Eight-node Quadrilateral Element Research Articles

Precise analysis of the static bending response of functionally graded doubly curved shallow shells via an improved finite element shear model

Doubly curved shallow shells, common in aerospace and civil engineering, pose significant challenges in predicting bending responses due to their complex geometry and material properties. Therefore, this research focused on the development of an efficient eight-node quadrilateral isoparametric element model based on a new improved first-order shear deformation theory (IFSDT) to analyze the bending deflection and stresses distribution of Functionally Graded (FG) doubly curved shallow shells. The present IFSDT obviates the necessity of a shear correction factor, which has challenges in determining for FG doubly curved shallow shells, by adopting a simpler assumption about transverse shear stresses. This assumption guarantees that the model precisely predicts the parabolic shear stresses distribution throughout the thickness of the FG shell while also meeting the requirements for free traction conditions on the top and lower surfaces of the shell. In the present analysis, five distinct types of FG doubly curved shallow shells are modeled: flat plates, spherical shells, hyperbolic parabolic shells, cylindrical shells, and elliptical paraboloid shells. A power law is employed to describe the gradual changes in material properties through the thickness direction. Numerical results for displacements and stresses are obtained for various shell types, materials, power-law factors, radii of curvature, and boundary conditions under uniform and sinusoidal loads. Comprehensive comparison and convergence studies are conducted to assess the robustness and accuracy of the proposed numerical model. This study also contributes new numerical results that are valuable for future research. Ultimately, the proposed numerical model offers a robust and straightforward tool for engineers and researchers engaged in the design and analysis of complex shell structures.

Design Optimization of Flexural and Eigenvalue Characteristics of Perforated and Skewed Bio-Composite Functionally Graded Panels

The development of novel materials is of prime importance in the biomedical areas, especially for bone replacements and dental implants. This study uses finite element modeling of skewed functionally graded bio-composite structures with perforations to examine flexural and eigenfrequency responses. Here, the bio-composite structure comprises Titanium (Ti) and Hydroxyapatite (HAp). Voigt’s micromechanical material model via power-law distribution and Representative volume element (RVE) method is utilized to compute the in-homogenous material properties in the thickness direction. The strain field is based on the equivalent single-layer first-order shear deformation theory with five degrees of freedom, and the governing equations are obtained using principle of minimum potential energy. A computer algorithm is developed in a 2D finite element platform using eight-noded quadrilateral elements. The mesh stability of the skewed and perforated structure is confirmed through the smart mesh technique, and the present model’s accuracy is demonstrated by comparing it with the available reported results. Numerous examples reveal the significance of material gradation, skewness, aspect ratios, perforations, and support conditions on bio-composite functionally graded structures’ flexural and eigenfrequency responses. Furthermore, parametric optimization is conducted to minimize the flexural response and maximize the fundamental frequency by employing the response surface methodology (RSM), followed by determining optimal process parameters and valuable findings.

Dynamic analysis of CNT functionally graded piezolaminated structures using third-order shear deformation theory

Piezoelectric Carbon Nanotube Reinforced Functionally Graded (P-CNT-FG) structures have great application potential in aerospace for vibration control. The precise modeling technique for P-CNT-FG plates and shells is a big challenge, especially for moderate thick and thick P-CNT-FG structures. This paper develops a Finite Element (FE) model for P-CNT-FG plates and shells. The FE model is derived by Reddy’s third-order shear deformation theory with consideration of various CNT distribution forms, CNT volume fractions, boundary conditions and arbitrary fiber orientations. Eight-node quadrilateral elements are developed for P-CNT-FG structures with seven mechanical degrees of freedom (DOFs) at each node. The mathematical model is first validated by a simply supported square plate, a cross-ply plate with sinusoidally distributed loading and a CNT plate with PZT layers. Afterwards, vibration and dynamic responses of P-CNT-FG plates have been presented, later the effective of parameters on static and dynamic response of P-CNT-FG smart structures are carried out by the present model.

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.

Damage propagation analysis of CFRP laminate by quasi-3D XFEM using hexahedral elements

A quasi-three-dimensional extend finite element method (XFEM) is applied to damage propagation analyses of carbon fiber reinforced plastics (CFRPs) laminate. An eight-node quadrilateral interface element and an eight-node hexahedral continuum element enriched with only the Heaviside function are used to model delamination and matrix cracks, respectively. Zig-zag enhanced cohesive zone model or the zig-zag cohesive zone model is introduced to a delamination and a matrix crack. The implicit static method is then used to solve system equations considering material nonlinearity. The Weibull damage criterion is used to predict fiber breakage when necessary. The results of damage propagation analyses were compared with experimental results obtained using open-hole tension tests, quasi-static indentation tests, and compression-after-impact tests. It was shown that the quasi-three-dimensional XFEM is an effective method for damage propagation analysis of CFRP laminates.

Influence of Fiber Angle on Steady-State Response of Laminated Composite Rectangular Plates.

Significant advances in the field of composite structures continue to be made on a variety of fronts, including theoretical studies based on advances in structural theory kinematics and computer models of structural elements employing advanced theories and unique formulations. Plate vibration is a persistently interesting subject owing to its wider usage as a structural component in the industry. The current study was carried out using the Co continuous eight-noded quadrilateral shear-flexible element having five nodal degrees of freedom, which is ground on first-order shear deformation theory (FSDT). For small strain and sufficiently large deformation, the geometric nonlinearity is integrated using the Von Kármán assumption. The governing equations in the time domain are solved employing the modified shooting technique along with an arc-length and pseudo-arc-length continuation strategy. This work explored the effect of fiber angle on the steady-state nonlinear forced vibration response. To explain hardening nonlinearity, the strain and stress fluctuation throughout the thickness for a rectangular laminated composite plate is determined. The cyclic fluctuation of the steady-state nonlinear normal stress during a time period at the centre of the top/bottom surfaces is also provided at the forcing frequency ratio of peak amplitude in a nonlinear response. Because of the variation in restoring forces, the frequency spectra for all fiber angle orientations show significantly enhanced harmonic participation in addition to the fundamental harmonic.

Compression after impact test analysis of CFRP laminate by quasi three-dimensional XFEM using hexahedron element

A quasi-three-dimensional extend finite element method (XFEM) is applied to damage propagation analyses of carbon fiber reinforced plastics (CFRPs) laminate. An eight-node quadrilateral interface element and an eight-node hexahedral continuum element enriched with only the Heaviside function are used to model delamination and matrix cracks, respectively. Zig-zag enhanced cohesive zone model is introduced to a delamination and a matrix crack. The implicit static method is then used to solve system equations considering material nonlinearity. The results of damage propagation analyses were compared with experimental results obtained by the quasi-static indentation tests, and compression-after-impact tests. It was shown that the quasi-three-dimensional XFEM is an effective method for damage propagation analysis of CFRP laminates.

Numerical modeling for cantilever sandwich smart structures with partially covered constrained viscoelastic layer

A finite element formulation using zig-zag hypothesis is developed for sandwich smart structures with partially smart constrained layer damping (SCLD) treatment. In order to achieve higher accuracy, both of the viscoelastic and elastic layers of the sandwich structures are based on the Reissner–Mindlin theory. An eight-node quadrilateral element with seven mechanical degrees of freedom (DOFs) is considered for the finite element (FE) analysis. Using Hamilton’s principle, a piezoelectric coupled dynamic FE model is obtained. This FE model is validated by frequency and loss factor analysis of an elastic–viscoelastic–elastic plate. Furthermore, the effects of various parameters on the damping characteristics of the cantilever sandwich smart structures are carried out, e.g. coverage percentage and position of SCLD, thickness of the viscoelastic or piezoelectric layer, curvature of structures, and shear modulus and loss factor of the viscoelastic layer.

Static and dynamic analyses of FGPM cylindrical shells with quadratic thermal gradient distribution

Functionally graded material (FGM) has great potential in the application of thin-walled structures, since delamination failure can be suppressed. Functionally graded piezoelectric material (FGPM) is a special kind of FGM, which couples with elastic and electric fields. The modeling of FGPM coupled smart structures under nonlinear thermal load is still a challenge task. This paper develops an electro-thermo-elastic finite element (FE) model of FGPM integrated smart structures based on the first-order shear deformation hypothesis. The FE model proposes two different nonlinear configurations of thermal gradient distribution through the thickness, which are compared with linear thermal gradient distribution. An eight-node quadrilateral shell element is developed for finite element analysis. The present FE model is firstly validated by a PVDF bimorph beam under nonlinear thermal loads. Later, the model is applied to the static and dynamic simulation of FGPM cylindrical shells. The results show that thermal distribution configurations have great impact on structural response, which should be carefully chosen in the simulation of thermal coupled problems.

TORSIONAL SHEAR STRESS IN PRISMATIC BEAMS WITH ARBITRARY CROSS-SECTIONS USING FINITE ELEMENT METHOD

Determining the shear stress of a structural element caused by torsion is a vital problem. The analytical solution of the Saint-Venant torsion is only suitable for simple cross-sections. The numerical method to evaluate the shear stress of complicated cross-sections is indispensable. Many numerical methods have been studied by scientists. Among these studies, Gruttmann proposed an excellent numerical method, which inherited the Saint-Venant theory. However, the use of isoparametric four-noded quadrilateral elements made the method not to reach the best optimization. The objective of this paper is to improve Gruttmann‘s method by using isoparametric eight-noded quadrilateral elements. MATLAB is the language for programming the numerical method. The validated examples have demonstrated that the author’s numerical method is more effective than Gruttmann‘s method.

Acoustic fluid–structure study of 2D cavity with composite curved flexible walls using graphene platelets reinforcement by higher-order finite element approach

In the present study, acousto-vibration analysis of 2D fluid-filled cavities/tanks having flat and curved flexible walls is made using a trigonometric function based shear deformable theory andthe Helmholtz wave model for fluid domain. The governing equation formed here is solved through higher-order finite element approach. The walls are modeled by C1 continuous 3-noded beam element and the fluid is idealized using an eight-noded quadrilateral element. Structural and coupled frequencies are evaluated for fluid-filled cavities with rigid/flexible vertical walls along with flat/curved beam on top.The sound pressure level is also predicted in the fluid domain due to a steady-state mechanical harmonic load on the top of the cavity. This investigation is conducted for metallic cavities and then extended to graphene platelets reinforced cavity. The effect of degree of fluid–structure coupling is examined assuming different fluid domains. Considering a wide range of cavity geometry and material parameters such asthickness ratio, curved beamangle, graded porosity and graphene platelets, porosity coefficient, loading of GPL, fluid medium, a comprehensive investigation is depicted to highlight their impacts on vibro-acoustic nature of fluid-filled cavities. It is observed that the dynamic characteristics of rigid and flexible wall cavities are significantly different from each other.

Local and non-local damage model with extended stress decomposition for concrete

In this work, a new damage model for concrete is proposed with an extension of the stress decomposition (limited to biaxial cases), to capture shear damage due to the opposite signed principal stresses. To extract the pure shear stress, the assumption is made that one component of the shear stress is a minimum absolute of the two principal stresses. The opposite signed principal stresses are decomposed into shear stress and uniaxial tensile/compressive stress. A local model is implemented in Abaqus UMAT and it is further extended to a non-local model by utilization of the gradient theory. The concept of three length scales (tension, compression, and shear) is kept the same as the recently proposed nonlocal damage model by the authors. The nonlocal model is implemented in the Abaqus UEL-UMAT subroutine with an eight-node quadrilateral user-defined element, having five degrees of freedom at corner nodes (displacement in X/Y direction and tensile/compressive and shear nonlocal equivalent strain) and two degrees of freedom at internal nodes. Some examples of a local model including uniaxial and biaxial loading are addressed. Also, five examples of mixed crack mode and mode-I cracking are presented to comprehensively show the performance of this model.

On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates

This paper presents a novel C0 higher-order layerwise finite element model for static and free vibration analysis of functionally graded materials (FGM) sandwich plates. The proposed layerwise model, which is developed for multilayer composite plates, supposes higher-order displacement field for the core and first-order displacement field for the face sheets maintaining a continuity of displacement at layer. Unlike the conventional layerwise models, the present one has an important feature that the number of variables is fixed and does not increase when increasing the number of layers. Thus, based on the suggested model, a computationally efficient C0 eight-node quadrilateral element is developed. Indeed, the new element is free of shear locking phenomenon without requiring any shear correction factors. Three common types of FGM plates, namely, (i) isotropic FGM plates; (ii) sandwich plates with FGM face sheets and homogeneous core and (iii) sandwich plates with homogeneous face sheets and FGM core, are considered in the present work. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume power laws of the constituents. The equations of motion of the FGM sandwich plate are obtained via the classical Hamilton’s principle. Numerical results of present model are compared with 2D, quasi-3D, and 3D analytical solutions and other predicted by advanced finite element models reported in the literature. The results indicate that the developed finite element model is promising in terms of accuracy and fast rate of convergence for both thin and thick FGM sandwich plates. Finally, it can be concluded that the proposed model is accurate and efficient in predicting the bending and free vibration responses of FGM sandwich plates.

Improvement on eight-node quadrilateral element (IQ8) using twice-interpolation strategy for linear elastic fracture mechanics

Improvement on eight-node quadrilateral element (IQ8) using twice-interpolation strategy for linear elastic fracture mechanics

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.

Nonlinear Free Flexural Vibration of Curvilinear Fibre Composite Laminates Using a Higher-Order Element

In the present work, the nonlinear free flexural vibration of thick curvilinear fiber composite laminates is investigated using a higher-order shear flexible eight-noded quadrilateral element developed considering the variation of in-plane and transverse displacement through the thickness. The formulation includes both the geometric nonlinearity and inertia effects. The governing equations, derived based on Lagrange’s equations of motion, are solved iteratively through an eigenvalue approach. The formulation is tested against various problems for which the solutions are available in the literature. A detailed analysis is made to assess the influence of fiber angles, lamination schemes, boundary conditions, thickness, and aspect ratios on the nonlinear frequency ratio at large amplitude vibrations of the laminates. A comparative study is also done along with the first-order and simple higher-order theory deduced from the present model by neglecting the thickness stretching effects. The present analysis shows the degree of hardening behavior getting affected noticeably compared to those of the traditional straight fibers, thus exhibiting the occurrence of drop off in frequency ratio and redistribution of mode shapes at certain amplitudes of vibration.

Comparative study on using static and dynamic finite element models to develop FWD measurement on flexible pavement structures

The deflection basin obtained through backcalculation analysis is compared with the measured deflection basin to determine the moduli of each pavement layer. Most computer programs use multi-layered elastic theory (MET) to perform backcalculation for determining deflection basin. Other structural analysis techniques, such as finite element method (FEM) and finite difference method (FDM), can be used to model flexible pavement structures when conducting FWD tests. Unlike FEM, MET analysis does not take into account nonlinear materials and dynamic loading. This study aims to develop a better finite element (FE) model by using the static and dynamic analyses in the ANSYS computer program. A comparative study was conducted by using varying sizes of model geometry and different types of elements and sizes to determine how they affect the developed FE model. The results of the analyses show that transient dynamic analysis is the best method for simulating FWD test. The percentage of errors between FE deflection basin and measured deflection basin range between 0.94 and 5.01%. Model geometry of 5000 × 5000 mm is sufficient to produce a good deflection basin which approximates the measured deflection. To ensure the accuracy of the developed model, the information on material properties must be valid. Additionally, finer and higher order elements should be used close to the loading region, for instance four or eight-node quadrilateral element (CAX4 or CAX8) with quadratic interpolation function.

Improved graded finite elements with applications in simulating non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials

Graded finite elements (GFEs) provide a promising way for simulating functionally graded materials. Nevertheless, the existing GFE method takes the conventional isoparametric transform functions as a unified representation for the material non-homogeneity in each element regardless of the practical global distribution forms of material properties. This inevitably leads to a certain difference between the local formulation of material property variation in the elements and the corresponding global gradation patterns. In order to eliminate this difference, an improved GFE algorithm is proposed in the present article. The property distribution in the element is formulated by substituting the isoparametric transform of the coordinates directly into the corresponding global gradation functions. Therefore, the local property distribution is always consistent with the global one. Both the improved six-node triangular elements (T6) and eight-node quadrilateral elements (Q8) are developed for non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials, respectively. Exact solutions in three special cases are presented to make comparison with the numerical results, and the accuracy of the proposed algorithm is verified. It is demonstrated that the improved GFE is an effective method for the numerical simulation of functionally graded materials.

Analysis of strain gradient Reissner–Mindlin plates using a C0 four-node quadrilateral element

The weak form of Reissner–Mindlin plate model in the context of the classical continuum theory requires C0-continuity of the lateral deflection as well as rotation fields. However, due to the presence of higher-order derivatives, the same plate model based on strain gradient elasticity demands C1-continuity of interpolation functions for a standard finite element (FE) formulation. On the other hand, it is known that C0 interpolation functions result in more stable and cost-effective elements. Accordingly, a four-node quadrilateral gradient-enhanced plate element using C0 interpolation of the field variables is introduced. The main idea is to start with an eight-node quadrilateral element on which any field variable and its spatial derivatives are independently interpolated via standard C0 shape functions. By introducing appropriate algebraic constraints, the degrees of freedom corresponding to the midside nodes of the eight-node element are then eliminated and a quadrilateral element with four nodes on its corners is constructed. Accuracy and performance of the proposed element in several examples and for a wide range of parameters is investigated. It is shown that the new element can successfully capture the size-dependent behaviour of plates at small scales, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin as well as thick plates in an arbitrary mesh, and is free of shear locking. Furthermore, the introduced element can reproduce the results of the Reissner–Mindlin plate model based on the classical continuum theory when the plate thickness is far greater than the material length scale parameter.

Finite element analysis of thermal stresses for laminated composites in terms of a new C0-type Reddy's theory

ABSTRACTThis article proposed an accurate C0-type Reddy's theory including effects of the transverse normal thermal strain to study the thermal behaviors of laminated composite plates. Although transverse normal thermal strain was taken into account, displacement variables are not increased as thermal loads can be included in the generalized force vector. Based on the proposed model, an eight-node quadrilateral isoparametric element is presented, in which the interelement C0 continuity conditions are satisfied. Numerical results show that the proposed model can produce accurate responses of laminated composite plates under temperature loads.