Shell Finite Element Formulation Research Articles

Elastoplastic bifurcation analysis for the lateral–torsional buckling of straight beams

This paper deals with the lateral–torsional buckling of elastoplastic steel beams under pure bending considering various cross-section geometries. Critical buckling moment expressions for elastic beams are well documented. In contrast, less attention has been devoted to the lateral–torsional buckling phenomenon in elastoplastic regime. When considering short beams (with sufficiently low slenderness), plastic zones will develop before buckling occurs. The problem being faced here is however much more complex than the classical problem of an elastoplastic beam subjected to an axial compressive force. Under such loading, plasticity occurs instantaneously and evolves uniformly over the cross-section and along the beam length whereas, in the other case, plasticity will spread gradually and the plastic zones will change with the loading. The main objective of the present paper is to develop original analytical solutions for the elastoplastic lateral–torsional buckling of perfectly straight beams. A general formulation based on a previously developed 3D plastic bifurcation theory is first proposed and then applied to the cases of two specific cross-section geometries, namely rectangular and I profiles. For the sake of simplicity, transverse shear strains are neglected in the analysis. The von Mises yield criterion with a linear isotropic hardening is adopted. In the case of a rectangular cross-section, closed-form solutions are obtained for the elastoplastic critical buckling moment, which is shown to depend on the geometric and material parameters of the beam, including the yield stress. For validation purposes, all the analytical solutions are compared against the results of numerical computations performed with an in-house program based on a shell finite element formulation. Analytical and numerical results are in very good accordance, making the analytical method presented here an efficient and precise tool to analyze the elastoplastic lateral–torsional buckling phenomenon.

Shell finite element formulation for geometrically nonlinear analysis of curved thin-walled pipes

A family of shell finite elements is developed for the geometrically nonlinear analysis of pipe bends. The constitutive description follows the Saint-Venant-Kirchhoff model. The first Piola–Kirchhoff stress and the conjugate gradient of the virtual displacement fields are adopted within the framework of the virtual work principle. Three C1 continuous schemes are used to interpolate the displacement fields in the longitudinal direction while Fourier series are used for circumferential interpolation. Eigenvalue analyses are conducted to assess the ability of the elements to represent rigid body motion. Comparisons with other shell and elbow models demonstrate the accuracy and versatility of the formulation.

Shell finite element formulation for geometrically nonlinear analysis of straight thin-walled pipes

A family of new geometrically nonlinear finite elements is formulated for the simulation of the structural response in the elastic regime of straight pipes with circular cross-sections under various loading conditions. The first Piola–Kirchhoff stress tensor is employed within the principle of virtual work framework in conjunction with a total Lagrangian approach. The Green–Lagrange strain tensor is adopted to capture the finite deformation-small strain effects. The formulations are based on the kinematic assumptions of the thin shell theory and capture the follower effects due to pressure load. Three schemes are proposed to interpolate the displacement fields in the circumferential direction: (1) a Fourier series expansion, (2) a quartic spline interpolation, and (3) a mixed interpolation combining Fourier series and splines. The performance and prediction accuracy of the elements are assessed through comparisons with finite element models based on shell and elbow elements in ABAQUS under various loading conditions. The results demonstrate the ability of the elements to predict the displacement and stress fields. In particular, the element based on Fourier series interpolation is shown to provide accurate predictions.

Hybrid-Mixed Low-Order Finite Elements for Geometrically Exact Shell Models: Overview and Comparison

We critically review and compare fifteen mixed and mixed-hybrid nonlinear shell finite element formulations with 4 nodes in order to identify those that are closest to the “optimal” one. We consider formulations that are based either on Assumed Natural Strain concept, Enhanced Assumed Strain concept, Hellinger–Reissner functional or Hu–Washizu functional, and those that effectively combine several of the mentioned approaches. Most of the formulations are state-of-the-art, but some are also presented for the first time. We show that the flexibility of mixed-hybrid formulations requires careful consideration of theoretical and numerical aspects in the process of design of a high-performance element. We make extensive numerical experiments in order to assess convergence properties, mesh distortion sensitivity, and computational speed (that is associated with the ability to achieve large steps during the solution search) of the considered formulations. We show that the application of the idea to combine mixed-hybrid formulations of the Hellinger–Reissner and Hu–Washizu type with the Assumed Natural Strain and/or the Enhanced Assumed Strain concepts generates the formulations that perform excellent. We also identify the formulations that are currently closest to the “optimal” one.

Interlaminar stress analysis of composite shell structures using a geometrically nonlinear layer-wise shell finite element

This work aims to calculate interlaminar stress distribution through the thickness of multilayered composite shell structures by employing a novel nonlinear layer-wise shell finite element formulation. Adapting the Mindlin– Reissner theory in each layer, the shear-deformable layer-wise shell element presents the interlaminar shear stress distributions by increasing the number of layers. The interlaminar normal stress distribution is then determined using the finite difference solution of the general form of equilibrium equation in the non-orthogonal curvilinear grid along the Gaussian points. Two boundary conditions at the bottom and the top surfaces are satisfied by adopting the linear Lagrange interpolation function. The developed formulation is assessed through some illustrative problems solved using a proprietary finite element computer program. The results compare very well with those available in the literature and those obtained by simulations with the commercial finite element software Ansys.

Modeling of deformation of battery cells using thick shell element formulation

We describe a new approach for modeling nonlinear deformation and stress distribution of battery cells using a new thick shell finite element formulation with a through-thickness calculation of stresses and strains that satisfy equilibrium conditions. Battery cells are transversely layered materials that contain numerous thin layers in a repeating sequence. The layers are made of materials with significantly different mechanical responses. Explicit discretization of the layers is computationally impractical except for very small domains, while homogenized material models cannot account for stress and strain variations and partition through the thickness. The new formulation allows for calculation of the transverse and interlayer stresses based on the material properties of the individual cell layers. The application to problems where the transverse stresses have strong influence is also possible.

Coupled thermo-mechanical finite element models with node-dependent kinematics for multi-layered shell structures

Node-dependent Kinematic (NDK) shell finite element (FE) formulations are presented for the steady-state thermo-mechanical analysis of laminated structures. The displacements and temperature change are treated as primary variables in the FE models and are directly solved through the coupled thermo-mechanical models. The enforcement of distributed temperature boundary conditions on the top or the bottom surface of hierarchical shell elements is conducted through the Linear Least Squares. The effectiveness of the proposed FE approaches is verified by comparing the results against those from the literature. The application of adaptive refinement approach based on the hierarchical elements and NDK to build FE models with optimal efficiency is demonstrated through numerical examples.

Constrained shell finite element method for elastic buckling analysis of thin-walled members

Abstract The objective of this paper is to develop a constrained shell Finite Element Method (cFEM) based on a force approach for elastic buckling analysis of thin-walled members. The new cFEM is able to separate the general deformation of thin-walled members into the three fundamental deformation mode classes, namely Global (G), Distortional (D), and Local (L), to enable the modal decomposition and identification. In this paper, four force-based mechanical criteria are defined to separate these mode classes. These mechanical criteria are implemented with the general shell finite element formulation without any special treatment of the element formulation. The constraint matrices of the G, D, and L mode classes are then constructed. Numerical examples are presented to demonstrate the capabilities of the new cFEM in modal decomposition and identification. In particular, the modal decomposition and identification results are compared with the cFSM solutions. Applicability of the new cFEM to other shell FE formulation and different loading conditions are illustrated. All these numerical examples demonstrate the potential of the developed cFEM in taking advantages of the modeling capability of the existing shell FE method.

Preliminary structural analysis of arch dams using a flat shell finite element formulation

Preliminary structural analysis of arch dams using a flat shell finite element formulation

Experimental and numerical study of geometrically nonlinear behavior of corrugated laminated composite shells using a nonlinear layer-wise shell FE formulation

This paper presents experimental and numerical studies on the geometrically nonlinear behavior of corrugated laminated composite shells (CLCS) under quasi-static loading along the corrugated direction. A geometrically nonlinear layer-wise shell finite element formulation is adopted to study the behavior of CLCS under large deformation by modeling of incremental different moduli in the tensile and compressive regimes through the thickness, where the spatial location of neutral axis shifts with deformation. A master curve is presented to estimate the value of compressive modulus from given tensile and flexural moduli. Using the prepreg autoclave method, the paper also describes practical challenges in the manufacturing of CLCS and reveals significant influence of thickness on the nonlinear elastic behavior of two thin and moderately thick CLCS. The proprietary layer-wise shell FE formulation is verified with solid-finite-element modeling and employing a developed user material (USERMAT) subroutine in the commercial software ANSYS. Resulting improvements in numerical modelling are assessed in both general and local behaviors.

A dynamic ductile failure analysis of shell structures using a nonlocal XFEM method with experimental validation

This paper presents a finite element continuous-discontinuous approach for the dynamic ductile failure analysis of shell structures. The continuum damage model based on continuous displacements is used in the continuous stage to describe the diffuse micro-cracking in ductile failure of high-strength steel before a macro-crack is formed. In the context of a fully integrated shear deformable shell formulation, a nonlocal modeling procedure based on a projection of mid-plane reference surface is introduced to regularize the element-wise strain fields induced by the continuum damage model. In the discontinuous stage, an incorporation of velocity discontinuities in shell finite elements is pursued by XFEM method when the damage variable exceeds a critical value and the transition from a continuous to a discontinuous model is permitted. A phantom-node approach is employed in the XFEM method to simplify the numerical treatment of velocity discontinuities in the shell finite element formulation. Several numerical benchmarks are examined using the explicit dynamics analysis and the results are compared with the experimental data to demonstrate the effectiveness and accuracy of the proposed method.

Viscoelastic material damping characteristics of carbon nanotubes based functionally graded composite shell structures

This work deals with the study of viscoelastic modeling and vibration analysis of functionally graded nanocomposite shell panels where carbon nanotubes are reinforced in the polymer matrix based on the functionally graded distributions of carbon nanotubes. Five types of grading of carbon nanotubes (such as UD, FGX, FGV, FGO, and FGΛ) in the thickness directions have been considered in order to investigate the vibration damping performance of such composite shell panels. A detailed mathematical formulation for the determination viscoelastic properties is presented. The Mori–Tanaka micromechanics in conjunction with weak interface theory has been developed for the mathematical formulations of the viscoelastic modeling of carbon nanotubes based polymer matrix phase. An eight-noded shell element with five degrees-of-freedom per node has been formulated to study the vibration damping characteristics of various panels made by such functionally graded nanocomposite materials. The shell finite element formulation is based on the transverse shear effects as per the Mindlin’s hypothesis, and stress resultant-type Koiter’s shell theory. Impulse and frequency responses of such structures have been performed to study the effects of various important parameters (such as volume fraction of carbon nanotubes, interfacial condition, agglomeration, temperature, geometries of shell panel) on the dynamic responses. Obtained results demonstrate that quick vibration mitigation may be possible using such carbon nanotubes based proposed composite materials.

Viscoelastic modelling and dynamic characteristics of CNTs-CFRP-2DWF composite shell structures

Abstract The carbon nanotubes (CNTs) are well known for its application in the areas of advanced composite materials for their improved elastic properties. The large specific interfacial surface area owing to tiny dimensions of CNTs may greatly escalation the interfacial sliding which may promote the dissipation of energy in a dynamic situation that makes them very ideal for damping applications in engineering structures/systems. The present article investigates the viscoelastic modelling and dynamic responses of the CNTs – based carbon fiber reinforced two dimensional woven fabrics (CNTs-CFRP-2DWF) composite spherical shell panels where CNTs are reinforced in the polymer matrix phase. The Mori – Tanaka (MT) micromechanics in conjunction with weak interface (WI) theory has been developed for the mathematical formulations of the viscoelastic modelling of CNTs-based polymer matrix phase. Further, strength of material (SOM) method has been employed to formulate viscoelastic material behavior of the yarn and finally the viscoelastic properties of the representative unit cell (RUC) is established based on the unit cell method (UCM). An eight-node shell element with five degree of freedom per node has been formulated to study the vibration damping characteristics of spherical shell structures made by CNTs-CFRP-2DWF composite materials. The shell finite element formulation is based on the transverse shear effect as per the Mindlin's hypothesis, and stress resultant-type Koiter's shell theory. Frequency and temperature dependent material properties of such CNTs-CFRP-2DWF composite materials have been obtained and analysed. Impulse and frequency responses of such structures have been performed to study the effects of various important parameters such as volume fraction of CNTs, interfacial condition, agglomeration, temperature, geometries of shell panel on the material properties and such dynamic responses. Obtained results demonstrate that quick vibration mitigation may be possible using such CNTs-CFRP-2DWF composite material which is desirable to overcome the drawbacks of conventional CFRP woven fabric composite materials.

Free vibration analysis of carbon nanotube-reinforced functionally graded composite shell structures

This paper deals with free vibration analysis of functionally graded composite shell structures reinforced by carbon nanotubes. Uniform and three distributions of carbon nanotubes which are graded in the thickness direction of the structure are considered. The effective material properties are determined via a micro-mechanical model using some efficiency parameters. The equations of motion are developed based on a discrete double directors shell finite element formulation which introduces the transverse shear deformations via a higher-order distribution of the displacement field. Comparison studies are carried out for various functionally graded composite shell structures reinforced by carbon nanotubes in order to highlight the applicability and the efficiency of the proposed model in the prediction of the vibrational behavior of such shell structures.

An efficient formulation for thin-walled beams curved in plan

An efficient formulation is developed for the elastic analysis of thin-walled beams curved in plan. Using a second-order rotation tensor, the strain values of the deformed configuration are calculated in terms of the displacement values and the initial curvature by adopting the right extensional strain measure. The principle of virtual work is then used to obtain the nonlinear equilibrium equations, based on which a finite element beam formulation is developed. The accuracy of the method is confirmed through comparison with test results, shell finite element formulations and other curved beam formulations from the literature. It is also shown that the results of the developed formulation are very accurate for the cases where initial curvature is large.

Linear and quadratic solid–shell finite elements SHB8PSE and SHB20E for the modeling of piezoelectric sandwich structures

ABSTRACTIn this article, hexahedral piezoelectric solid–shell finite element formulations with linear and quadratic interpolation, denoted by SHB8PSE and SHB20E, respectively, are proposed for the modeling of piezoelectric sandwich structures. Compared to conventional solid and shell elements, the solid–shell concept reveals to be very attractive, due to a number of well-established advantages and computational capabilities. More specifically, the present study is devoted to the modeling and analysis of multilayer structures that incorporate piezoelectric materials in the form of layers or patches. The interest in this solid–shell approach is shown through a set of selective and representative benchmark problems. These include numerical tests applied to various configurations of beam, plate and shell structures, both in static and vibration analysis. The results yielded by the proposed formulations are compared with those given by state-of-the-art piezoelectric elements available in ABAQUS, in particular, the C3D20E quadratic hexahedral finite element with piezoelectric degrees of freedom.

Non-linear rotation-free shell finite-element models for aortic heart valves

Hyperelastic material models have been incorporated in the rotation-free, large deformation, shell finite element (FE) formulation of (Stolarski et al., 2013) and applied to dynamic simulations of aortic heart valve. Two models used in the past in analysis of such problem i.e. the Saint-Venant and May-Newmann–Yin (MNY) material models have been considered and compared. Uniaxial tests for those constitutive equations were performed to verify the formulation and implementation of the models. The issue of leaflets interactions during the closing of the heart valve at the end of systole is considered. The critical role of using non-linear anisotropic model for proper dynamic response of the heart valve especially during the closing phase is demonstrated quantitatively. This work contributes an efficient FE framework for simulating biological tissues and paves the way for high-fidelity flow structure interaction simulations of native and bioprosthetic aortic heart valves.

A B-splines-based GBT formulation for modeling fire behavior of restrained steel beams

An implicit finite element model based on the generalized beam theory (GBT) is developed to simulate the fire behavior of restrained steel beams. The model considers both material and geometric nonlinearities. A bilinear stress–strain curve is used to represent the material behavior at elevated temperatures. The GBT cross-section deformation modes are utilized to construct the mid-surface displacements of each wall, where the longitudinal variations of the mode amplitudes are approximated by the cubic B-splines. The bending related displacements are established by using assumptions of the Kirchhoff's plate theory. The Green–Lagrangian strain tensor is employed to represent the deformation at each quadrature point, and therefore the geometric nonlinearity can be included. A weak-form of the nonlinear equilibrium equations is derived by the Principle of Virtual Work and is solved by employing an implicit iterative procedure. The current model can capture local deformation because higher-order cross-section deformation modes are used. The fidelity of the developed GBT formulation has been evaluated by comparing the predictions with three benchmark results available in the literature, and case studies have been performed. It is found that the local buckling can influence the overall behavior of beams in fire, which is different to the finding of the conventional beam-element-based analysis. The current GBT approach is an alternative to the shell finite element formulation, but exhibits higher computational efficiency.

A seven-parameter spectral/hp finite element formulation for isotropic, laminated composite and functionally graded shell structures

In this paper we apply high-polynomial order spectral/hp basis functions in the numerical implementation of a novel 7-parameter continuum shell finite element formulation using general quadrilateral finite elements. The implementation is applicable to the analysis of isotropic, functionally graded and laminated composite shells undergoing fully geometrically nonlinear mechanical response. The shell finite element formulation is constructed in a purely displacement-based setting such that no mixed variational procedures are employed and full numerical integration of all quantities appearing in the virtual work statement is performed using high-order Gauss–Legendre quadrature rules. An efficient procedure for numerically integrating the discrete weak formulation through the shell thickness is implemented; hence no thin- or shallow-shell type restrictions are imposed on the element. For the case of laminated composites, we introduce a discrete tangent vector field, defined on the approximate shell mid-surface, that permits the use of skewed and/or arbitrarily curved elements in the numerical simulation of complex shell structures. To reduce computer memory requirements in the numerical implementation we adopt element-level static condensation, wherein, the interior degrees of freedom of each element are implicitly eliminated prior to assembly of the global sparse finite element coefficient matrix; an approach that results in storage requirements that are on par with traditional low-order finite element implementations. The accuracy and overall robustness of the developed shell element are illustrated through the solution of several nontrivial benchmark problems taken from the literature. These numerical studies provide evidence that the proposed shell element may be used to obtain accurate locking-free results and that large load increments can be employed (in the numerical simulation of shell structures undergoing very large deformations).

A shell finite element formulation to analyze highly deformable rubber-like materials

In this paper, a shell finite element formulation to analyze highly deformable shell structures composed of homogeneous rubber-like materials is presented. The element is a triangular shell of any-order with seven nodal parameters. The shell kinematics is based on geometrically exact Lagrangian description and on the Reissner-Mindlin hypothesis. The finite element can represent thickness stretch and, due to the seventh nodal parameter, linear strain through the thickness direction, which avoids Poisson locking. Other types of locking are eliminated via high-order approximations and mesh refinement. To deal with high-order approximations, a numerical strategy is developed to automatically calculate the shape functions. In the present study, the positional version of the Finite Element Method (FEM) is employed. In this case, nodal positions and unconstrained vectors are the current kinematic variables, instead of displacements and rotations. To model near-incompressible materials under finite elastic strains, which is the case of rubber-like materials, three nonlinear and isotropic hyperelastic laws are adopted. In order to validate the proposed finite element formulation, some benchmark problems with materials under large deformations have been numerically analyzed, as the Cook's membrane, the spherical shell and the pinched cylinder. The results show that the mesh refinement increases the accuracy of solutions, high-order Lagrangian interpolation functions mitigate general locking problems, and the seventh nodal parameter must be used in bending-dominated problems in order to avoid Poisson locking.