Four-node Quadrilateral Element Research Articles

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Formulating strain-based quadrilateral membrane finite elements with drilling rotations

Membrane finite elements with drilling degrees of freedom have sparked interest in many research works since they can be conveniently combined with plates to form shell elements. This study presents two non-conforming strain-based four-node quadrilateral membrane elements, SBQ13 and SBQ13E, for static analysis. SBQ13 partially satisfies the equilibrium equations, while SBQ13E completely fulfils the force balance equations. Both elements carry drilling rotations at each node. One difficulty when formulating quadrilateral elements is the singularity in the transformation matrix, which is addressed in this study by utilising the properties of singular matrices. Both quadrilateral elements were tested in several benchmark problems. It has been found that both elements passed the higher-order patch test but failed the constant stress patch test. Nevertheless, SBQ13 produced accurate responses in most numerical tests, but SBQ13E unreasonably overestimated the solution. Solving the eigenvalue problem revealed that the SBQ13E element has a near-zero energy deformation mode, which might explain the anomaly. Although fulfilling equilibrium does not always enhance solution accuracy, it is essential to overcome volumetric locking. Apart from the newly developed elements, this paper presents several new ideas that may apply to strain-based element formulations.

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.

Band gap study of periodic piezoelectric micro-composite laminated plates by finite element method and its application in feedback control

Phononic crystals are materials or structures with periodic variations of elastic constants and density. Elastic waves in phononic crystals cannot propagate within a certain frequency range, called the band gap, due to the interference of their internal structure. This article investigates how fiber orientation affects the band gap properties of periodic micro-composite laminated plates. We examined the impact of five common composite laminate configurations on the system’s band gap and discovered that the antisymmetric configuration had superior high-frequency band gap performance. Based on this finding, we further explored how the antisymmetric angle influenced the system’s band gap. Since the study was conducted at a micro scale, micro structural effects were considered. Therefore, we used the modified coupled stress elastic dynamics and a four-node quadrilateral non-conforming element to discretize the micro composite laminated plate model, which had the nodal compatibility of high-order elasticity theory. Moreover, we employed feedback control to the proposed structure to dynamically adjust the band gap width according to different practical needs. This research supports the design of periodic micro-composite laminated plates and the piezoelectric feedback control system, which can control the vibration and wave propagation behaviors of micro structures using coupled piezoelectric sensors and actuators. The piezoelectric feedback control employed two control methods: direct proportional control and acceleration control. This article compared the single and multiple control methods of these two approaches to determine the optimal control strategy.

Dynamic response of magneto-electro-elastic composite plates lying on visco-Pasternak medium subjected to blast load

This is the first endeavor to present a finite element approach to predict free oscillation and transient characteristics of magneto-electro-elastic (MEE) composite rectangular and elliptical plates resting on the visco-Pasternak medium in a hygro-thermal environment subjected to blast load. The variation of electric and magnetic potentials along the thickness direction of the MEE plate is determined via the Maxwell equation and magneto-electric boundary condition. The governing equations of motion for the MEE plate are obtained using refined higher-order shear plate theory and Hamilton's principle. Stiffness matrices, mass matrices, damping matrices, and force vectors of the plate are derived using a four-node quadrilateral element with eight degrees of freedom per node approximated using C1-order non-conforming Hermite and Lagrange functions. In addition, the Navier close-form solutions for rectangular plates with simply supported boundary conditions are used as a useful comparison tool for the numerical solutions. Dynamic response results of plates were obtained using Newmark's direct integration written by Matlab's programming. The accuracy of the method is verified through numerical comparison with confidence statements. The results show that the applied magnetic and electric potential highly inspires the transient responses. These results can be used in vibration control studies of structures and structures subjected to explosive or low-velocity impact loads.

A two-dimensional four-node quadrilateral inverse element for shape sensing and structural health monitoring

The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional four-node quadrilateral inverse finite element formulation is presented. The element is suitable for thin structures under in-plane loading conditions. To validate the accuracy and demonstrate the capability of the inverse element, four different numerical cases are considered for different loading and boundary conditions. iFEM analysis results are compared with regular finite element analysis results as the reference solution, and very good agreement is observed between the two solutions demonstrating the capability of iFEM approach.

Towards composite suspension control arm: Conceptual design, structural analysis, laminate optimization, manufacturing, and experimental testing

The automotive industry needs composite materials to decrease the weight of new-generation vehicles whilst increasing their strength. In this study, one of the critical (load bearing) components of automobiles, i.e., the suspension control arm made of steel, is fully redesigned for its suitable manufacturing using composite materials. To this end, innovative mechanical simulation methods are developed and coupled to perform the design, analysis, and optimization of the automotive suspension control arm. The main design/optimization criteria are set to reduce no less than 75% weight of the metal control arm and increase its safety by at least 60% by using composite materials and a new geometry suitable for mass production. To predict the deformation-stress state of the control arm, a four-node quadrilateral shell element is implemented based on the kinematics of refined zigzag theory (RZT). Once verified numerically, the computer implementation of the RZT is combined with the optimization algorithm to achieve the optimum laminate stacking sequence of the control arm. Accordingly, prototypes of the composite control arms with optimum lamination plans are manufactured and then experimentally tested under the loading and constraint conditions defined at the conceptual design stage. The numerical and full-scale experimental results are compared, and the RZT models are comprehensively validated. Hence, the advantages of the overarching design-analysis-optimization strategy presented herein are revealed for redesigning and manufacturing automobile parts from composite materials.

Energy harvesting enhancement of nonuniform functionally graded piezoelectric beam using artificial neural networks and Lichtenberg algorithm

In this paper, we present a novel optimization approach employing Artificial Neural Networks (ANN) to enhance electric power generation in functionally graded piezoelectric tapered harvesters (FGPEH). The FGPEH consists of a host beam composed of functionally graded material, with piezoelectric layers covering its top and bottom surfaces. We formulate a finite element model using Kirchhoff plate assumptions and a four-node quadrilateral element to derive the electromechanical governing equation for the proposed harvester. To maximize the output voltage of the harvester, we optimize five material and geometrical parameters, including tapering parameters, host beam material properties, and the thickness of the piezoelectric layer. The optimization cost is significantly reduced by employing an ANN predictive model, trained on data obtained from numerous random simulations of the numerical model. The trained model achieved an optimal predicted response that surpassed the best database response by 323%, all while reducing computational costs by 80%, utilizing a novel lightning-inspired algorithm known as the Lichtenberg Algorithm (LA). The coupling of ANN and LA in the optimization process results in two designs. The first design, a single non-uniform FGPEH, increases harvested energy by 63%. The second design, a dual non-uniform FGPEH, further enhances energy harvesting by nearly 72%.

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.

Analysis of a Steel-Concrete Composite Plate resting on Axial Bars using the Finite Element Method

This study applied finite element analysis to a steel-concrete composite plate resting on the axial bar, using a four-node quadrilateral finite element for a steel-concrete composite plate combined with a truss element. Plate displacements and deformations were used to formulate the finite element method based on the first-order plate theory, also known as the Mindlin plate theory. The finite element analysis to combine steel-concrete composite plates and axial bars was implemented in MATLAB. Numerical examples were used in detail, showing a very small difference between the current study and the SAP2000 results.

Four-node quadrilateral C 0-element based on cell-based smoothed strains strategy and third-order shear deformation theory for functionally graded carbon nanotube reinforced composite plates

This study indicates the analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates using a four-node quadrilateral element related to the C0-type of Reddy’s third-order shear deformation theory (C0 HSDT) and cell-based smoothed strains (CS) strategy. Reddy’s theory is surely taking the advantages and desirable properties of the third-order shear deformation theory. Besides, FG-CNTRC plates with advanced material properties are changed from the bottom to top surface with four kinds of carbon nanotube (CNTs). Numerical results and comparison with other reference solutions suggest that the benefits of the present element are accuracy and efficiency in analysis of FG-CNTRC plates.

Mixed finite elements based on superconvergent patch recovery for strain gradient theory

Strain gradient theory is one of the most powerful candidates to simulate micro/nano-sized structures considering size effect. However, too many degrees of freedom (DOFs) have been placed to satisfy the C1-continuity condition for the development of finite elements based on the strain gradient theory, greatly reducing the practicality of these elements. In this work, four-node quadrilateral and eight-node hexahedral elements are developed to simulate size effect based on strain gradient theory. Since only displacement DOFs are placed on each node without using displacement gradient DOFs, there is dramatic reduction in computational cost. The displacements are approximated with shape functions of standard finite elements. The displacement gradient field is approximated with linear polynomials and the coefficients are determined using the superconvergent patch recovery. It was found that the superconvergent properties of Barlow points are still valid even though the gradient recovery is performed during the calculation of the element stiffness matrix. The preconditioned conjugate gradient method is utilized to solve the system equations. Numerical examples demonstrate the accuracy and efficiency of the proposed elements. Size effect observed in experiments is captured by using the developed finite elements.

Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation

This article develops a novel finite element formulation based on nonlocal theory to analyze the vibration of viscoelastic orthotropic nanoplates resting on the variable viscoelastic foundation (VEF). The mechanical properties of nanoplate are assumed to be viscoelastic orthotropic according to Kelvin’s model. The variable VEF consists of two layers: a shear layer with constant stiffness, the other layer is described as a system composed of alternating damping and springs, in which the spring stiffness and the damping coefficient vary x-axis only. Based on Hamilton’s principle, a refined higher-order shear deformation plate theory (HSDT) and nonlocal theory are used to establish motion equations of the nanoplates. A four-node quadrilateral finite element is created by combining two interpolation functions including Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement. The obtained results in our work are verified through reliable publications. A series of factors influencing the vibration of orthotropic nanoplates resting on the variable VEF is discovered and discussed.

Novel finite element modeling for free vibration and buckling analysis of non-uniform thickness 2D-FG sandwich porous plates using refined Quasi 3D theory

A novel four-node quadrilateral element with eleven degrees of freedom per node is approximated using the C1 -order non-conforming Hermite and Lagrange functions and the novel refined Quasi-3D plate hypothesis for buckling and free vibration analysis of non-uniform thickness bi-directional functionally graded sandwich porous (BFGSP) plates resting on variable elastic foundations (VEF) in a hygro-thermal environment. Material and mechanical properties that change in both the length and thickness directions with three different laws of porosity, which are made up of a fully ceramic core layer and two bi-directional functionally graded (2D-FG) material ones, may be used a lot in the aerospace engineering and military industries. In order to analyze the buckling and free oscillation behaviors shown by the plate with arbitrary boundary conditions, a series of mathematical methods created in Matlab’s software are used. The computation program’s correctness is verified by comparing numerical findings to dependable assertions. In addition, a comprehensive analysis of the impact of factors on the buckling and free oscillation responses is conducted. The findings reveal that the novel porosity patterns, the hygro-thermal environment, the elastic medium characteristics, and the boundary conditions have a substantial impact on the mechanical behaviors of the non-uniform thickness 2D-FG sandwich porous plates. The results of this article can be used as a useful reference for engineers when calculating and designing structures of this type in engineering practice.

Pitting corrosion identification approach based on inverse finite element method for marine structure applications

This paper presents a state-of-the art inverse finite element method (iFEM) for full-field, real-time structural health monitoring (SHM) of corroding structures. The study focuses on detecting the location and identifying the extent of pitting corrosion, which is one of the most hazardous forms of corrosion due to the difficulties regarding its prediction and detection. For this purpose, high-fidelity finite element (FE) models of corroded samples with complex-shaped, semi-ellipsoidal, and cylindroconical defects are developed to replicate the strain field of pitting corroded structures. The strain fields produced by the FE models, verified against experimental data, are reconstructed by the iFEM model using a robust and practical four-node quadrilateral inverse-shell element, iQS4. A strain-based damage criterion is defined to obtain the damage distribution and the location of corrosion pits on dog-bone tensile test samples with full sensor iFEM and coupled iFEM-GA reduced sensor technique. Finally, in a practical example of a structure with complex geometry, the proposed method's performance is assessed through detecting the location of corrosion damage on a cylindrical marine structure with axial stiffeners.

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.

Non-linear finite element analysis of seismically excited sloped wall tank

The non-linear seismic behavior of a ground-supported sloped wall tank is analyzed in the present study using a finite element model. A combination of three-node triangular and four-node quadrilateral finite elements is used to discretize the liquid domain. The velocity potential formulation is adopted for the liquid domain with a mixed Eulerian-Lagrangian scheme. By comparing the current numerical results with the existing experimental results, the competency of the present non-linear model is verified. The fundamental sloshing frequencies obtained from the present non-linear model are found closer to the experimental results compared to the linear counterpart. Non-linear seismic characteristics of liquid sloshing inside a ground-supported sloped wall tank are investigated under six different earthquakes categorized as low-, intermediate-, and high-frequency contents. Also, the dynamic impulsive and convective response components of hydrodynamic base shear force and pressure are quantified successfully. The comparison of the non-linear hydrodynamic response with the linear counterpart shows a significant difference in results which reasonably clarifies the limited capabilities of the linear model for response prediction as well as designing sloped wall tanks subjected to seismic excitation. The developed non-linear numerical model in the present study can be utilized for tuning and designing structure-coupled sloped wall TLDs.

Post-buckling analysis of shell-like structures using an implicit dynamic time integration scheme

PurposeThis paper aims to investigate post-buckling responses of shell-like structures using an implicit conservative-decaying time integration dynamic scheme.Design/methodology/approachIn this work, the authors have proposed the use of a four-node quadrilateral flat shell finite element with drilling rotational degree of freedom within the framework of an updated Lagrangian formulation mutually with an implicit conservative-dissipative time integration dynamic scheme.FindingsSeveral numerical simulations were considered to evaluate the accuracy, robustness, stability and the capacity of the considered time integration scheme to dissipate numerical noise in the presence of high frequencies. The obtained results illustrate a very satisfying performance of the implicit conservative-dissipative direct time integration scheme conjointly with the quadrilateral flat shell finite element with drilling rotation.Originality/valueThe authors have investigated the potential of the implicit dynamic scheme to deal with unstable branches after limit points in the non-linear post-buckling response of shell structures with no need for structural damping. The capability of the studied algorithm to study buckling and post-buckling behaviour of thin shell structures is illustrated through several numerical examples.

Sensor Placement Optimization for Shape Sensing of Plates and Shells Using Genetic Algorithm and Inverse Finite Element Method

This paper reports the first investigation of the inverse finite element method (iFEM) coupled with the genetic algorithm (GA) to optimize sensor placement models of plate/shell structures for their real-time and full-field deformation reconstruction. The primary goal was to reduce the number of sensors in the iFEM models while maintaining the high accuracy of the displacement results. Here, GA was combined with the four-node quadrilateral inverse-shell elements (iQS4) as the genes inherited through generations to define the optimum positions of a specified number of sensors. Initially, displacement monitoring of various plates with different boundary conditions under concentrated and distributed static/dynamic loads was conducted to investigate the performance of the coupled iFEM-GA method. One of these case studies was repeated for different initial populations and densities of sensors to evaluate their influence on the accuracy of the results. The results of the iFEM-GA algorithm indicate that an adequate number of individuals is essential to be assigned as the initial population during the optimization process to ensure diversity for the reproduction of the optimized sensor placement models and prevent the local optimum. In addition, practical optimization constraints were applied for each plate case study to demonstrate the realistic applicability of the implemented method by placing the available sensors at feasible sites. The iFEM-GA method's capability in structural dynamics was also investigated by shape sensing the plate subjected to different dynamic loadings. Furthermore, a clamped stiffened plate and a curved shell were also considered to assess the applicability of the proposed method for the shape sensing of complex structures. Remarkably, the outcomes of the iFEM-GA approach with the reduced number of sensors agreed well with those of the full-sensor counterpart for all of the plate/shell case studies. Hence, this study reveals the superior performance of the iFEM-GA method as a viable sensor placement strategy for the accurate shape sensing of engineering structures with only a few sensors.

Research of the thermoelastic performance of the functionally graded sandwich slabs with convective-radiative boundary conditions based on FVM

<p indent="0mm">In this study, a two-dimensional control volume finite element method (CV-FVM) has been developed for the thermoelastic coupling problem of sandwich structures with functionally graded material core under a convection-radiation heat transfer environment. Based on a four-node quadrilateral element, the discrete process of the governing equation and the nonlinear heat transfer boundary are given in detail. Based on the staggered grid technology, the variables are stored at the node of the element, and the material parameters are defined at the center of the element. To ensure the convergence of the temperature solution, the radiation heat transfer boundary condition is handled by the linearization technique. The numerical method developed in this study is validated for the heat conduction problem of sandwich structures with an exponential core. The results showed that CV-FVM results are consistent with the analytical solution. The influences of the geometric and physical parameters, such as the Biot number, radiation conduction coefficient, environmental temperature ratio, panel thickness, and gradient index of the functionally graded core, on the thermoelastic performance of the structure are investigated. The results showed that the Biot number of the heating surface has only a slight effect on the thermoelastic response of the structure, whereas the Biot number of the cooling surface has a significant effect on the thermoelastic response. The cooling surface is more sensitive to the change in the environmental temperature than the heating surface. For sandwich structures with an exponential core, increasing the thickness of the upper panel can reduce the large gradient change of the stress at the junction between the upper panel and the core. The stress with a linear variation of the properties (<italic>p</italic> = 1) is smoother than that with a nonlinear variation of the properties (<italic>p</italic> = 0.1 and 10), and the large gradient change of the stress of the structure can be reduced significantly (<italic>p</italic> = 1).