A finite element formulation for sliding beams,Part I

We use the updated Lagrangian and the co-rotational finite element methods to obtain solutions for geometrically non-linear flexible sliding beams. Finite element formulations are normally carried out for fixed domains. Since the sliding beam is a system of changing mass, first we discretize the system by introducing a variable-domain beam element and model the sliding beam by a fixed number of elements with changing length. Second, we transform the system governing equations of motion to a fixed domain and use conventional finite elements (fixed size and number) to discretize the system. Then our investigation is followed by a comparison between two formulations. Finally, we use the co-rotational method in conjunction with a variable domain beam element to obtain the discretized system equations. To do so, we consider the beam to slide with respect to a fixed channel and later we consider a formulation in which the beam remains at rest and the channel slides with a prescribed velocity. We show that both formulations end up with identical discretized equations of motion. © 1998 John Wiley & Sons, Ltd.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

Dispersion analysis of a non‐conforming finite element method for the Helmholtz and elastodynamic equations

We investigate the dispersive properties of a non‐conforming finite element method to solve the two‐dimensional Helmholtz and elastodynamics equations. The study is performed by deriving and analysing the dispersion relations and by evaluating the derived quantities, such as the dimensionless phase and group velocities. Also the phase difference between exact and numerical solutions is investigated. The studied method, which yields a linear spatial approximation, is shown to be less dispersive than a conforming bilinear finite element method in the two cases shown herein. Moreover, it almost halves the number of points per wavelength necessary to reach a given accuracy when calculating the mentioned velocities in both cases here presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C¹ continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C¹ continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second-order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.     

Finite element methods for the analysis of softening plastic hinges in beams and frames

This paper presents new finite element methods for the analysis of localized failures in plastic beams and frames in the form of plastic hinges. The hinges are modeled as discontinuities of the generalized displacements of the underlying Timoshenko beam/rod theory. Hinges accounting for a discontinuity in the transversal and longitudinal displacements and the rotation field are developed in this context. A multi-scale framework is considered in the incorporation of the dissipative effects of these discontinuities in the large-scale problem of a beam and a general frame. A localized softening cohesive law relating these generalized displacements with the stress resultants acting at the level of the cross section is effectively introduced in the frame response. The resulting models, referred to as localized models, are then able to capture the localized dissipation observed in the localized failures of these structural members, avoiding altogether the inconsistencies observed for classical models in the stress resultants with strain softening. The constructive approach followed in the development of these models leads naturally to the formulation of enhanced strain finite elements for their numerical approximation. In this context, we develop new finite elements incorporating the singular strains associated to the plastic hinges at the element level. A careful analysis is presented so the resulting finite elements avoid the phenomenon of stress locking, that is, an overstiff response in the softening of the hinge, not allowing for the full release of the stress. The accurate approximation of the kinematics of the hinges requires a strain enhancement linking the jumps in the deflection and the rotation fields, given the coupled definition of the transverse shear strain in these two fields. Different enhanced strain elements, involving different base finite elements and different enhancement strategies, are considered and analyzed in detail. Their performance are then compared in several representative numerical simulations. These analyses identify optimally enhanced finite elements for the accurate modeling the localized failures observed in common framed structures.Financial support for this research has been provided by the ONR under grant no. N00014-00-1-0306 with UC Berkeley. This support is gratefully acknowledged.     

A novel finite element formulation for frictionless contact problems

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

A layerwise finite element formulation for vibration and damping analysis of sandwich plate with moderately thick viscoelastic core

Abstract

Most previous studies of viscoelastic sandwich plates were based on the assumption that damping was only resulting from shear deformation in the viscoelastic core. However, extensive and compressive deformations in the viscoelastic core should also be considered especially for sandwich plates with moderately thick viscoelastic core. This paper presents a finite element formulation for vibration and damping analysis of sandwich plates with moderately thick viscoelastic core based on a mixed layerwise theory. The face layers satisfy the Kirchhoff theory while the viscoelastic core meets a general high-order deformation theory. The viscoelastic core is modeled as a quasi-three-dimensional solid where other types of deformation such as longitudinal extension and transverse compression are also considered. To better describe the distribution of the displacement fields, auxiliary points located across the depth of the sandwich plate are introduced. And based on the auxiliary points, the longitudinal and transverse displacements of the viscoelastic core are interpolated independently by Lagrange interpolation functions. Quadrilateral finite elements are developed and dynamic equations are derived by Hamilton’s principle in the variation form. Allowing for the frequency-dependent characteristics of the viscoelastic material, an iterative procedure is introduced to solve the nonlinear eigenvalue problem. The comparison with results in the open literature validates the remarkable accuracy of the present model for sandwich plates with moderately thick viscoelastic core, and demonstrates the importance of the higher-order variation of the transverse displacement along the thickness of the viscoelastic core for the improvement of the analysis accuracy. Moreover, the influence of the thickness and stiffness ratios of the viscoelastic core to the face layers on the damping characteristics of the viscoelastic sandwich plate is discussed.     

A Galerkin/least‐squares finite element formulation for consolidation

A Galerkin/least‐squares (GLS) finite element formulation for problem of consolidation of fully saturated two‐phase media is presented. The elimination of spurious pressure oscillations appearing at the early stage of consolidation for standard Galerkin finite elements with equal interpolation order for both displacements and pressures is the goal of the approach. It will be shown that the least‐squares term, based exclusively on the residuum of the fluid flow continuity equation, added to the standard Galerkin formulation enhances its stability and can fully eliminate pressure oscillations. A reasonably simple framework designed for derivation of one‐dimensional as well as multi‐dimensional estimates of the stabilization factor is proposed and then verified. The formulation is validated on one‐dimensional and then on two‐dimensional, linear and non‐linear test problems. The effect of the fluid incompressibility as well as compressibility will be taken into account and investigated. Copyright © 2001 John Wiley & Sons Ltd.     

On the thermal problem for composite beams using a finite element semi-discretisation

The paper deals with steady state thermo-elastic problems in beam-like structures and it is composed of three theoretical sections. The first part presents a two-dimensional finite element procedure to compute the temperature distribution within a beam cross section subjected to prescribed boundary conditions. It allows the beam cross section to be modelled taking into account any kind of thermal anisotropy or inhomogeneity.

The second part is devoted to the structural thermo-elastic problem in a beam having arbitrary non-homogeneous, anisotropic material properties over the cross section but constant along the axis; the extension of a well-known semi-discretisation procedure to take into account anisotropic thermal expansion coefficients is presented. In this way it is possible to compute strains and stresses related to temperature distributions on the cross section computed, using the method outlined in the first part of the paper.

The third part describes the procedure to evaluate thermal equivalent loads suitable for a three dimensional frame analysis.

Some examples are presented and the results are compared either with their theoretical counterparts or with numerical results obtained from a full three-dimensional finite element analysis.     

An accurate two-node finite element for shear deformable curved beams

An accurate two-node (three degrees of freedom per node) finite element is developed for curved shear deformable beams. The element formulation is based on shape functions that satisfy the homogeneous form of the partial differential equations of motion which renders it free of shear and membrane locking. The element is demonstrated to converge to the results obtained from a shear deformable straight beam when the beam becomes shallower. Numerical examples were performed to demonstrate the accuracy and efficiency with respect to previously published formulations. © 1998 John Wiley & Sons, Ltd.     

A cohesive finite element formulation for modelling fracture and delamination in solids

In recent years, cohesive zone models have been employed to simulate fracture and delamination in solids. This paper presents in detail the formulation for incorporating cohesive zone models within the framework of a large deformation finite element procedure. A special Ritz-finite element technique is employed to control nodal instabilities that may arise when the cohesive elements experience material softening and lose their stress carrying capacity. A few simple problems are presented to validate the implementation of the cohesive element formulation and to demonstrate the robustness of the Ritz solution method. Finally, quasi-static crack growth along the interface in an adhesively bonded system is simulated employing the cohesive zone model. The crack growth resistance curves obtained from the simulations show trends similar to those observed in experimental studies     

A stabilized finite element formulation for the transonic small‐disturbance system

A stabilized finite element formulation for the transonic small‐disturbance system of equations is developed and used to solve a variety of problems in transonic aerodynamics. An adaptive mesh refinement technique and a common discontinuity capturing operator are used to resolve regions with large gradients in the velocity field. The scheme works well in both flow regimes, subsonic and supersonic, and captures shocks naturally. Agreement with available experimental observations and theoretical approximations is very good. Copyright © 2001 John Wiley & Sons, Ltd.     

Vibration analysis of beams with non‐local foundations using the finite element method

In this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element formulation for the simulation of propagating delaminations in layered composite structures

This paper presents a finite element method to simulate growing delaminations in composite structures. The delamination process, using an inelastic material law with softening, takes place within an interface layer having a small, but non‐vanishing thickness. A stress criterion is used to detect the critical points. To prevent mesh‐dependent solutions a regularization technique is applied. The artificial viscosity leads to corresponding stiffness matrices which guarantee stable equilibrium iterations. The essential material parameter which describes the delamination process is the critical energy release rate. The finite element calculations document the robustness and effectivity of the developed model. Extensive parameter studies are performed to show the influence of the introduced geometrical and material quantities. Copyright © 2001 John Wiley & Sons, Ltd.     

Mixed finite element approximations based on 3‐D hp‐adaptive curved meshes with two types of H(div)‐conforming spaces

Two stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical‐like region, and constant polynomial degree distribution k, show L² errors of order k+1 or k+2 for the potential variable, while keeping order k+1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case of hp‐adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.