Kirchhoff Plate Element Research Articles

Topology optimization of plate structures using plate element-based moving morphable component (MMC) approach

A topology optimization approach for designing the layout of plate structures is proposed in this article. In this approach, structural mechanical behavior is analyzed under the framework of Kirchhoff plate theory, and structural topology is described explicitly by a set of moving morphable components. Compared to the existing treatments where structural topology is generally described in an implicit manner, the adopted explicit geometry/layout description has demonstrated its advantages on several aspects. Firstly, the number of design variables is reduced substantially. Secondly, the obtained optimized designs are pure black-and-white and contain no gray regions. Besides, numerical experiments show that the use of Kirchhoff plate element helps save 95–99% computational time, compared with traditional treatments where solid elements are used for finite element analysis. Moreover the accuracy of the proposed method is also validated through a comparison with the corresponding theoretical solutions. Several numerical examples are also provided to demonstrate the effectiveness of the proposed approach.

Strain gradient differential quadrature Kirchhoff plate finite element with the C2 partial compatibility

This paper constructs a four-node Kirchhoff plate element considering dilatation, deviatoric stretch and rotation gradient effects to address the general boundary value problems of size-dependent isotropic thin micro-plates. This element benefits from the merits of differential quadrature method (DQM) and finite element method (FEM) and possesses four nodal displacement parameters at each node, i.e., deflection, its two first partial derivatives and one second mixed partial derivative with respect to two in-plane coordinates. To guarantee the C2 partial compatibility among neighboring elements, we establish a novel DQ-based geometric mapping scheme relating the deflection values at Gauss-Lobatto quadrature points to the displacement parameters at four nodes. By applying the DQ rule, the Gauss-Lobatto quadrature rule and the developed mapping scheme, the total potential energy of a generic gradient-elastic Kirchhoff plate element is represented as a function of nodal displacement parameters. The element formulation is derived using the minimum total potential energy principle. Several numerical examples are provided to demonstrate the validity of the proposed method and explore the static bending, free vibration and critical buckling behavior of thin micro-plates. It is validated that the size-dependence of vibration and critical buckling mode shapes of thin micro-plates can be observed in some cases.

Shear deformable plate elements based on exact elasticity solution

The 2-D approximation functions based on a general exact 3-D plate solution are used to derive locking-free, rectangular, 4-node Mindlin (i.e., first-order plate theory), Levinson (i.e., a third-order plate theory), and Full Interior plate finite elements. The general plate solution is defined by a biharmonic mid-surface function, which is chosen for the thick plate elements to be the same polynomial as used in the formulation of the well-known nonconforming thin Kirchhoff plate element. The displacement approximation that stems from the biharmonic polynomial satisfies the static equilibrium equations of the 2-D plate theories at hand, the 3-D Navier equations of elasticity, and the Kirchhoff constraints. Weak form Galerkin method is used for the development of the finite element model, and the matrices for linear bending, buckling and dynamic analyses are obtained through analytical integration. In linear buckling problems, the 2-D Full Interior and Levinson plates perform particularly well when compared to 3-D elasticity solutions. Natural frequencies obtained suggest that the optimal value of the shear correction factor of the Mindlin plate theory depends primarily on the boundary conditions imposed on the transverse deflection of the 3-D plate used to calibrate the shear correction factor.

A dynamic analysis approach for identifying the elastic properties of unstitched and stitched composite plates

The identification of elastic properties from the dynamic behavior of structure has long been an alternative method to mechanical test. This method can avoid the variability of properties due to extraction of the specimen and achieve better homogenization by involving the whole structure. The mode classification is proposed and integrated in the identification process in this paper and helps to reduce the calculation time.Firstly, the measured frequencies are provided by vibration test while the calculated frequencies are provided by a finite element model with discrete Kirchhoff plate element. Then a mode classification process is conducted to accelerate the identification. Every constant is identified by minimizing the difference between the calculated and the measured frequencies of the group that it dominates. This method has been applied to two types of composite plate: unstitched plate and dispersedly stitched plate which presents both macro and meso heterogeneity. The limited dimension along the width direction of the stitch makes the modulus inaccessible by mechanical test and thus this dynamic method turns to be indispensable.

Strain Gradient Finite Element Analysis on the Vibration of Double-Layered Graphene Sheets

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C1 continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.

Hybrid Stress Finite Element Methods for Plate and Shell Structures

This paper is a survey of the use of assumed stress hybrid finite element methods for analyzing plate and shell structures. It describes the construction of Kirchhoff plate element by complementary energy principle, and of Reissner-Mindlin plate element by Hellinger-Reissner principle. Special hybrid stress elements are used to model joining of elements which are not coplanar. Such elements include the so-called semiLoof plate elements and 3D-solid elements tailored for analyzing thin plates and shells.

On the linear analysis of plates and shells using a new-16 degrees of freedom flat shell element

In this paper, we present a new quadrilateral discrete Kirchhoff flat shell element (called DKQ16) with 16 degrees of freedom (three displacements U, V, W at each corner, a rotation θ s at each mid-side) for the linear analysis of plates and shells. This new element is formulated on the basis of the so-called rational element method proposed by Zhong et al. [Zhong WX, Zeng J. J Computat Struct Mech Appl 1996;13:1–8 [in Chinese]]. In this new formulation, the rational quadrilateral plane element (RQ4) is employed for the membrane part and a new discrete Kirchhoff plate element DKQ8 proposed by the present authors [Batoz JL, Hammadi F, Zheng CL, Zhong WX, submitted for publication] is taken for the bending part. The DKQ16 element is one of the most simple quadrilateral flat shell elements. It can be combined with the DKT12 (or Morley) element. Numerical results for some typical problems demonstrate the overall good performance of the new shell element.

Finite element free flexural vibration analysis of arbitrary plates

In the analysis of plates with arbitrary geometry, the isoparametric element is widely used because of its capability to model an arbitrary geometry successfully. But the isoparametric element tends to suffer from the problem of shear locking when applied to very thin plates which are common to many practical structures. Though techniques such as reduced integration or selective integration are used to avoid this effect, they do not necessarily guarantee the solution to the problem and they may lead to the generation of zero energy spurious mechanisms. In this paper a new approach for the analysis of plates of arbitrary shapes is used which successfully models the arbitrary shape of the plate like isoparametric element. But in this new element as the shear deformation is ignored by using Kirchhoff plate element all the above discrepancies inherent in the isoparametric element are completely avoided. Thus is developed for the first time a rectangular element through which a Kirchhoff plate of any arbitrary shape can be analysed.

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

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

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

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

Quadrilateral finite elements for analysis of thick and thin plates

We discuss two quadrilateral plate elements applicable in the analysis of both thick and thin plates. The elements are based on Reissner-Mindlin plate theory and an enhanced displacement interpolation, which enables the consistent loading vector to be constructed. The constraint on the constant shear strain is enforced explicitly thus eliminating the shear locking phenomena in the analysis of thin plates. As a by-product of this work, we take a new look at a well-known discrete Kirchhoff plate element.

Plate quadrilateral finite element with incompatible modes

AbstractA four‐node quadrilateral plate element with a set of incompatible modes is presented in this work. In an analysis of thin plates, the element exhibits a similar performance as the well known discrete Kirchhoff plate element. The computational effort associated with the element is also comparable to that associated with the discrete Kirchhoff element. However, as opposed to the discrete Kirchhoff plate element, the presented plate element is based on the Reissner–Mindlin plate theory and can be used in a thick plate analysis as well.

Discrete Kirchhoff plate element and work equivalent loads

Discrete Kirchhoff plate element and work equivalent loads

On the pituitary tumuours induced by I 131 in mice : Susumu Watanabe, Fumio Hirose and Hiroyuki Takemoto, Department of Pathology, Hiroshima University School of Medicine

The identification of elastic properties from the dynamic behavior of structure has long been an alternative method to mechanical test. This method can avoid the variability of properties due to extraction of the specimen and achieve better homogenization by involving the whole structure. The mode classification is proposed and integrated in the identification process in this paper and helps to reduce the calculation time.Firstly, the measured frequencies are provided by vibration test while the calculated frequencies are provided by a finite element model with discrete Kirchhoff plate element. Then a mode classification process is conducted to accelerate the identification. Every constant is identified by minimizing the difference between the calculated and the measured frequencies of the group that it dominates. This method has been applied to two types of composite plate: unstitched plate and dispersedly stitched plate which presents both macro and meso heterogeneity. The limited dimension along the width direction of the stitch makes the modulus inaccessible by mechanical test and thus this dynamic method turns to be indispensable.