A rational elasto‐plastic spatially curved thin‐walled beam element

Torsion is one of the primary actions in members curved in space, and so an accurate spatially curved‐beam element needs to be able to predict the elasto‐plastic torsional behaviour of such members correctly. However, there are two major difficulties in most existing finite thin‐walled beam elements, such as in ABAQUS and ANSYS, which may lead to incorrect predictions of the elasto‐plastic behaviour of members curved in space. Firstly, the integration sample point scheme cannot capture the shear strain and stress information resulting from uniform torsion. Secondly, the higher‐order twists are ignored which leads to loss of the significant effects of Wagner moments on the large twist torsional behaviour. In addition, the initial geometric imperfections and residual stresses are significant for the elasto‐plastic behaviour of members curved in space. Many existing finite thin‐walled beam element models do not provide facilities to deal with initial geometric imperfections. Although ABAQUS and ANSYS have facilities for the input of residual stresses as initial stresses, they cannot describe the complicated distribution patterns of residual stresses in thin‐walled members. Furthermore, external loads and elastic restraints may be applied remote from shear centres or centroids. The effects of the load (and restraint) positions are important, but are not considered in many beam elements. This paper presents an elasto‐plastic spatially curved element with arbitrary thin‐walled cross‐sections that can correctly capture the uniform shear strain and stress information for integration, and includes initial geometric imperfections, residual stresses and the effects of the load and restraint positions. The element also includes elastic restraints and supports, which have to be modelled separately as spring elements in some other finite thin‐walled beam elements. Comparisons with existing experimental and analytical results show that the elasto‐plastic spatially curved‐beam element is accurate and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

Two kinds of tube elements for transient thermal–structural analysis of large space structures

Two kinds of thin‐walled tube elements are presented for transient thermal–structural analysis of large space structures by the finite element method. Not only the average temperature, but also the perturbation temperature in the cross‐section of the tube is considered in the present elements. These two temperatures are decoupled in the deduction about the new elements and the non‐linear analysis is restricted to solving the equations of average temperature. Therefore, the magnitude of the non‐linear analysis can be reduced by the presented method. The main difference between the two kinds of thin‐walled tube elements is in the shape functions of the temperature along the circumference of cross‐section. Corresponding to the transient temperature field, quasistatic thermo‐elastic analysis is also introduced. Three examples are shown and the effectiveness of the new elements is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of Laminated and Sandwich Composite Structures Using Solid-like Shell Elements

A new solid-like shell element was formulated which is suitable for analysis of laminated and sandwich composite structures. Then, a multiscale analysis technique was implemented to the shell element formulation so that micro-level stresses and strains (i.e. stresses and strains in reinforcing fibers and the binding matrix) in those structures can be computed. The shell element has three displacement degrees of freedom per node like a 3-D solid element. Therefore, the shell elements can be stacked easily on top of one another like 3-D solid elements in order to represent multiple layers through the thickness of laminated and sandwich structures. The effect of a thin resin or adhesive layer in laminated and sandwich composite structures was investigated on both static and the dynamic responses of the structures using the developed shell elements. The study showed an apparent effect of the resin/adhesive layer even though it is very thin. As a result, the present shell element can be used effectively to include those thin layers in finite element analysis models of laminated and sandwich composite structures.     

Buckling of composite thin walled beams by refined theory

This work presents the buckling analysis of laminated composite thin walled structures by the 1D finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation (CUF). In the present study, the refined beam theories are obtained on the basis of Taylor-type expansions. The finite element analysis has been chosen to easily handle arbitrary geometries as well as boundary conditions. Buckling behavior of laminated composite beam and flat panels are analyzed to illustrate the efficacy of the present formulation and various types of buckling modes are observed depending on the geometrical and material parameters. It is observed that the lower order models are unable to deal with torsion.     

A bilinear shell element based on a refined shallow shell model

A four‐node shell finite element of arbitrary quadrilateral shape is developed and applied to the solution of static and vibration problems. The element incorporates five generalized degrees of freedom per node, namely the three displacements of the curved middle surface and the two rotations of its normal vector. The stiffness properties of the element are defined using isoparametric principles in a local co‐ordinate system with axes approximately parallel to the edges of the element. The formulation is based on a modern, refined variant of the shallow shell models found from the classical books on shell theory. In addition, the bending behavior of the element is improved with numerical modifications, which include mixed interpolation of the membrane and transverse shear strains. The numerical experiments show that the element is able to compete in accuracy with the highly reputable bilinear elements of the commercial codes ABAQUS and ADINA. The new formulation even outperforms its commercial rivals in problems with strong layers such as vibration problems or problems with concentrated loads. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of 3D problems using a new enhanced strain hexahedral element

The now classical enhanced strain technique, employed with success for more than 10 years in solid, both 2D and 3D and shell finite elements, is here explored in a versatile 3D low‐order element which is identified as HIS. The quest for accurate results in a wide range of problems, from solid analysis including near‐incompressibility to the analysis of locking‐prone beam and shell bending problems leads to a general 3D element. This element, put here to test in various contexts, is found to be suitable in the analysis of both linear problems and general non‐linear problems including finite strain plasticity. The formulation is based on the enrichment of the deformation gradient and approximations to the shape function material derivatives. Both the equilibrium equations and their variation are completely exposed and deduced, from which internal forces and consistent tangent stiffness follow. A stabilizing term is included, in a simple and natural form. Two sets of examples are detailed: the accuracy tests in the linear elastic regime and several finite strain tests. Some examples involve finite strain plasticity. In both sets the element behaves very well, as is illustrated in numerous examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A coupled analysis method for structures with independently modelled finite element subdomains

A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented, assessed, and demonstrated. This method provides a means of coupling local and global finite element models whose nodes do not coincide along their common interface. In general, the method provides a means of coupling structural components (e.g., wing and fuselage) which may have been modelled by different analysts. In both cases, the need for transition modelling, which is often tedious and complicated, is eliminated. The coupling is accomplished through an interface for which three formulations are considered and presented. These formulations are: collocation, discrete least-squares, and hybrid variational. Several benchmark problems are analysed and it is shown that the hybrid variational formulation provides the most accurate solutions.     

Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures

The substructured finite element/extended finite element (S-FE/XFE) approach is used to compute stress intensity factors in large aircraft thin walled structures containing cracks. The structure is decomposed into a ‘safe’ domain modeled with classical shell elements and a ‘cracked’ domain modeled using three-dimensional extended finite elements. Two applications are presented and discussed, supported by validation test cases. First a section of stiffened panel containing a through-thickness crack is investigated. Second, small surface cracks are simulated in the case of a generic ‘pressure membrane’ with realistic crack configurations. These two semi-industrial benchmarks demonstrate the accuracy, robustness and computational efficiency of the substructured finite element/extended finite element approach to address complex three-dimensional crack problems within thin walled structures.     

A finite element formulation for thermoelastic damping analysis

We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well‐known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8‐node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three‐dimensional 20‐node iso‐parametric thermoelastic element, which is suitable to model massive structures. For the 8‐node element the dissipation along the plate thickness has been taken into account by introducing a through‐the‐thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled‐field elements. Copyright © 2008 John Wiley & Sons, Ltd.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

Modeling of cement hydration in high performance concrete structures with hybrid finite elements

A hybrid finite element formulation is used to model the hygro‐thermo‐chemical process of cement hydration in high performance concrete. The temperature and the relative humidity fields are directly approximated in the domain of the element using naturally hierarchical bases independent of the mapping used to define its geometry. This added flexibility in modeling implies the independent approximation of the heat and moisture flux fields on the boundary of the element, the typical feature of hybrid finite element formulations. The formulation can be implemented using coarse and, eventually, unstructured meshes, which may contain elements with high aspect ratios, an option that can be advantageously used in the simulation of the casting of concrete structural elements. The resulting solving system is sparse and well suited to adaptive refinement and parallelization. It is solved coupling a trapezoidal time integration rule with an adaptation of the Newton–Raphson method designed to preserve symmetry. The relative performance of the formulation is assessed using a set of testing problems supported by experimental data and results obtained with conventional (conform) finite elements. Copyright © 2015 John Wiley & Sons, Ltd.     

Isoparametric elemetns for cross-sepctional properties and stress analysis of beams

This paper presents an isoparametric finite element formulation for the torsion and the flexure due to end shears for the beam cross-sections of arbitrary shape. Isoparametric line elements are developed using this formulation for the beam cross-sections consisting of very thin wall open or close multicells. Isoparametric transition elements are also developed for the beam cross-sections consisting of both thin wall sections and solid like sections. Numerical examples are presented to demonstrate the accuracy and the applications of such elements.     

Extension of MITC to higher‐order beam models and shear locking analysis for compact,thin‐walled,and composite structures

A class of mixed interpolated beam elements is introduced in this paper under the framework of the Carrera Unified Formulation to eliminate the detrimental effects due to shear locking. The Mixed Interpolation of Tensorial Components (MITC) method is adopted to generate locking‐free displacement‐based beam models using general 1D finite elements. An assumed distribution of the transverse shear strains is used for the derivation of the virtual work, and the full Gauss‐Legendre quadrature is used for the numerical computation of all the components of the stiffness matrix. Linear, quadratic, and cubic beam elements are developed using the unified formulation and applied to linear static problems including compact, laminated, and thin‐walled structures. A comprehensive study of how shear locking affects general beam elements when different classical integration schemes are used is presented, evidencing the outstanding capabilities of the MITC method to overcome this numerical issue. Refined beam theories based on the expansion of pure and generalized displacement variables are implemented making use of Lagrange and Legendre polynomials over the cross‐sectional domain, allowing one to capture complex states of stress with a 3D‐like accuracy. The numerical examples are compared to analytic, numerical solutions from the literature, and commercial software solutions, whenever it is possible. The efficiency and robustness of the proposed method demonstrated throughout all the assessments, illustrating that MITC elements are the natural choice to avoid shear locking and showing an unprecedent accuracy in the computation of transverse shear stresses for beam formulations.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

Higher‐order beam analysis of box beams connected at angled joints subject to out‐of‐plane bending and torsion

The difficulty in the analysis of thin‐walled beams by a beam theory comes from slowly decaying end effects associated with warping and distortion. However, a beam theory without considering such effects yields inaccurate solutions especially near beam ends. Numerical analysis using a higher‐order beam theory capable of representing such effects is now available, but the analysis of a series of box beams connected by angled joints still remains an unsolved problem because of the lack of a matching condition at the joint. The objectives of this investigation are to develop a field‐variable‐matching technique at an angled joint through a higher‐order beam theory and to implement it in the finite element formulation. Thin‐walled box beams in consideration are assumed to be subject to out‐of‐plane bending and torsion. Thus, the minimization of three‐dimensional displacement mismatch is used to relate the field variables at a joint intersection. The minimization condition turns out to represent coupling effects of different deformation kinematics such as torsion, bending, distortion and warping. Point‐wise displacement matching is not possible with a higher‐order beam theory. The validity of the proposed technique was verified by a finite element analysis using two‐node higher‐order beam elements applied to some benchmark problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid equilibrium element for folded plate and shell structures

This paper extends hybrid equilibrium formulation concepts, previously used with success for planar problems, to the analysis of folded plates and curved shells. A 2D hybrid equilibrium flat shell quadrilateral element is formulated for linear analysis, where detailed consideration is given to the implication of slope discontinuity when the element is used for non‐planar domains. Benchmark plate bending, folded plate and curved shell problems are modelled using equilibrium and conforming elements for comparison. In models of the latter two problems, torsional moments may be released along lines of slope discontinuity, and the effects of this assumption for the folded plate are studied by analysing a third type of model composed of 3D solid brick elements. The comparisons demonstrate an excellent performance from the new hybrid equilibrium analysis method for folded plates and curved shells. Copyright © 2013 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.