. In addition, the proposed formulation is sufficiently versatile to represent with the same level of accuracy the behaviour of thin‐to‐thick laminated shells as well as of strongly heterogeneous sandwich structures. Copyright © 2008 John Wiley & Sons, Ltd.     

New finite element model for analysis of Kirchhoff plate

A new kind of approach to formulate an isotropic thin plate bending element is presented. The strain energy of the element is decomposed into two parts: an integral concerning the first strain invariant and a line integral around the elemental boundary. The former can be discretized by quasi-conforming technique¹ and the latter can be directly calculated using the interpolation of the deflection and its normal slope along the element boundary. By this method, an assumed first strain invariant quadrilateral element (AFSIQ) is derived. The procedure of formulating the element and the numerical examples show that the new kind of element not only simplifies the formulation considerably but also has excellent accuracy.     

An adaptive finite element method for second‐order plate theory

We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P₂‐approximations on simplices for the out‐of‐plane deformations, using C⁰‐continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Two simple and efficient displacement‐based quadrilateral elements for the analysis of composite laminated plates

Two simple 4‐node 20‐DOF and 4‐node 24‐DOF displacement‐based quadrilateral elements named RDKQ‐L20 and RDKQ‐L24 are developed in this paper based on the first‐order shear deformation theory (FSDT) for linear analysis of thin to moderately thick laminates. The deflection and rotation functions of the element sides are obtained from Timoshenko's laminated composite beam functions. Linear displacement interpolation functions of the standard 4‐node quadrilateral isoparametric plane element and displacement functions of a quadrilateral plane element with drilling degrees of freedom are taken as in‐plane displacements of the proposed elements RDKQ‐L20 and RDKQ‐L24, respectively. Due to the application of Timoshenko's laminated composite beam functions, convergence can be ensured theoretically for very thin laminates. The elements are simple in formulation, and shear‐locking free for extremely thin laminates even with full integration. A hybrid‐enhanced procedure is employed to improve the accuracy of stress analysis, especially for transverse shear stresses. Numerical tests show that the new elements are convergent, not sensitive to mesh distortion, accurate and efficient for analysis of thin to moderately thick laminates. Copyright © 2004 John Wiley & Sons, Ltd.     

Constitutive model for quasi‐static deformation of metallic sandwich cores

All‐metal sandwich construction holds promise for significant improvements in stiffness, strength and blast resistance for built‐up plate structures. Analysis of the performance of sandwich plates under various loads, static and dynamic, requires modelling of face sheets and core with some fidelity. While it is possible to model full geometric details of the core for a few selected problems, this is unnecessary and unrealistic for larger complex structures under general loadings. In this paper, a continuum constitutive model is proposed as an alternative means of modelling the core. The constitutive model falls within the framework of a compressible rate‐independent, anisotropic elastic–plastic solid. The general form of the model is presented, along with algorithmic aspects of its implementation in a finite element code, and selected problems are solved which benchmark the code against existing codes for limiting cases and which illustrate features specific to compressible cores. Three core geometries (pyramidal truss, folded plate, and square honeycomb) are considered in some detail. The validity of the approach is established by comparing numerical finite element simulations using the model with those obtained by a full three‐dimensional meshing of the core geometry for each of the three types of cores for a clamped sandwich plate subject to uniform pressure load. Limitations of the model are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Refined triangular discrete Kirchhoff plate element for thin plate bending,vibration and buckling analysis

A refined triangular discrete Kirchhoff thin plate bending element RDKT which can be used to improve the original triangular discrete Kirchhoff thin plate bending element DKT is presented. In order to improve the accuracy of the analysis a simple explicit expression of a refined constant strain matrix with an adjustable constant can be introduced into its formulation. The new element displacement function can be used to formulate a mass matrix called combined mass matrix for calculation of the natural frequency and in the same way a combined geometric stiffness matrix can be obtained for buckling analysis. Numerical examples are presented to show that the present methods indeed, can improve the accuracy of thin plate bending, vibration and buckling analysis. © 1998 John Wiley & Sons, Ltd.     

Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory

In this study, free vibration of laminated skew plates was investigated. Discrete singular convolution (DSC) method is used for numerical solution of vibration problems. The straight‐sided quadrilateral domain is mapped into a square domain in the computational space using a four‐node element by using the geometric transformation. Typical results are presented for different geometric parameters and boundary conditions. It is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate. The results obtained by DSC method are compared with those obtained by analytical and numerical approaches. It is shown that reasonable accurate results are obtained. Present work also indicates that the method of DSC is a promising and potential approach for computational mechanics. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

An improved four‐node hybrid‐mixed element based upon Mindlin's plate theory

Based on the mixed shear projected (MiSP) approach (Reference [54]: Int. J. Numer. Meth. Engng 1998; 42 :1149–1179), an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, such as the skew plate (30°) problem, have often shown poor convergence when low‐order elements (3‐ or 4‐node element) are developed using linear approximations for kinematic variables. To put right this weakness, we propose a high‐order interpolation for rotational dofs which results in more rich bending curvatures. The mid‐side rotational nodes are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypotheses. The derived 4‐node quadrilateral element, called MiSP4+, is free of shear locking and passes all patch tests for thick and thin plates in an arbitrary mesh. Copyright © 2002 John Wiley & Sons, Ltd.     

A cell‐based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner–Mindlin plates

The cell‐based strain smoothing technique is combined with discrete shear gap method using three‐node triangular elements to give a so‐called cell‐based smoothed discrete shear gap method (CS‐DSG3) for static and free vibration analyses of Reissner–Mindlin plates. In the process of formulating the system stiffness matrix of the CS‐DSG3, each triangular element will be divided into three subtriangles, and in each subtriangle, the stabilized discrete shear gap method is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three subtriangles. The numerical examples demonstrated that the CS‐DSG3 is free of shear locking, passes the patch test, and shows four superior properties such as: (1) being a strong competitor to many existing three‐node triangular plate elements in the static analysis; (2) can give high accurate solutions for problems with skew geometries in the static analysis; (3) can give high accurate solutions in free vibration analysis; and (4) can provide accurately the values of high frequencies of plates by using only coarse meshes. Copyright © 2012 John Wiley & Sons, Ltd.     

A hybrid‐mixed four‐node quadrilateral plate element based on sampling surfaces method for 3D stress analysis

The hybrid‐mixed assumed natural strain four‐node quadrilateral element using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the plate body N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed hybrid‐mixed four‐node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the 3D stress analysis of thin and thick plates because the SaS formulation gives the possibility to obtain solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

A triangular plate element for thermo‐elastic analysis of sandwich panels with a functionally graded core

A sandwich construction is commonly composed of a single soft isotropic core with relatively stiff orthotropic face sheets. The stiffness of the core may be functionally graded through the thickness in order to reduce the interfacial shear stresses. In analysing sandwich panels with a functionally gradient core, the three‐dimensional conventional finite elements or elements based on the layerwise (zig‐zag) theory can be used. Although these elements accurately model a sandwich panel, they are computationally costly when the core is modelled as composed of several layers due to its grading material properties. An alternative to these elements is an element based on a single‐layer plate theory in which the weighted‐average field variablescapture the panel deformation in the thickness direction. This study presents a new triangular finite element based on {3,2}‐order single‐layer theory for modelling thick sandwich panels with or without a functionally graded core subjected to thermo‐mechanical loading. A hybrid energy functional is employed in the derivation of the element because of a C¹ interelement continuity requirement. The variations of temperature and distributed loading acting on the top and bottom surfaces are non‐uniform. The temperature also varies arbitrarily through the thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved three‐node hybrid‐mixed element for Mindlin/Reissner plates

Based on the mixed shear projected (MiSP) approach [6], an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, as skew plate (30°) problem for instance, have often shown poor convergence when low order elements (3‐ or 4‐node element) are developed using linear approximations for kinematic variables. To put right this weakness, we propose a high‐order interpolation for rotational dofs which results in more rich bending curvatures. The mid‐side rotational nodes are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypothesises. The derived 3‐node triangular element, called MiSP3+, is free of shear locking and passes all patch‐tests for thick and thin plates in an arbitrary mesh. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical and experimental studies on aluminum sandwich plates of variable thickness

Abstract

The elastic flexural behavior of static deformation and free vibration of sandwich plates of variable thickness is investigated numerically and experimentally. In the analysis, the face plates are treated as Marguerre shells, and the core is assumed to be an antiplane core and to provide resistance to transverse shear and normal stresses only. Displacement continuity conditions are used at the interfaces between face plates and the core to derive the displacement field. Energy formulations are obtained and solved by the isoparametric finite element method. The numerical results are obtained to compare with the results in the existing literature and to show the effects of taper constant and face plate thickness on deflections and natural frequencies. Finally, experimental works based on the method of holographic interferometry are conducted to confirm the theoretical findings. Experimental and numerical data agree quite well in this work.