Triangular Plate Element Research Articles

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

AbstractFree vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.

Interlaminar shear stresses around an internal part-through hole in a laminated composite plate

Abstract The equilibrium/compatibility method, which is a semi-analytical post-processing method, is employed for computation of hitherto unavailable through-thickness variation of interlaminar (transverse) shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of an internal part-through circular cylindrical hole weakening an edge-loaded laminated composite plate. A C o -type triangular composite plate element, based on the assumptions of transverse inextensibility and layer-wise constant shear-angle theory (LCST), is utilized to first compute the in-plane stresses and layer-wise through-thickness average interlaminar shear stresses, which serve as the starting point for computation of through-thickness distribution of interlaminar shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of the part-through hole. The same stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the bi-layer interface circumferential re-entrant corner line singularity arising out of the internal part-through hole, and are found to violate the interfacial compatibility condition. The computed interlaminar shear stress can vary from negative to positive through the thickness of a cross-ply plate in the neighborhood of this kind of stress singularity.

A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.

A higher-order plate element for accurate prediction of interlaminar stresses in laminated composite plates

Lower-order finite elements have been conventionally used to solve a wide range of practical problems involving laminated composite plates. However, these elements are unable to provide accurate distributions of stress resultants in structures with free edges and stress singularities. Thus, the objective of this paper is to present a higher-order triangular plate element that is based on the third-order shear deformation theory and a layer-wise plate theory of Reddy for the bending analysis of laminated composite plates. It will be shown herein that the proposed element is able to accurately predict the shear forces and twisting moments as well as the transverse shear stresses across the thickness of a laminated composite plate. Unlike the usual deflection criterion used for establishing the convergence performance of most finite elements, the present study employs a stress analysis which is pivotal in the design phase of a structural component. Finally, the developed element is shown to be able to handle complex problems involving stress singularities more efficiently as compared to some widely used lower-order finite elements.

New rotation-free finite element shell triangle accurately using geometrical data

A new triangle shell element is presented. The advantages of this element are threefold: simplicity, generality and geometrical accuracy. The formulation is free from rotation degrees of freedom. The triangle here presented can be used regardless of the mesh topology, thus generality is conserved for any mesh-represented surface. From an original first order approach we evolve to a third order geometric description. The higher degree geometric description is based on the Bézier triangles concept, a very well known geometry in the domain of CAGD [G. Farin, Curves and Surfaces for CAGD. A Practical Guide, fifth ed., Morgan Kaufmann Publishers, San Francisco, CA, 2002]. Using this concept we show the path to reconstruct a general third order interpolating surface using only the three coordinates at each node. This work takes as starting point the nodal implementation of a basic triangle shell element [E. Oñate, F. Zárate, Rotation-free triangular plate and shell elements, Int. J. Numer. Methods Engrg. 47 (2000) 557–603]. In order to use an exact formula for the curvature, the normal directions at each node and the way to characterize them are proposed. Then, the geometrical properties and the mechanical behavior of the surface created are introduced. Finally, different examples are presented to depict the versatility and accuracy of the element.

The stability of stars of triangular equilibrium plate elements

AbstractEquilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models, but when formulating equilibrium models questions of stability, or admissibility of loads, are of major concern. This paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and/or flexural actions. Patches of elements that share a common vertex are considered, and such patches are termed stars. Stars may be used in global analyses as assemblies of elements forming macro‐elements, or in local analyses. The conditions for stability, or the existence and number of spurious kinematic modes, are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration. The procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form. Examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links. Copyright © 2008 John Wiley & Sons, Ltd.

Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

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.

Nonlinear Panel Flutter of Composite Sandwich Plates with Thermal Effect

ABSTRACTBy considering the total transverse displacement of a sandwich plate as the sum of the displacement due to bending of the plate and that due to shear deformation of the core, a high precision higher order triangular plate element is developed for the nonlinear panel flutter analysis of thermally buckled sandwich plates. Von Karman large deformation assumptions and quasi-steady aerodynamic theory are employed for the analysis. Newmark numerical time integration method is applied to solve the nonlinear governing equations in time domain. Results show that temperature will increase both the maximum displacement and motion speed of the plate. But the maximum displacement and velocity of the plate will not vary much with the aerodynamic pressure. Buckle pattern change phenomenon occurred in some specific case will increase the flutter boundary and change the flutter motion type of the plate. Temperature gradient increases the overall stiffness of the plate, which in turn stabilizes the sandwich panel and increases the flutter boundary of the plate.

Computation of stress resultants in plate bending problems using higher-order triangular elements

This paper is concerned with the development of a higher-order triangular plate element based on the first-order shear deformation plate theory. The present study brings out the shortcomings of conventional lower-order finite element interpolation when applied for the bending analysis of plates with free edges. The stress resultants, especially the transverse shear forces and twisting moments, obtained using the lower-order displacement finite elements fail to satisfy the natural (or force) boundary conditions accurately. On the other hand, the plate element with higher-order interpolation of the field variables enables the accurate prediction of stress resultants for plates. The superior performance of the higher-order plate bending element is demonstrated through bending analyses of plates of various shapes and free edges and examining the distribution of the stress resultants and the satisfaction of the natural boundary conditions.

A new finite element for free vibration analysis of stiffened composite plates

A new 6-noded stiffened triangular plate element for the analysis of stiffened composite plates based on Mindlins deformation plate theory has been developed. The stiffened plate element is a combination of basic triangular element and bar component. The element can accommodate any number of arbitrarily oriented stiffeners and obviates the use of mesh lines along the stiffeners. Free vibration analyses of stiffened laminated plates have been carried out with this element and the results are compared with those published. The finite element results show very good matching with the experimental ones.

An isostatic assumed stress triangular element for the Reissner–Mindlin plate‐bending problem

AbstractThe paper describes a new assumed stress triangular element for Reissner–Mindlin plates, called TIP3, with three nodes and three degrees of freedom per node. The kinematics is constructed by means of the so‐called linked interpolation ruled by technically significant degrees of freedom (i.e. one transversal displacement and two rotations per node) without using additional bubble modes. The static representation starts from a moment–shear uncoupled polynomial approximation and is constrained to satisfy some equilibrium conditions in order to reduce the stress parameters to a minimum number. The resulting element is locking free, presents correct rank and passes the bending patch test even for very thin plates. The good performances of the element are demonstrated by several comparisons with other triangular plate elements available in the literature. Copyright © 2007 John Wiley & Sons, Ltd.

Weakening effects of internal part-through elliptic holes in homogeneous and laminated composite plates

Abstract A C 0 -type triangular composite plate element, based on the assumptions of transverse inextensibility and layer-wise constant shear-angle theory (LCST) is employed to analyze an edge-loaded laminated plate weakened by an internal (or embedded) part-through elliptical hole. A homogenous isotropic plate weakened by a similar part-through elliptical hole can be analyzed as a special case of the above. Numerical results for a cross-ply laminate compromised by the presence of an embedded part-through elliptical hole indicate the existence of severe cross-sectional warping in the vicinity of the hole and plate boundaries. Furthermore, three-dimensional nature of the stress concentration factor in the neighborhood of the internal part-through elliptic hole boundary is clearly exhibited. Besides, very high stress concentration factors are found in the layer weakened by the part-through hole. Finally, for a cross-ply laminate compromised by the presence of an embedded part-through elliptic hole in the middle layer, in addition to the singularity at the circumferential line of intersection of the part-though elliptic hole with the material of the middle layer in the form of a circumferential bimaterial interface corner, there is a free-edge singularity at the bimaterial interface at the plate boundaries. These two stress singularities interact so that the effect of the part-through hole never dies down. This is unlike what is shown for a homogeneous isotropic plate weakened by the same part-through hole.

Rigid body concept for geometric nonlinear analysis of 3D frames, plates and shells based on the updated Lagrangian formulation

In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [ k g] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [ k g] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is featured by the fact that the formulation is simple, the expressions are explicit, and all kinds of actions are considered in the stiffness matrices. The robustness of the procedure is demonstrated in the solution of several benchmark problems involving the postbuckling response.

A reliable three‐node triangular plate element satisfying rigid body rule and incremental force equilibrium condition

This paper proposes a simple method for deriving the geometric stiffness matrix (GSM) of a three‐node triangular plate element (TPE). It is found that when the GSM of the element is combined into the global one of the structure, this structural stiffness matrix becomes symmetric and satisfies both the rigid body rule and incremental force and moment equilibrium (IFE) conditions, which are basically two fundamental conditions for analysis of mechanics. The former condition has been widely used in the community of mechanics; while the latter one, to our best knowledge, has never been considered. Advantages with the GSM derived are that derivations only need simple matrix operations without cumbersome non‐linear virtual strain energy derivations and tedious numerical integrations and more appealingly, this derived GSM can be explicitly given for applications. In addition, based on IFE and the rigid body rule conditions, a reasonable GSM for the three‐node TPE must be asymmetric; however, an asymmetric matrix usually gives rise to tedious numerical calculation especially in geometrically nonlinear problems and further, greatly influences computation efficiency. Fortunately, the skew‐symmetric parts of the derived GSM can be canceled out once they are merged into the global stiffness matrix of the structure. In this regard, this structural stiffness matrix becomes a symmetric one and thus enhances its effectiveness. Finally, several examples are provided for validating the robustness of the derived GSM.

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

AbstractA 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.

A new triangular layered plate element for the non-linear analysis of reinforced concrete slabs

A new 3-node, 18-DOF triangular layered plate element is developed in this paper for the geometric and material non-linear analysis of isotropic plates and reinforced concrete slabs under service loads. The proposed model is a combination of Allman's 3-node, 9-DOF triangular membrane element with drilling degrees of freedom and the refined non-conforming 3-node, 9-DOF triangular plate-bending element RT9 in order to account for the coupling effects between membrane and bending actions. The element is modelled as a layered system of concrete and equivalent smeared steel reinforcement layers, and perfect bond is assumed between the concrete layers and the smeared steel layers. The maximum normal stress criterion is employed to detect cracking of the concrete, and a smeared fixed crack model is assumed. Both geometric non-linearity with large displacements but moderate rotations and material non-linearity, which incorporates tension, compression, concrete cracking and tension stiffening, are included in the model. An updated Lagrangian approach is employed as a solution strategy for the non-linear finite element analysis and a numerical example of reinforced concrete slab is given to demonstrate the efficacy of this robust element. Copyright © 2005 John Wiley & Sons, Ltd.

Free Vibration of Thermally Buckled Composite Sandwich Plates

Abstract A high precision triangular plate element is developed for the free vibration analysis of thermally buckled composite sandwich plates. Due to an uneven thermal expansion in the two principal material directions, the buckling mode of the plate may change from one pattern to another in the postbuckling region for certain fiber orientation and aspect ratio of the plate. Because of this buckle pattern change, the sequence of natural frequencies of the plate is also suddenly altered. By examining the buckling and free vibration modes of the plate, a clear picture of buckle pattern change and vibration mode shifting is presented. Numerical results show that if the shape of a free vibration mode is similar to the plate buckling mode then the natural frequency of that mode will drop to zero when the temperature reaches the buckling temperature.

An efficient higher-order theory and finite element for laminated plates subjected to thermal loading

In the present paper, the global–local higher-order theory is simply derived, which satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. Moreover, the number of unknowns of this theory is independent of the layer numbers of the laminate. Based on the global–local higher-order theory, the discrete Kirchhoff element PDKT and refined triangular plate element PRT9 are presented for predicting interlaminar stresses and displacements in laminated plates subjected to thermal loading. The two triangular elements satisfy the interelement C 1 continuity conditions. The numerical examples show that the in-plane stresses and transverse shear stresses can be accurately calculated by the direct constitutive equation approach. The equilibrium equation approach is employed to predict transverse normal stresses.

A triangular hybrid equilibrium plate element of general degree

Hybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.