Four-node Quadrilateral Element Research Articles

Nonlinear Bending Analysis of Functionally Graded Plates Using SQ4T Elements based on Twice Interpolation Strategy

This paper develops a computational model for nonlinear bending analysis of functionally graded (FG) plates using a four-node quadrilateral element SQ4T within the context of the first order shear deformation theory (FSDT). In particular, the construction of the nonlinear geometric equations are based on Total Lagrangian approach in which the motion at the present state compared with the initial state is considered to be large. Small strain-large displacement theory of von Karman is used in nonlinear formulations of the quadrilateral element SQ4T with twice interpolation strategy (TIS). 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 structures.

Effects of the Radius of Curvature on Natural Frequency and Mode Shape

In this study, the effect of radius of curvature on the natural frequency of isotropic square thin plate is investigated. The models examined are obtained by simply changing the radius of curvature by keeping the length constant from the flat plate to the semicircle model. The free vibration analysis of the plates fixed on two straight edges is performed by the finite element method. As a finite element type, the four-node quadrilateral rectangular element type, which has a total of 24 generalized coordinates, is used. Out-of-plane theory and in-plane theory are used together to form the curve model. The accuracy and validity of the theory are controlled with the data obtained from the finite element program. The effect of the curvature radius on the first five natural frequencies and the mode shapes of these natural frequencies is given in tables and graphs.

Application of the particle finite element method for large deformation consolidation analysis

Purpose Large deformation problems are frequently encountered in various fields of geotechnical engineering. The particle finite element method (PFEM) has been proven to be a promising method to solve large deformation problems. This study aims to develop a computational framework for modelling the hydro-mechanical coupled porous media at large deformation based on the PFEM. Design/methodology/approach The PFEM is extended by adopting the linear and quadratic triangular elements for pore water pressure and displacements. A six-node triangular element is used for modelling two-dimensional problems instead of the low-order three-node triangular element. Thus, the numerical instability induced by volumetric locking is avoided. The Modified Cam Clay (MCC) model is used to describe the elasto-plastic soil behaviour. Findings The proposed approach is used for analysing several consolidation problems. The numerical results have demonstrated that large deformation consolidation problems with the proposed approach can be accomplished without numerical difficulties and loss of accuracy. The coupled PFEM provides a stable and robust numerical tool in solving large deformation consolidation problems. It is demonstrated that the proposed approach is intrinsically stable. Originality/value The PFEM is extended to consider large deformation-coupled hydro-mechanical problem. PFEM is enhanced by using a six-node quadratic triangular element for displacement and this is coupled with a four-node quadrilateral element for modelling excess pore pressure.

Multilayered triangular and quadrilateral flat shell elements based on the Refined Zigzag Theory

The paper presents a class of C0-continuous, flat shell elements based on the Refined Zigzag Theory (RZT) for the analysis of multilayered and curved composite and sandwich structures. The use of the interdependent interpolation strategy allows eliminating the shear-locking phenomenon and introducing the drilling rotation necessary to complete the set of classical nodal degrees of freedom (three displacements and three rotations). Additional kinematic variables are present in RZT, the zigzag rotations around the in-plane axes that measure the normal distortion typical of multilayered structures. An additional “drilling” zigzag rotation is therefore included among the nodal degrees of freedom in order to properly model curved and built-up structures. A stabilization procedure is adopted to suppress spurious zero-energy modes. A three-node triangular and a four-node quadrilateral flat shell element are formulated with 9 degrees of freedom per node. Example problems involving flat and curved multilayered structures are presented and discussed in order to assess the accuracy and convergence properties of the presented elements. Both static response predictions and free vibrations analyses are considered and the comparison is made with analytic RZT solutions, high-fidelity 3D finite element models and FSDT-based flat shell elements.

A cell-based smoothed CBS finite element formulation for computing the Oldroyd-B fluid flow

In this paper, a cell-based smoothed finite element method (CS-FEM) incorporating characteristic-based split (CBS) scheme is developed to compute viscoelastic fluid flow. The system of the Navier–Stokes and Oldroyd-B constitutive equations is decoupled in tandem with equal low-order approximation of the triple primitive variables. The spatial discretization is based upon CS-FEM which readily smooths all gradient-related entries of the governing equations over bilinear four-node quadrilateral elements. Moreover, the cell-based smoothing concept is applied to evaluation of viscoelastic fluid force. Four benchmark problems are replicated to demonstrate the applicability and robustness of the smoothed CBS finite element formulation. A good agreement is exposed between the present and previous results.

Seismic Control of Structures Using Sloped Bottom Tuned Liquid Damper

Tuned liquid damper (TLD) as an economical passive damper has gained importance because of its efficiency in suppressing the undesirable motion of a structure due to devastating earthquakes when it is rigidly attached to the structure. This phenomenon has been studied previously, but mostly limited to rectangular- or cylindrical-shaped TLD. The present study is aimed at determining the potential of the sloped bottom tuned liquid damper to suppress the unwanted vibration of the structure subjected to earthquake ground motion. Three different earthquake ground motions characterized by low-, intermediate- and high-frequency contents are selected for the dynamic analysis of the TLD–structure coupled system. A parametric study on the sloped bottom TLD and its interaction with the sustaining structural system is conducted. The structural equations, which are coupled with fluid equations, are solved by the Newmark-[Formula: see text] method with average acceleration, while the fluid equations are solved using velocity potential-based finite element method. The liquid domain in the TLD is comprised of a combination of three-node triangular and four-node quadrilateral elements. The result shows the reduction in structure response is significant with less amount of water in sloped bottom TLD rather than rectangular TLD.

Nonnodal Extended Finite-Element Method for Crack Modeling with Four-Node Quadrilateral Elements

AbstractA nonnodal extended finite-element method (NXFEM) was developed to predict two-dimensional dynamic failure with four-node quadrilateral elements. A new nonnodal enrichment scheme is present...

The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes

This paper generalizes the cell-based smoothed finite element method (CS-FEM) to viscoelastic fluid flows under the fractional-step umbrella. The characteristic-based split (CBS) scheme is employed to decouple the Navier–Stokes equations and the Oldroyd-B constitutive model. Moreover, the discrete elastic-viscous-split-stress (DEVSS) procedure and stabilized pressure gradient projection (SPGP) are introduced into a second-order CBS framework. The projections of velocity and pressure gradients therein combine to form the DEVSS-G/CBS(B)-SPGP algorithm that further stabilizes the discretized model. Both proposed schemes are suitable for use in CS-FEM because they accept equal low-order interpolations for primitive variables. The spatial discretization is based upon bilinear four-node quadrilateral elements where all gradient-related items are smoothed without any methodological difficulties. In particular, the cell-based smoothing concept enables accurate evaluation of viscoelastic fluid force. The developed methods are validated against previously published data for several benchmark problems. A satisfactory agreement exists between the present and earlier results.

Improving the bending response of four-node quadrilateral and eight-node hexahedral elements for Cosserat elasticity problems

PurposePoor bending response is a major shortcoming of lower-order elements due to excessive representation of shear stress/strain field. Advanced finite element (FE) formulations for classical elasticity enhance the bending response by either nullifying or filtering some of the symmetric shear stress/strain modes. Nevertheless, the stress/strain field in Cosserat elasticity is asymmetric; consequently any attempt to nullify or filter the anti-symmetric shear stress/strain modes may lead to failure in the constant couple-stress patch test where the anti-symmetric shear stress/strain field is linear. This paper aims at enhancing the bending response of lower-order elements for Cosserat elasticity problems.Design/methodology/approachA four-node quadrilateral and an eight-node hexahedron are formulated by hybrid-stress approach. The symmetric stress is assumed as those of Pian and Sumihara and Pian and Tong. The anti-symmetric stress components are first assumed to be completely linear in order to pass the constant couple-stress patch test. The linear modes are then constrained with respect to the prescribed body-couple via the equilibrium conditions.FindingsNumerical tests show that the hybrid elements can strictly pass the constant couple-stress patch test and are markedly more accurate than the conventional elements as well as the incompatible elements for bending problems in Cosserat elasticity.Originality/valueThis paper proposes a hybrid FE formulation to improve the bending response of four-node quadrilateral and eight-node hexahedral elements for Cosserat elasticity problems without compromising the constant couple-stress patch test.

Formulation and evaluation of a new four-node quadrilateral element for analysis of the shell structures

Shell structures are lightweight constructions which are extensively used by engineering. Due to this reason presenting an appropriate shell element for analysis of these structures has become an interesting issue in recent decades. This study presents a new rectangular flat shell element called ACM-SQ4 obtained by combining bending and membrane elements. The bending element is a well-known plate bending element called ACM which is based on the classical thin-plate theory and the membrane element is an unsymmetric quadrilateral element called US-Q4θ, the test function of this element is improved by the Allman-type drilling DOFs and a rational stress field is used as the element’s trial function. Finally, some numerical benchmark problems are used to evaluate the performance of the proposed flat shell element. The obtained results show that despite its simple formulation, the proposed element has reasonable accuracy and acceptable convergence in comparison with other shell elements.

An Improved Four-Node Element for Analysis of Composite Plate/Shell Structures Based on Twice Interpolation Strategy

This paper deals with numerical linear analyses of laminated composite plate/shell structures. The method is based on a four-node quadrilateral element, namely, SQ4T, within the framework of the first-order shear deformation theory (FSDT). This element is established by incorporating the twice interpolation strategy (TIS) into a traditional four-node finite element to build the membrane, bending and shear stiffness matrices. Many desirable characteristics of this efficient numerical method are shown as continuous nodal gradients, higher-order polynomial basis, no increase in number of the degree of freedom of the system. The performance of the proposed element is validated and demonstrated through several numerical benchmark problems. Convergence studies and comparison with other existing solutions in the literature suggest that the present element is efficient, accurate and free of lockings.

Evaluation of dynamic characteristics of liquid sloshing in sloped bottom tanks

Sloshing characteristics in partially filled slo-ped bottom container is investigated numerically. Galer-kin finite element method is used to derive the modal equation of motion for sloshing in 2D tank. The dynamic charcateristics of liquid sloshing in V-shaped bottom and the trapezoid tank are the critical investigations in the present work under the action of periodic ground motion. The liquid domain in V-shaped bottom and the trapezoid tank is discretized with a combination of three node triangular and four-node quadrilateral elements for this analysis. The reliability of the present numerical method is verified by comparing the results with the available analytical and experimental results for $$30^\circ $$ trapezoid tank and $$20^\circ $$ V-shaped bottom tank. Further, a parametric study on the V-shaped bottom and trapezoid tank is carried out to determine it’s dynamic behavior. The dynamic response of trapezoid tank under horizontal harmonic motion is manifested in terms of slosh wave displacement, base shear force at tank bottom and overturning base moment which remained limited for sloped bottom container.

A three-field smoothed formulation for partitioned fluid–structure interaction via nonlinear block-Gauss–Seidel procedure

This article presents a three-field smoothed formulation for partitioned fluid–structure interaction problems. The cell-based smoothed finite element method (CS-FEM) is adopted to solve the Navier–Stoke equations under the arbitrary Lagrangian–Eulerian description. To do this, the number and numbering of smoothing cells are deployed in line with those of Gaussian points in each four-node quadrilateral element for smoothed Galerkin weak form. In the meantime, CS-FEM is directly applied to nonlinear solid dynamics. A simple post-smoothing algorithm is introduced to enhance the quality of the updated mesh. Nonlinear block-Gauss–Seidel procedure is subsequently employed for the coupling of all smoothed individual fields. Two popular examples are presented to demonstrate the applicability and robustness of the proposed methodology.

Buckling analysis of plain-woven fabric structure using shell element and a one cell-based integration scheme in smoothed finite element method

A one smoothing cell integration scheme in the strain smoothing technique in finite elements (referred as SFEM) was proposed to evaluate the nodal train fields of a four-node quadrilateral (Q4) shell element, which is based on the first-order shear deformation theory of plate (FSDT). A mixed interpolation of tensorial components (MITC) approaches for Q4 transverse shear strains also applied to eliminate a shear locking phenomenon that may occur when the thin plate/shell elements are geometrically distorted in curved geometries of fabric sheet. The numerical eigenvalues of buckling analysis of a plain-woven fabric sample, of which physical and mechanical parameters extracted from Kawabata evaluation system for fabrics (KES-FB), obtained a higher efficiency in numerical computation and approximated to Q4 shell element implemented in the finite element method (FEM).

Impact of Integration on Straining Modes and Shear-Locking for Plane Stress Finite Elements

Stiffness matrix of the four-node quadrilateral plane stress element is decomposed into normal and shear components. A computer program is developed to obtain the straining modes using adequate and reduced integration. Then a solution for the problem of mixing straining modes is found. Accuracy of the computer program is validated by a closed-form stiffness matrix, derived for the plane rectangular as well as square element. It is shown that method of integration has no effect on the straining modes, but it influences the eigenvalues of the bending modes. This effect is intensified by increasing the element aspect ratio, confirming the occurrence of shear locking.

Eigenvalue analysis for plain-woven fabric structure using shell element and one smoothing cell in the smoothed finite element method

An efficient four-node quadrilateral (Q4) shell element based on the first-order shear deformation theory of plate (FSDT) and the strain smoothing technique in finite elements (referred as SFEM) was proposed for eigenvalue analysis of plain-woven fabric structure. A one smoothing domain (or cell) integration scheme in SFEM was proposed to evaluate the nodal train fields of Q4 shell elements. The numerical result of eigenvalue analysis, which was in the case of free vibration analysis, approximated to that one implemented in the finite element method (FEM) but gave a higher efficiency in computation in terms of central processing unit (CPU) time and numerical implementation.

Thermal-mechanical crack propagation in orthotropic composite materials by the extended four-node consecutive-interpolation element (XCQ4)

A numerical study of thermal-mechanical crack growth in orthotropic composite materials by the recently developed extended nodal gradient finite element method is presented. The extended four-node consecutive-interpolation quadrilateral element (XCQ4) used in this work is the conventional four-node quadrilateral element, but it is enhanced by additional terms of averaged nodal gradients to smoothen the derivative fields and increase the accuracy of solution, while keeping the same total number of degrees of freedom (DOFs) as that of the traditional element. Inspired by enrichment partition-of-unity method, the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack are mathematically described through known enrichment functions. Once the stress fields are obtained, stress intensity factors (SIFs) can thus be evaluated by interaction integral technique. The SIFs are then used to predict crack growth direction. Unlike isotropic media, material orientation plays a critical role and that has to be considered when modeling the evolution of crack in orthotropic composite materials under thermo-mechanical loading condition. In addition, we introduce for the first time an improved way for the enriched approximation of discontinuous temperature field in cracked orthotropic media by taking into account the effect of material orientation. Several numerical examples for thermo-mechanical fracture problems in orthotropic composites with both isothermal and adiabatic loading conditions are analyzed to show the accuracy and performance of the present XCQ4. The computed results of the SIFs are compared with reference solutions derived from experiments, the standard extended four-node quadrilateral element (XQ4), and other numerical methods available in literatures.

Nonlinear buckling analysis of variable stiffness composite plates based on the reduced order model

Abstract The variable-stiffness fiber composite plates which have an enhanced design flexibility, largely rely on laminate optimizations to maximize the buckling performance. The corresponding computational efficiency becomes a key issue, in particular when the nonlinear structural behavior is considered. The finite element method based on a full nonlinear analysis is a standard technique for nonlinear structural analysis, however the high computational complexities generated from both the incremental-iterative procedure and the very refined mesh needed for the discrete modeling of curved fibers, are still a decisive cost factor on modern computers. In this work, the Koiter-Newton method is further extended to nonlinear buckling analysis, including the pre and post buckling stage, of variable stiffness composite plates. A four-node quadrilateral element based on the classical laminated plate theory is developed in framework of the von Karman kinematics, for the finite element implementation of the proposed asymptotic method . The reduced order model , with or without imperfections, is constructed using the improved Koiter’s asymptotic expansion , for both the symmetrical and unsymmetrical laminates. The nonlinear response curve of loaded structure can be traced automatically, using the nonlinear predictor and corrections both generated from the reduced order model. This leads to a fairly large step size in the path-tracing process, compared to that for the classical Newton method . The reduced order model largely reduces the computational burden produced by the high-density FE mesh for the varied fiber path. Numerical results indicate the overall high quality and efficiency of the proposed method.

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.

Application of Smoothed Finite Element Method to Two-Dimensional Exterior Problems of Acoustic Radiation

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshfree methods. In the SFEM-Q4, only the values of shape functions (not the derivatives) at the quadrature points are needed and the traditional requirement of coordinate transformation procedure is not necessary to implement the numerical integration. Consequently, no additional degrees of freedom are required as compared with the original FEM. In addition, the original “overly-stiff” FEM model for acoustic problems (governed by the Helmholtz equation) is properly softened due to the gradient smoothing operations implemented over the smoothing domains and the present SFEM-Q4 possesses a relatively appropriate stiffness of the continuous system. Therefore, the well-known numerical dispersion error for Helmholtz equation is decreased significantly and very accurate numerical solutions can be obtained by using relatively coarse meshes. In order to truncate the unbounded domains and employ the domain-based numerical method to tackle the acoustic radiation in unbounded domains, the Dirichlet-to-Neumann (DtN) map is used to ensure that there are no spurious reflections from the far field. The numerical results from several numerical examples demonstrate that the present SFEM-Q4 is quite effective to handle acoustic radiation problems and can produce more accurate numerical results than the standard FEM.