Quadrilateral Plate Element Research Articles

The effective finite element method for free and forced vibration analysis of 2D-FGSW plates lying an elastic foundation

The purpose of this study is to analyze the influence of geometric parameters and material properties on the free and forced vibration of bi-directional functionally graded sandwich (2D-FGSW) plates lying on an elastic foundation (EF) using an effective finite element method (FEM) based on the trigonometric shear deformation theory (TrSDT). The mechanical properties of 2D-FGSW are variable in both the x- and z-directions. The EF includes the springer stiffness k 1 and the shear layer stiffness k 2. Moreover, the impacts of structural damping and periodic loading on the behavior of 2D-FGSW plates are considered. Element matrices are created using a four-node quadrilateral plate element (Q4) with eight degrees of freedom per node. The effectiveness and accuracy of the current formulation are confirmed by contrasting the acquired results with those of earlier works. Then, many studies are carried out to illustrate how input parameters might affect the vibration characteristics of 2D-FGSW plates lying on EF. In addition, the eigenfrequencies of different types of 2D-FGSW plates with a center hole are investigated.

Buckling analysis of composite plates surface bonded with graphene‐reinforced piezoelectric actuators

AbstractGraphene is a highly conductive and exceptional material that has shown potential for significantly improving the piezoelectric and mechanical properties of piezoelectric matrix. This paper examines the stability of composite plates surface bonded with graphene‐reinforced composite piezoelectric (GRCP) actuators. However, the significant differences in material characteristics at the interfaces will pose some challenges in analyzing the buckling behavior of piezoelectric composite plates by means of existing higher‐order shear deformation theories. To address this issue, a refined plate theory is developed for the buckling analysis of piezoelectric composite plates. The developed theory includes a new interlaminar shear stress field that precisely describes the distribution of interlaminar shear stresses, which differs from earlier higher‐order theories. It should be emphasized that the finite element formulation can be simplified by removing the second‐order derivatives of in‐plane displacement parameters from interlaminar shear stresses. In terms of the developed theory, a four‐node C0 quadrilateral plate element is introduced to examine the buckling behavior of piezoelectric composite plates. Furthermore, the modified interlaminar shear stress field is absorbed into the strain energy, significantly improving the ability to predict the critical loads of composite plates with GRCP actuators. The refined plate model is evaluated through the utilization of three‐dimensional (3D) elasticity solutions and results obtained from other associated theories. Numerical results demonstrate that the refined plate model is capable of producing favorable results, and a comprehensive examination is conducted to analyze the effects of significant parameters on the buckling behaviors of piezoelectric composite plates. This study sets a solid foundation for future research and development on the application of graphene‐reinforced composite piezoelectric actuators in composite structures.Highlights Buckling analysis of composite plates with GRCP actuators is conducted. A refined plate theory with modified interlaminar shear stresses is developed. The ability to predict critical loads can be significantly improved. The finite element formulation can be simplified based on the proposed model. Comprehensive parametric studies on buckling behavior are caried out.

An accurate and efficient 4-noded quadrilateral plate element for free vibration analysis of laminated composite plates using a refined third-order shear deformation plate theory

An accurate, efficient and reliable four-noded quadrilateral plate element is developed for the free vibration analysis of laminated composite plates. The element formulation is based on a refined third-order shear deformation plate theory (TSDT) and the quasi-conforming element technique. The equations of motion and the variational consistent mass matrix are derived through Hamilton’s principle. The resulting composite plate element is free from shear locking. Furthermore, both its element stiffness and mass matrices are computed explicitly. The convergence and accuracy of the present element are evaluated by the standard static and dynamic examples of isotropic plates. The accuracy of the computed natural frequencies of laminated composite plates is verified by the examples with various lamination schemes, elastic modulus ratios, aspect ratios and boundary conditions and by the comparison with other results. The effect of the associated boundary conditions in different TSDTs is also evaluated through numerical results. The influence of the higher-order and coupling mass matrices on the frequencies of higher-mode vibration is conducted too. The numerical results clearly demonstrate that the present plate element is not only very efficient but also capable of yielding accurate natural frequencies of both the fundamental and the higher-mode vibrations of laminated composite plates.

Three new triangular thick plate/shell elements based on absolute nodal coordinate formulation

In this paper, three new triangular fully parameterized thick plate/shell elements are proposed based on the absolute nodal coordinate formulation. The volume parameters and corresponding independent volume gradient vectors are used to simplify the element construction. The node-wise structural parameter is adopted to circumvent the slope discontinuity issue associated with the volume gradient vectors. Two three-node elements inherit the interpolation strategy in the membrane direction of a well verified thin plate element. For the thickness direction, linear and quadratic interpolation are used in the three-node elements. Inspired by the quadrilateral plate elements with gradient vectors only along thickness direction, a six-node triangular element is presented with quadratic interpolation in both membrane and thickness direction. Three static analyses with large deformation and one dynamic analysis are conducted to evaluate the performance of the new elements. It concludes that the three-node element with linear interpolation in the thickness direction performs not well due to the incompatible/low interpolation in the thickness direction. The other two elements show good accuracy in all the numerical tests. Though the six-node element exhibits slightly higher accuracy than the quadratic three-node element, its DOFs are much more than the quadratic three-node element, which makes it less attractive.

A hybrid first/third-order plate theory for finite element analysis of sandwich plates with a transversely compressible core

Sandwich structures with thick soft cores subjected to transverse loads are usually resulting in deformations with various modes, including bending, twisting and compressing. In addition, the compression of the core results in a significant influence of the structural response. All classical plate theories are inadequate in predicting the behavior sandwich structures with thick soft core due to the assumption of no thickness change during deformation. To overcome this issue, a hybrid first/third-order shear deformation hypothesis with consideration of compressive effect is proposed for sandwich structures with thick soft cores. The first-order shear deformation theory is applied to both the top and bottom layers, while a refined third-order shear deformation theory that releases the assumption of no thickness change is developed for the thick soft cores. An eight-node quadrilateral plate element with twelve degrees of freedom including compressible displacements is created for finite element modeling. Based on the proposed refined hybrid first/third-order theory, a finite element model is developed using the Hamilton’s principle. The numerical model is first validated by the computational results of sandwich structures using commercial software. Then, the influences of various parameters on static displacements, vibrations and compressive deformations are systematically investigated, including e.g. boundary conditions, thickness of the core and face layers, elastic modulus of the core layer and the plate size. The present finite element modeling approach provides a powerful tool to accurately investigate the mechanical response of sandwich structures.

A new higher-order mixed four-node quadrilateral finite element for static bending analysis of functionally graded plates

In this paper, a new C0 four-node quadrilateral plate element (MiQ4) is proposed for the static bending analysis of functionally graded (FG) plates based on a C0 higher-order shear deformation theory. By introducing a quadratic shape function for higher-order terms and a constant shear result, which is eliminated at the element level, the transverse shear stresses are free on two surfaces of the plates. The present finite element model has high accurateness and a fast rate of convergence, especially for both thin and very thin plates. In addition, the proposed element is insensitive to distorted mesh, hence, it is compatible with the investigation of complex geometrical plates. Although the proposed element does not need any shear correction coefficients and selected or reduced integrations, the shear-locking phenomenon does not appear even for very thin plates. Therefore, the computation cost and time are reduced and the spurious deformation or hourglass modes are avoided. A large number of examples have been exhibited that help to validate the convergence rate, correctness, and reliability of the proposed element. Then the present element is applied for analysis of FG structures to demonstrate the potential applications of the present plate element. Significantly, the paper provides a good element that can be used in many areas of engineering for the design, testing, analysis, and optimization of complex structures to reduce the computation cost and time as well as increase the reliability of the numerical results.

Improving the Bergan Wang Quadrilateral plate element by using the UI approach for analysis of isotropic and FGM sandwich plates

This work presents the improvement of Bergan Wang Quadrilateral (BWQ) element using the UI (unified and integrated) approach. In this approach, a new formulation of quadrilateral plate element is developed with a single independent variable just like what Bergan and Wang applied to develop a non-conforming BWQ element. The difference between the two approaches lies in the formulation of transverse displacement w, wherein the UI approach transverse displacement w is expressed as the sum of transverse bending, and shear displacements (wb and ws) and the transverse bending displacement wb becomes the single independent variable. The transverse shear displacement ws is defined as the second derivative of the transverse bending displacement wb. The rotations, curvatures, and shear strains are the first, second, and third derivatives of the bending displacement wb. The resulting non-conforming element, called UI-Q4 element, has proper rank, no shear locking problem, and passes the patch test. To examine the element convergence and accuracy, standard benchmark problems of isotropic plates, i.e. square plates, Razzaque and Morley skew plates, and circular plates, are used. The UI-Q4 element shows very good performance, which is observed by comparing it with MITC4, DSQK and BWQ elements. Moreover, the application of UI-Q4 element for analysing functionally graded material (FGM) sandwich plate analysis also performs well and delivers results which are quickly converged to the reference values.

Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM

The static bending and buckling behaviors of bi-directional functionally graded (BFG) plates with porosity are investigated in this paper. An improved first-order shear deformation theory with an assuming parabolic distribution shear stresses is developed to describe the displacement, strain, and stress fields of the plates. The significant novelty of the proposed theory is that the transverse shear stresses equal to zero at two free surfaces of the BFG plates. Therefore, no shear correction factor is required as in other first-order shear deformation theory. A four-node quadrilateral plate element (IMQ4) is developed based on the improved first-order shear deformation theory, mixed finite element method (FEM) and Hamilton's principle for analysis of BFG plates. Several comparison studies are provided to demonstrate the precision and robustness of the proposed plate element IMQ4. Then the proposed plate element, IMQ4, is employed to analyze the bending and buckling responses of the BFG plates. Some new numerical results on the flexural and buckling behaviors of BFG plates are achieved via a deep parametric study.

Spectral element modeling and experimental investigations on vibration behaviors of imperfect plate considering irregular hole and curved crack

• A general spectral element modeling method is firstly developed. • The method is applied for imperfect plate considering irregular hole and curved crack. • Modal tests considering irregular hole and curved crack are firstly implemented. • Spectral element model has excellent predictive capability. A weak-form spectral element modeling and the corresponding experimental studies in this work are concerned to clarify free vibration mechanism for the plate of arbitrary shape having built-in complex hole and crack. To effectively address the discontinuity problem at irregular hole as well as crack of varying curvature, some quadrilateral plate elements with variable curvature edges are rigidly coupled together to form the investigated structural system. A flexible coordinate mapping strategy is here employed for satisfying the adaptability requirements of the introduced displacement functions to the sophisticated solution domain. Accordingly, the complete vibration equation of the physical model is solved based on the well-known first-order scheme of plate in conjunction with 2D spectral element functions, where the boundary constraints and compatibility relations in the form of energy quantification are imposed through the penalty function method. After that, modal experiments considering six unconstrained imperfect plates with pre-configured crack and hole, are incorporated with a series of acceptable solutions, provided by the available literature and some associated models of FEM, for disclosing the excellent predictive capability of the spectral element methodology in vibration performance. The reported eigenfrequencies and eigenmodes respecting arbitrarily shaped plate introducing intricate hole and curved crack are the first presented vibration solutions, and meaningful benchmarks for evaluating correctness of other numerical or analytical approaches.

Development of a Pseudo-Nine-Node Thick Plate Element

A highly accurate new quadrilateral thick plate element based on the Reissner–Mindlin theory is developed in this paper. The proposed element possesses nine nodes for improved accuracy, but the mid-sides and center nodes are condensed to form the element stiffness matrix so that it behaves as a four-node element externally, which is more convenient for practical engineering applications. The behavior of this new element, PLATE-Q9, is free from shear locking problems, and it can degenerate to the thin plate solution. The performance of this new plate element is compared with similar quadrilateral plate elements using static condensation under highly distorted mesh conditions. The numerical problems with some popular plate elements under adverse condition are highlighted, and some strange results are obtained. It is also demonstrated that the proposed quadrilateral element can still perform well under these special cases, and is able to pass through all the patch tests.

Simulation of Dynamics During Deployment of Foldable Origami Structures

Foldable origami-based structures are a type of deployable structures that are increasingly applied in the space and building industries. When folded, the small size of such structures facilitates transportation and storage. Meanwhile, the properties of their larger deployed state may be of interest to different applications. A stable working condition is established by locking the structure in its deployed state, as in the process of deployment, the driving forces may generate a dynamic effect, thus leading to instability of the system. Hence, the study of dynamic characteristics of such structures, including trajectory, duration, velocity, and acceleration is of paramount importance. In this paper, based on the general dynamic equation and Lagrange’s equations of the first kind, the finite element method is adopted to investigate the dynamic deployment of foldable plate structures in terms of the generalized nodal coordinates. The proposed geometric description of a quadrilateral plate element is based on a folding plate composed of refined triangular elements, which are used to approximate the real shells in the structure. Subsequently, a MATLAB framework is developed on the basis of the element using the Newmark integration and the Newton–Raphson iteration method to simulate the deployment process of the structure. Comparisons between MATLAB results and ADAMS results verify the reliability of the framework in analyzing the dynamic deployment of the foldable origami-based structures with sufficient accuracy.

Porous Functionally Graded Plates: An Assessment of the Influence of Shear Correction Factor on Static Behavior

The known multifunctional characteristic of porous graded materials makes them very attractive in a number of diversified application fields, which simultaneously poses the need to deepen research efforts in this broad field. The study of functionally graded porous materials is a research topic of interest, particularly concerning the modeling of porosity distributions and the corresponding estimations of their material properties—in both real situations and from a material modeling perspective. This work aims to assess the influence of different porosity distribution approaches on the shear correction factor, used in the context of the first-order shear deformation theory, which in turn may introduce significant effects in a structure’s behavior. To this purpose, we evaluated porous functionally graded plates with varying composition through their thickness. The bending behavior of these plates was studied using the finite element method with two quadrilateral plate element models. Verification studies were performed to assess the representativeness of the developed and implemented models, namely, considering an alternative higher-order model also employed for this specific purpose. Comparative analyses were developed to assess how porosity distributions influence the shear correction factor, and ultimately the static behavior, of the plates.

Bending and free vibration behaviors of composite plates using the C0-HSDT based four-node element with in-plane rotations

In this paper the smoothed four-node element with in-plane rotations MISQ24 is combined with a C0-type higher-order shear deformation theory (C0-HSDT) to propose an improved linear quadrilateral plate element for static and free vibration analyses of laminated composite plates. This improvement results in two additional degrees of freedom at each node and require no shear correction factors while ensuring the high precision of numerical solutions. Composite plates with different lay-ups, boundary conditions and various geometries such as rectangular, skew and triangular plates are analyzed using the proposed element. The obtained numerical results are compared with those from previous studies in the literature to demonstrate the effectiveness, the reliability and the accuracy of the present element.
 Keywords:
 composite laminated plates; bending problems; free vibration; C0-HSDT; MISQ24.

An accurate zigzag theory for bending and buckling analysis of thick laminated sandwich plates with soft core

An accurate and computationally attractive zigzag theory is developed for bending and buckling analysis of thick laminated soft core sandwich plates. The kinematic assumptions of the proposed zigzag theory are obtained by superimposing a nonlinear zigzag function on the first-order shear deformation theory. In order to obtain the accurate transverse shear stresses, a preprocessing approach based on the three-dimensional equilibrium equations and the Reissner mixed variational theorem is used. It is significant that the second-order derivatives of in-plane displacement variables have been removed from the transverse shear stresses, such that the finite element implementation is greatly simplified. Thus, based on the proposed zigzag model, a computationally efficient four-node C0 quadrilateral plate element with linear interpolation function is proposed for bending and buckling analysis of soft core sandwich 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. Moreover, the accurate transverse shear stresses can be involved in the strain energy which can actively improve the accuracy of critical loads. 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.

A Family of C0 Quadrilateral Plate Elements Based on the Refined Zigzag Theory for the Analysis of Thin and Thick Laminated Composite and Sandwich Plates

The present work focuses on the formulation and numerical assessment of a family of C0 quadrilateral plate elements based on the refined zigzag theory (RZT). Specifically, four quadrilateral plate elements are developed and numerically tested: The classical bi-linear 4-node element (RZT4), the serendipity 8-node element (RZT8), the virgin 8-node element (RZT8v), and the 4-node anisoparametric constrained element (RZT4c). To assess the relative merits and drawbacks, numerical tests on bending (maximum deflection and stresses) and free vibration analysis of laminated composite and sandwich plates under different boundary conditions and transverse load distributions are performed. Convergences studies with regular and distorted meshes, transverse shear-locking effect for thin and very thin plates are carried out. It is concluded that the bi-linear 4-node element (RZT4) has performances comparable to the other elements in the range of thin plates when reduced integration is adopted but presents extra zero strain energy modes. The serendipity 8-node element (RZT8), the virgin 8-node element (RZT8v), and the 4-node anisoparametric constrained element (RZT4c) show remarkable performance and predictive capabilities for various problems, and transverse shear-locking is greatly relieved, at least for aspect ratio equal to 5 × 102, without using any reduced integration scheme. Moreover, RZT4c has well-conditioned element stiffness matrix, contrary to RZT4 using reduced integration strategy, and has the same computational cost of the RZT4 element.

Geometrically nonlinear analysis of CNT-reinforced functionally graded composite plates integrated with piezoelectric layers

The paper develops a geometrically nonlinear finite element model with large rotation based on the first-order shear deformation (FOSD) hypothesis for static and dynamic analyses of piezoelectric integrated carbon nanotube reinforced functionally graded (P-CNT-FG) composite structures. A constant electric field distribution through the thickness of plate is considered. An eight-node quadrilateral plate element with five mechanical degrees of freedom (DOFs) and one electric degree of freedom is developed for finite element analysis. Four typical forms of CNT distributions are included in the model, namely uniform, V-shaped, O-shaped, and X-shaped distributions. The nonlinear model considers fully geometrically nonlinear strain-displacement relations and large rotations of the shell direction of plate. Using the Hamilton’s principle, a nonlinear dynamic model including dynamic and sensory equations is obtained. The proposed nonlinear model is validated by a frequency analysis of a simply supported P-CNT-FG composite plate. Furthermore, the effects of various parameters on the static and dynamic behavior are investigated, e.g. CNT-reinforcement orientation, CNT distribution, the number of laminate layers and volume fraction.

Compliant assembly deviation analysis of large-scale thin-walled structures in different clamping schemes via ANCF

PurposeThe structure stiffness is greatly affected by the fixture constraints during assembly due to the flexibility of large-scale thin-walled structures. The compliant deformation of structures is usually not consistent for the non-uniform stiffness in various clamping schemes. The purpose of this paper is to investigate the correlation between the assembly quality and the clamping schemes of structures with various initial deviations and geometrical parameters, which is based on the proposed irregular quadrilateral plate element via absolute nodal coordinate formulation (ANCF).Design/methodology/approachTwo typical clamping schemes are specified for the large-scale thin-walled structures. Two typical deviation modes are defined in both free and clamping states in the corresponding clamping schemes. The new irregular quadrilateral plate element via ANCF is validated to analyze the compliant deformation of assembled structures. The quasi-static force equilibrium equations are extended considering the factors of clamping constraints and geometric deviations.FindingsThe initial deviations and geometrical parameters strongly affect the assembly deviations of structures in two clamping schemes. The variation tendencies of assembly deviations are demonstrated in details with the circumferential clamping position and axial clamping position in two clamping schemes, providing guidance to optimize the fixture configuration. The assembly quality of structures with deviations can be improved by configuration synthesis of the clamping schemes.Originality/valueTypical over-constraint clamping schemes and deviation modes in clamping states are defined for large-scale thin-walled structures. The plate element via ANCF is extended to analyze the assembly deviations of thin-walled structures in various clamping schemes. Based on the proposed theoretical model, the effects of clamping schemes and initial deviations on the deformation and assembly deviation propagation of structures are investigated.

Enhancement to four-node quadrilateral plate elements by using cell-based smoothed strains and higher-order shear deformation theory for nonlinear analysis of composite structures

This paper improves four-node quadrilateral plate elements by using cell-based strain smoothing enhancement and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of composite structures. Small strain-large displacement theory of von Kármán is used in nonlinear formulations of four-node quadrilateral plate elements that have strain components smoothed or averaged over the sub-domains of the elements. From the divergence theory, the displacement gradients in the smoothed strains are transformed from the area integral into the line one. The behavior of composite structures follows the third-order shear deformation theory. The solution of the nonlinear equilibrium equations is obtained by the iterative method of Newton–Raphson with the appropriate convergence criteria. The present numerical results are compared with the other numerical results available in the literature in order to demonstrate the effectiveness of the developed element. These results also contribute a better knowledge and understanding of nonlinear bending behaviors of these composite structures.

An implicit strong $$\mathrm {G}^{1}$$-conforming formulation for the analysis of the Kirchhoff plate model

In this work, we present a quadrilateral plate element for the Kirchhoff plate bending model that satisfies the continuity requirements in implicit way. The element is designed on the basis of the rational Gregory’s enhancement of the bi-cubic Coons patch. This Coons-Gregory patch is based on the boundary data set of a surface, that accounts for both the displacement and the edge rotation along the sides of the element. In this way, an implicitly conforming interpolation with 20-dofs per element is obtained. The Coons-Gregory patch ensures $$\mathrm {G^1}$$ -conformity only for the case of structured meshes. Numerical examples show that the proposed formulation is highly efficient with respect to accuracy, rate of convergence and robustness.

Compliant Assembly Variation Analysis of Scalloped Segment Plates With a New Irregular Quadrilateral Plate Element Via ANCF

The accurate calculation of deformation during assembly process is important for deviation propagation of large-scale thin-walled hemisphere structures with manufacturing deviations due to the nonuniformed material properties and nonlinear geometrical behavior. In this study, a new irregular quadrilateral plate element based on the absolute nodal coordinate formulation (ANCF) is proposed to discretize the scalloped segment plates with shape deviations. The high-order shape functions of the new element are developed by considering the variable geometrical boundaries. The generalized elastic forces (GEFS) of the new elements for anisotropic and orthotropic materials are derived based on continuum mechanics approach. The bending deviation mode is defined and the evaluation indexes for assembly quality of thin-walled hemisphere structures are proposed. The force equilibrium equations are employed to study the deformation during assembly process for large-scale thin-walled hemisphere structures with multiple scalloped segment plates. The numerical results are compared with that from experimental data and abaqus. The correlation between the assembly quality and the bending deviation, the clamping methods, the geometrical parameters, and the material properties of structures is also investigated.