Four-node Element Research Articles

The strength calculation of the transport aircraft control system element by the finite element method

The work is devoted to determining the stress-strain state of the element of the control system of the transport aircraft, which serves to transmit motion and is an element of the mechanism for deflecting the rudder and removing the load from the control wiring. If this element is destroyed, the control system will not function, as a result of which some elements of the electronic installation of the aircraft will be disabled. Analytical methods are not always effective when studying the stress state of complex parts and mechanisms of aircraft construction, although they are also used. To determine the stress-strain state in the work, a numerical method was used - the finite element method, which models the stress-strain state of mechanical systems with great accuracy, taking into account the boundary conditions and the given load. A four-node tetrahedral element was chosen as a finite element. Based on the analysis of the stress-strain state, dangerous places in the structure were identified, where there is a possibility of a crack appearing and, as a consequence, destruction.

An SFEM Abaqus UEL for Nonlinear Analysis of Solids

In this paper, three different smoothed finite element method (SFEM), viz., node-based smoothed finite element method (NS-FEM), face-based smoothed finite element method (FS-FEM) and [Formula: see text]-finite element method ([Formula: see text]-FEM) are adopted for 3D solids undergoing large deformation. The common feature of all these techniques is the introduction of smoothed strain which is written as a weighted average of the compatible strain field over smoothing domains. The choice of smoothing domain is what differentiates them. The spatial discretization can be based on the simplest and automatically genera-table four-node tetrahedral elements and aforementioned techniques have shown to yield accurate results even on a coarser discretization. To take the advantages of the SFEM, it is beneficial to the FEM community to have it implemented in the widely used Abaqus[Formula: see text] software. Such an implementation is challenging because the neighboring SFEM elements are interconnected in the smoothed strain matrices in the elemental level. In this work, the above-mentioned SFEM models are implemented in the commercial software Abaqus using the softwares’ user element (UEL) feature. The challenges during the definition and the assembly of the smoothing domains are effectively addressed in this work. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/3DSFEM. The implementation is validated against benchmark examples and the robustness is demonstrated with complicated real-life problems, viz., tire patch contact with road and simulation of human thumb.

Finite element method on topology optimization applied to laminate composite of fuselage structure

Abstract This research applies a numerical study of topology optimization of laminate composite structures by using a finite element method (FEM). In this methodology, the plies orientation is excluded from the optimization. The geometry-based optimization from frames of a MALE UAV fuselage structure is presented. The minimum strain energy with an optimization constraint of 20% of weight reduction is used in the objective function. Before the primary analysis, benchmark studies of topology optimization without considering orientations from previously published literature are performed. The convergence studies were taken to acquire the appropriate mesh size in the FEM technique, which utilized a four-noded shell element. The finite element analysis and optimization results showed that the structural design of the newly framed composite fuselage MALE UAV meets the structural strength requirements specified in the airworthiness standard STANAG 4671.

Quadrilateral element in mixed FEM for analysis of thin shells of revolution

The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.

Vibration response of functionally graded material sandwich plates with elliptical cutouts and geometric imperfections under the mixed boundary conditions

Abstract The present article investigates the effect of elliptical cutouts and geometric imperfections on the vibrational response of functionally graded material (FGM) sandwich plates. Generalised governing equations for the sandwich FGM (SFGM) plate are derived based on non-polynomial higher-order shear deformation theory. Geometric discontinuities have been incorporated as elliptical cutouts in the plates, and the various geometric imperfections are modelled using the generic function. The mathematical modelling has been carried out using the C0 continuity isoparametric finite element formulation by considering four-noded elements with seven degrees of freedoms per node. Convergence and validation studies have been performed to demonstrate the efficiency and accuracy of the present methodology. The influence of volume fraction index, geometric imperfections, and elliptical cutouts on the vibrational frequency of SFGM plates have been analysed under the mixed boundary conditions.

On the second Piola-Kirchhoff and Cauchy stress tensors in nonlinear shells subjected to displacement-dependent loads

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term “geometrically exact” means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant.

Higher order finite element models for transient analysis of strain gradient functionally graded microplates

In the study, three different higher order finite element models (HOFEMs) are developed to perform the transient analysis of functionally graded (FG) microplates subjected to different dynamic loads, namely, sinusoidally distributed step and exponential blast loads. A normal and shear deformable plate theory with five unknowns is used to present displacement field and modified strain gradient theory is employed for small-scale effect. A rectangular four-noded element and Newmark's method is used to solve transient analysis of FG microplates. The effects of boundary condition, type of dynamic loading, thickness to material length parameter, aspect ratio, gradient index and CPU time are investigated. It is found that two of the HOFEMs developed based on the C1 and C2 continuity requirements produced almost same numerical results. However, the one with 36 unknows per element yields slightly different results than the others. The HOFEM satisfying the C1 continuity requirement consumes the minimum computational time.

Generalized cyclic bond-slip model of rebar embedded in concrete under lateral pressure: Parameter calibration and numerical implementation

For some reinforced concrete members, the bond-slip between rebar and concrete is sometimes affected by the lateral pressure, e.g., the beam-column joint with the axial load induced by the column. For such cases, a cyclic bond-slip behavior taking into account the effect of lateral pressure is of great significance for seismic numerical analysis of these members/structures. So far, however, there is no available numerical approaches which can consider the cyclic bond-slip behavior of RC member under lateral pressure. Therefore, this study presents a generalized bond-slip model, which can consider the effect of cyclic loads and lateral pressure simultaneously. Based on the existing test data, the key parameters of the model are calibrated, so that the model can be used for arbitrary cyclic loading history and is easy to be implemented into existing displacement-based numerical code. Furthermore, a four-node zero-thickness bond element is developed in commercial software ABAQUS. The proposed model is implemented into the bond element and verified by the experimental results. The bond element is preliminarily applied to the force–displacement analysis of reinforcing bars in the beam-column joint, and the feasibility of the model in member-level analysis is discussed based on the numerical results.

SONLU ELEMAN MODELLEME TEKNİKLERİNİN KİRİŞLERİN DİNAMİK ANALİZİNE ETKİSİ

Analysis of the free and forced vibration responses of beams is one of the most critical problems to be examined in the design step of these structural members. The finite-element method which solves boundary value problems can be applied efficiently to vibration problems. In this study, the natural vibration frequency and damped and undamped transient analyses of the pinned-pinned beams are investigated. The well-known finite-element software packages, ANSYS and SAP2000, are used. The 2-D elastic beam which is based on the Euler-Bernoulli Beam theory, 3-D two-node and 3-D three-node beam elements which are based on Timoshenko beam theory, and four-node shell elements are used in ANSYS and the frame member is utilized in SAP2000. The effect of these elements on the dynamic behaviors of the isotropic beam is discussed. The results are given in tabular and graphical form for the free and forced vibration, respectively.

Improved chaos game optimization algorithm for optimal frequency prediction of variable stiffness curvilinear composite plate

The optimal fundamental frequency of the variable stiffness laminated composite (VSCL) plate are predicted using an improved chaos game optimization technique (ICGO). In this analysis, the design parameters related to the fibre orientations within the layers are considered the discrete integer, i.e. the values vary between [Formula: see text] and [Formula: see text]. The fibre orientations are changed linearly concerning the longitudinal geometry coordinate in each ply. The numerical solutions are computed by adopting a four-noded isoparametric element (five degrees of freedom per node). The proposed optimization algorithm’s effectiveness, reliability and accuracy have been verified by comparing its optimum results with those of other algorithms in the literature. Also, the effect of the different relevant structural parameters, i.e. the aspect ratios, the edge conditions and the number of layers, on the optimal design, are reported in detail.

Accurate mechanical buckling analysis of couple stress-based skew thick microplates

We investigate, for the first time, the elastic buckling of skew thick microplates under combined in-plane shear and compressive loading within the framework of modified couple stress theory. The displacement field of the microplates is described by a two-variable refined shear deformation theory, wherein the transverse deflection is partitioned into bending and shear components, resulting in a simple and universal elastic buckling model. The Euler-Lagrange equation is used to obtain the equations governing the motion of the skew thick microplates. Obtaining an analytical buckling solution for general boundary supported skew microplates is challenging; therefore, a C1-type four-node 32-DOF differential quadrature finite element is developed. The element stiffness and geometric stiffness matrices are derived according to the minimum potential energy principle. Significant parametric studies are conducted with different geometrical dimensions, boundary edges, in-plane loadings, and material length scale parameters. The buckling characteristics of skew microplates are investigated using a combination of unequal biaxial compression loads and in-plane compression and shear loads. Numerical results imply that the buckling modes are influenced by the combination of size effects and skew angle and not by the buckling loads alone.

FE mesh density influence on blast loading analysis

To examine the influence of mesh density on the blast loads of anti-landmines (ALMs) and improvised explosive devices (IEDs) under an armored vehicle, models with different FE mesh densities were numerically analysed. The multi-material arbitrary Lagrangian-Eulerian (MM-ALE) algorithm was used to simulate under armored vehicle explosions. Explicit dynamic analysis was performed in LS-DYNA. Blast loading simulation was performed using the ConWep (conventional weapon) loading model, which is implemented in LS-DYNA. The material characteristics of high-strength steel were used to model the protective plates on the armored vehicle. Protective plates were modeled in a V-shape and as a deformable solid with the Johnson-Cook plasticity model. In this paper, the FE model with different mesh densities was numerically tested to analyse the influence of mesh density on the obtained results. For the purposes of the modeling, different types of finite elements were used for the FE models. The floor of the vehicle was modeled with 3D hexahedral eight-noded finite elements, and the protective plates were modeled with four-noded plate finite elements. The maximal value of plastic strain and the maximal value of the displacement of the central node on the protective plate was chosen as the parameters to compare the obtained results.

Modelling and Analysis of Thermal Vibrations of Coated Fiber-Reinforced Composite Cylindrical Thin Shells

The thermal vibration characteristics of fiber-reinforced composite (FRC) cylindrical thin shells (FRCCTSs) coated with functionally graded porous graphene platelet (FGPGP) are investigated in this work, which is based on a theoretical model constructed by a mixed analytical and finite element method. Firstly, the porosity distributions of the FGPGP coating are assumed to be uniform or nonuniform along thickness direction with four porous forms of coating being taken into account. Next, the displacement field functions along with the axial, circumferential, and transverse directions are assumed on the basis of Love’s first-order approximation theory. Furthermore, this coated thin shell is discretized by the four-node shell element method to calculate the mass and stiffness matrices, with the artificial spring technology being adopted to simulate arbitrary boundary conditions. After the frequency parameters and dynamic responses are successfully solved, the proposed model with and without coating material is roughly validated by comparing with literature results at different boundary conditions without considering the temperature effect. Meanwhile, by utilizing the natural frequencies and vibration responses measured via a thermal vibration experiment bench, the comprehensive verification is performed within a temperature range of 20–200∘C. Finally, parametric studies are undertaken to study the influences of boundary condition, porosity distribution of coating, fiber layup pattern, the predefined thickness ratio, and elastic modulus ratio on the corresponding thermal vibration properties.

The usability and limitations of the various absolute nodal coordinate beam elements subjected to torsional and bi-moment loading

This paper explores the usability of several beam elements based on the Absolute Nodal Coordinate Formulation (ANCF) subjected to torsional and bi-moment loading scenarios. ANCF beam element solutions are compared against an analytical solution and beam and solid finite element solutions. When subjected to bi-moment loading, most of studied higher-order ANCF beam elements suffer from strong “non-smooth” displacement continuity between elements, and therefore they should be used with caution. The distribution of warping displacement was shown to be unacceptable for four-node higher-order ANCF beam elements. Numerical results revealed that the higher-order elements failed to characterize local loading scenarios such as a bi-moment. This problem can be alleviated using very fine beam discretization, which would be more computationally intensive.

On the improvement of four-node plane stress element

In this paper, a simple way is proposed to improve the overall performance of the four-node plane stress element. Numerical examples are given. Results show that the accuracy of both stress and displacement for a coarse mesh is raised by the proposed method. The reason is clearly given. The problem of mesh size dependent may be overcome in a certain degree if integration points are properly selected.

Numerical Modeling and Experimental Investigation of Thermal Shrinkage in Polymeric Cords. II: Numerical Implementation

ABSTRACTIn this Part II of a two-part paper, a way of modeling axisymmetric rebars to represent the thermal shrinkage of polymeric cords within a thermo-mechanically coupled algorithm is presented without the assumption of a smeared layer. The mechanical characteristics of the proposed approach are compared to the classical approach using only one four-node axisymmetric element with unit dimensions. In addition, the shrinkage behavior of a simplified model made only of plies is discussed. The deformation behavior of polymeric cords during the tire manufacturing process, from the in-molding where a tire is formed by a mold under high pressure and temperature to curing and cooling, is addressed. Finally, the predicted profiles with and without post-cure inflation are compared to the test results.

Prediction of buckling capacity of liquid-filled steel conical tanks considering field-measured imperfections

Geometric imperfections caused by fabrication and welding reduce the buckling capacity of steel conical tanks. Few studies have considered field-measured geometric imperfections in such type of shell structures. This study carries out elastoplastic finite element analyses (FEA) to evaluate the buckling capacity of a full-scale stiffened conical steel water tank by considering its imperfections extracted from high-resolution laser scan data. Both global and local imperfections are obtained by comparing the laser scan data with the nominal tank geometry. The global imperfection involves deviation of the actual central axis of the tank from its nominal axis and ovalization of the tank circumference, whereas the local imperfection involves local fluctuations of the tank geometry around the circumference at a given elevation. The FEA is conducted using the commercial package ANSYS where both geometric and material nonlinearities are considered in the analysis. A four-node shell element with a six degrees of freedom per node is utilized in the model. The analysis results shed light on the shapes and magnitudes of global and local imperfections in an existing stiffened steel conical tank and their impact on the buckling capacity of the tank. Furthermore, the adequacy of imperfection tolerances recommended in widely used design standards for steel tanks are examined and compared with those extracted from the field measurements. The analysis results suggest that the local imperfections can be more critical than the combination of global and local imperfections in terms of the buckling capacity of the conical steel tank.

Variational coupling of non-matching discretizations across finitely deforming fluid-structure interfaces.

This paper presents a stabilized monolithic method for coupling incompressible viscous fluids with finitely deforming elastic solids across non-matching interfacial meshes. Governing equations for the solid are written in the finite deformation Lagrangian frame using velocity field as the primary unknown, while the governing equations for the fluid are written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate large motions of the fluid-solid interfaces. Interface coupling terms are derived by embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework and locally resolving the fine-scale variational equations from the fluid and the solid subdomains along the common interface. The unique attribute of the proposed Variational Multiscale Discontinuous Galerkin (VMDG) method is a systematic procedure for deriving analytical expression for the traction Lagrange multiplier at the fluid-solid interface. The structure of the interface stabilization tensor emerges naturally and is shown to be a function of the boundary operators that are associated with the domain interior operators from the fluid and the solid subdomains. The derived stabilization tensor possesses the features of area-averaging and stress-averaging and evolves spatially and temporally with the evolving nonlinear fields at the interface. An analysis of the mathematical properties of the interface stabilization tensor is presented and subsequently verified numerically. The method is implemented using four-node tetrahedral elements for the fluid as well as the solid subdomains while employing equal-order interpolations for the various interacting fields. Benchmark problems are presented for the verification of the method for finitely deforming fluid-structure interfaces.

A three-dimensional hybrid immersed smoothed point interpolation method for fluid-structure interactions

A three-dimensional hybrid immersed smoothed point interpolation method (3D HIS-PIM) has been developed to handle fluid-structure interactions (FSI) with simple geometries. Incompressible viscous fluid is solved by semi-implicit characteristic-based split method with standard Galerkin discretization, smoothed point interpolation method is employed to solve the motion and deformation of nonlinear solids based on gradient smoothing technique, and the virtual fluid is introduced into the entire solid domain to calculate the FSI force. The proposed method can avoid the fluid mesh adjustments, and allows the use of four-node tetrahedral elements for both fluid and solid domains to simplify the pre-processing for mesh generations. The FSI force condition is imposed with a hybrid force, which uses a form of body force for pressure term and boundary force for shear force. The boundary force accords with the physical law and the body force can ensure the numerical stability. The proposed method works well for FSI problems with moving or largely deformable solids, and can achieve more accurate results using a hybrid force than the fully body force in the original method. A large range of mesh size ratios between fluid and solid meshes is allowed due to the treatment of FSI boundary conditions.

Modeling of partially delaminated composite plates resting on two-parameter elastic foundation with improved layerwise theory

In this work, we aim to build the mathematical model for delaminated composite plates resting on elastic foundation by means of improved layerwise theory and finite element implementation for free vibration analysis. The proposed model takes into account the transverse normal and shear deformation of the elastic foundation so as to investigate their effect on the dynamic characteristics of delaminated plates. The four-node plate element is used for the finite element formulations with Langrage interpolation functions for in-plane structural unknowns and Hermite cubic interpolation functions for out-of-plane structural unknowns. The delamination is simulated by means of Heaviside unit step functions allowing for the discontinuity of displacement fields. The effects of both longitudinal locations and interfacial locations of delamination are investigated. The results show that when the delamination located closer to the clamped end and at the middle plane, it has much more severe influence on its natural frequencies. The foundation parameters also have significant effect on the natural frequencies and frequency shift due to delamination. Different layup stacking sequences are also considered in this study.