Three-node Triangular Plate Elements Research Articles

A MITC3+ element improved by edge-based smoothed strains for analyses of laminated composite plates using the higher-order shear deformation theory

In this paper, a new three-node triangular plate element is proposed to analyze laminated composite plates based on the higher-order shear deformation theory (HSDT). Originating from the MITC3+ shell finite elements, the displacement fields of the HSDT are interpolated by usual linear functions of the three-node triangular element and a cubic supplemented function associated with a node located at the centroid of the element. The transverse shear strain fields are separately approximated according to the MITC3+ shear-locking removal technique. The edge-based smoothed (ES) strain method is employed to improve the in-plane strain fields. Applying the divergence theorem, the surface integration of the in-plane stiffness matrices is transformed into the line integration on the boundary of the smoothing domains. The performance of the so-called ES-MITC3+ plate elements is validated through several numerical examples. The numerical results of the static, frequency and buckling analyses when this new element is used converge to the exact solutions and agree well with those given in other references.

Free vibration analysis of laminated composite and sandwich plates based on a mixed zigzag theory

In this article, an accurate mixed zigzag theory (HZZTM) is proposed for free vibration analysis of laminated composite and sandwich plates. In-plane displacements in the proposed zigzag theory nonlinearly distribute through the thickness and exhibit an abrupt discontinuity of slope at interfaces, which coincide with three-dimensional analysis. In order to obtain the accurate transverse shear stresses, a preprocessing approach based on the three-dimensional (3D) equilibrium equations and the Reissner mixed variational theorem (RMVT) is used. It is significant that the second-order derivatives of in-plane displacement variables have been removed from the transverse shear stress fields, such that the finite element implementation is greatly simplified. Thus, based on the proposed zigzag model, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for free vibration analysis of thick laminated composite and sandwich plates. Moreover, the accurate transverse shear stresses are introduced in the dynamic equilibrium equation by employing the Hamilton’s principle, which can actively improve the accuracy of natural frequencies of the laminated composite plates with different thickness and material properties at each ply. Performance of the proposed model is assessed by comparing with several benchmark solutions. Agreement between the present results and the reference solutions is very good, and the proposed model only includes the seven displacement variables which can demonstrate the accuracy and effectiveness of the proposed model.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.

Improvement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysis

An improved three-node triangular plate finite element is presented for plate analysis. The key idea is to enhance the performance of the original MITC3 element with an edge-based strain smoothing technique. To address this derivation, the MITC3 is explicitly formulated to obtain constant gradient matrices. All strain fields are then averaged over edge-based integration domains defined by two nodes of an element’s edge and the centroid(s) of element(s) sharing the element’s edge. A stabilized coefficient factor is also used for the shearing stiffness. We name this element as ES-MITC3, and it is then employed to analyze various plate problems under static loading and free vibration. Numerical results show that the ES-MITC3 element is free of “shear locking” and performs well for static and vibration behaviors. The present element exhibits a good competitor with other several existing three-node triangular plate elements, even the four-node MITC4 element.

A refined three-node triangular element based on the HW variational theorem for multilayered composite plates

For the three-node triangular elements, the displacement parameters are linear variation within one element, so that the second and the third derivatives of displacement parameters are close to zero. Therefore, the three-node triangular element will encounter difficulty to accurately predict transverse shear stresses by integrating the three-dimensional equilibrium equation through the thickness direction. Thus, the few three-node triangular plate elements can accurately yield the transverse shear stresses of multilayered composite plates. A higher-order zig-zag model accounting for the zero transverse shear strain conditions at the top and the bottom surfaces is firstly presented in this work. Based on the proposed model, a refined three-node triangular plate element satisfying the requirement of C1 weak-continuity conditions in the inter-element is developed to accurately produce the distributions of displacements and stresses through the thickness of multilayered composite plates. To obtain the improved transverse shear stresses, the simplified three-dimensional equilibrium equation has been a priori used. It is significant that the higher-order derivatives of displacement parameters have been taken out from the expression of transverse shear stresses by employing the three-field Hu-Washizu (HW) variational principle, which is convenient for the finite element implementation. Performance of the present element is tested by comparing with three-dimensional elasticity solutions as well as the reference results evaluated using other models. A detailed study is conducted to highlight the influences of boundary condition, number of layers and aspect ratio on the static response of multilayered composite plates, and numerical results can show the accuracy and range of applicability for the proposed element.

Development of the Cell-based Smoothed Discrete Shear Gap Plate Element (CS-FEM-DSG3) using Three-Node Triangles

The paper presents the formulation and recent development of the cell-based smoothed discrete shear gap plate element (CS-FEM-DSG3) using three-node triangles. In the CS-FEM-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the original plate element DSG3 is used to compute the strains and to avoid the transverse shear locking. Then the cell-based strain smoothing technique (CS-FEM) is used to smooth the strains on these three sub-triangles. The numerical examples illustrate four superior properties of the CS-FEM-DSG3 including: (1) being a strong competitor to many existing three-node triangular plate elements in the static analysis; (2) giving high accurate solutions for problems with skew geometries in the static analysis; (3) giving high accurate solutions in free vibration analysis; (4) providing accurate values of high frequencies of plates by using only coarse meshes. Due to its superior and simple properties, the CS-FEM-DSG3 has been now developed for various analyses such as: flat shells, stiffened plates, functionally graded plates, composite plates, piezoelectricity composite plates, cracked plate and plates resting on the viscoelastic foundation subjected to moving loads, etc.

A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation

A cell-based smoothed finite element method using three-node Mindlin plate element (CS-FEM-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS-FEM-MIN3 is extended and incorporated with damping-spring systems for dynamic responses of sandwich and laminated composite plates resting on viscoelastic foundation subjected to a moving mass. The plate-foundation system is modeled as a discretization of three-node triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results. A parametric examination is also conducted to determine the effects of various parameters on the dynamic response of the plates on the viscoelastic foundation subjected to a moving mass.

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates

The cell-based strain smoothing technique is combined with the well-known three-node Mindlin plate element (MIN3) to give a so-called the cell-based smoothed MIN3 (CS-MIN3) for static and free vibration analyses of plates. In the process of formulating the system stiffness matrix of the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized MIN3 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 sub-triangles. The numerical examples demonstrated that the CS-MIN3 is free of shear locking, passes the patch test and shows four superior properties such as: (1) be 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, (4) can provide accurately the values of high frequencies of plates by using only coarse meshes.

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 simple nonlinear triangular plate element and strategies of computation for nonlinear analysis

According to the rigid body rule, for a solid member subjected to rigid body rotations, the initial forces acting on the member that form an equilibrating set must rotate following the rigid body rotations, while remaining unchanged in magnitude. Such a rule is physically intuitive and is employed in this paper to derive an approximate geometric stiffness matrix for a three-node triangular plate element (TPE) containing three translational and three rotational degrees of freedom (DOFs) at each node. An element such as this is attractive, since it can be easily used along with the 12-DOF beam element to simulate various plate and shell assemblies. Another advantage with the geometric stiffness matrix derived is that it can be explicitly given, which renders numerical integrations unnecessary. Finally, the element and procedure proposed are demonstrated to be robust in that solutions of good accuracy can always be obtained if a practically fine mesh has been used, and that the solutions converge rapidly to the exact one upon mesh refinement.