Curved Beam Finite Element Research Articles

Nonlocal nonlinear higher-order finite element model for static bending of curved nanobeams including graphene platelets reinforcement

The aim of this work is concerned with the nonlocal nonlinear bending characteristics of non-classical isotropic and composite curved beams based on graphene platelets reinforcement, viz., nano size/micro beams. The size-dependant effect prominent in micro/nanobeams is incorporated in the formulation based on the nonlocal theory using size-dependent continuum approach. The proposed structural model includes both moderately large displacement and rotations, respectively. The system governing equations are established in terms of displacement functions using the principle of virtual work statements. A three-noded shear deformable curved beam finite element using a higher-order sinusoidal function based beam theory is employed; the solutions are obtained by adopting direct iterative procedure. The efficacy of the model is tested against the analytical studies available for the nano-size beams. A detailed analysis is made to highlight the influence of various design variables like thickness ratio, curvature of beam, nonlocal parameter, and the support conditions on the nonlinear bending behavior of curved nano/micro-size beams.

Rigid body qualified curved beam approach for lateral torsional buckling of arches

The curved beam finite element approach based on the potential energy formulation is widely recognized as a versatile tool for lateral torsional buckling analysis of arches. However, there exist a variety of potentials, which were developed with distinct assumptions. The diversity of energy-based curved beam theories introduces confusions in practical applications. This paper aims to examine various potential energy formulations as to whether or not they obey the rigid body rule which is a prerequisite for valid second-order analysis. The induced moments at the ends of curved beam element that have been neglected from and/or the distributed moments that should be removed from available potentials are identified to help them become rigid body qualified. Based on the revised potential functional, a curved beam finite element formulation has been developed to effectively predict yet with less effort the out-of-plane critical load for circular and parabolic arches subject to various loading and constraint conditions. The results of numerical examples demonstrate that a potential energy formulation eligible for describing the out-of-plane behavior of curved beam is required to be rigid body qualified by fulfilling two aspects: (i) the neglected end moments that are imperative for satisfying the rigid body rule must be considered unless the virtual work done by them vanishes, and (ii) undesirable distributed moments that may be induced when undergoing the rigid body motion should be removed by considering all essential potential components.

A finite element model for static analysis of curved thin-walled beams based on the concept of equivalent layered composite cross section

In this study, an accurate 1D finite element model with low degrees of freedom (DOFs) is presented for the static extensional-shearing-bending analysis of curved thin-walled beams. In order to incorporate the nonclassical effects like transverse shear flexibility in the formulation, the concept of equivalent layered composite cross section (ELCS) is employed. Based on this new concept, the cross section of a thin-walled beam is replaced with an equivalent layered composite one. A global–local layered beam theory is employed to describe the displacement fields of the curved beams. A curved beam finite element with three nodes and 13 DOFs is developed for the approximation of the unknown variables of the displacement fields. For validation of the proposed model, the results of 1 D models of other researchers as well as the results of the 2D/3D finite element simulations (ABAQUS) are employed. Needlessness to the shear correction factor, satisfaction of zero-conditions of shear stresses on the upper and lower planes of the curved beam, high accuracy in the prediction of structural responses, and significant reduction in the time of computations are some of distinct features of the proposed finite element model.

Nonlinear bending of porous curved beams reinforced by functionally graded nanocomposite graphene platelets applying an efficient shear flexible finite element approach

In the present study, the nonlinear flexural bending of thick and thin porous curved composite beams reinforced and functionally graded by graphene platelets is carried out using a three-noded C1 continuous curved beam finite element developed introducing an efficient shear deformation theory based on trigonometric function. The nonlinearity through the strain–displacement relationship by introducing von Karman’s assumptions is considered. The nonlinear equilibrium equations resulting from minimum potential energy principle are numerically solved based on the direct iteration technique. The bending nonlinear features through the load–deflection relationship are presented by selecting various design parameters like long and short beams, shallow and deep curved cases, support conditions, the variation of graphene platelets in the metal foam and the existence of porosity. This investigation reveals that the level of nonlinearity gets noticeably affected by the depth coupled with the curved beam slenderness ratio.

A new two-noded curved beam finite element formulation based on exact solution

Despite being one of the simplest structural elements, beams are used in many engineering structures. One of the most common methods to analyze and design such structures is the finite element method. Even though many different shape functions for finite beam elements have been offered, still there is a need for a beam formulation that does not suffer from numerical errors, locking problems, and yields accurate results with minimum number of elements. For this reason, in this study, a finite curved beam element formulation is developed based on the exact analytical solution of the governing differential equation of planar curved beams. The axial extension and shear deformation effects are considered in the formulation. Since the stiffness matrix and consistent load vector are obtained from the exact solution, there is no locking problem with the formulation. Many numerical examples are solved to indicate the performance of the proposed element with any loading and boundary conditions. Beams with varying curvature and varying cross section are investigated along with the circular beams with constant cross section. The results show that the element formulation is superior to other elements in the literature with accuracy and wide range of applicability for arbitrarily shaped curved beams.

Efficient coupled polynomial interpolation scheme for out-of-plane free vibration analysis of curved beams

The performance of two curved beam finite element models based on coupled polynomial displacement fields is investigated for out-of-plane vibration of arches. These two-noded beam models employ curvilinear strain definitions and have three degrees of freedom per node namely, out-of-plane translation (v), out-of-plane bending rotation (θz) and torsion rotation (θs). The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler–Bernoulli beam elements using the force-moment equilibrium equations. Numerical performance of these elements for constrained and unconstrained arches is compared with the conventional curved beam models which are based on independent polynomial fields. The formulation is shown to be free from any spurious constraints in the limit of ‘flexureless torsion’ and ‘torsionless flexure’ and hence devoid of flexure and torsion locking. The resulting stiffness and consistent mass matrices generated from the coupled displacement models show excellent convergence of natural frequencies in locking regimes. The accuracy of the shear flexibility added to the elements is also demonstrated. The coupled polynomial models are shown to perform consistently over a wide range of flexure-to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios and are inherently devoid of flexure, torsion and shear locking phenomena.

A Dual Grid Curved Beam Finite Element Model of Piston Rings for Improved Contact Capabilities

A Dual Grid Curved Beam Finite Element Model of Piston Rings for Improved Contact Capabilities

An Arbitrary Cross Section, Locking Free Shear-flexible Curved Beam Finite Element

In this paper, a finite element model is proposed for the study of two-dimensional arch structures. Three- and five-node elements are developed irrispective of the shape of the arch and capable of analyzing thick to very thin structures using a modified Mindlin-Reissner theory. A compatibility displacement-based method is used: full integration is introduced to evaluate all energy terms and the convergence pattern is completely independent of the thickness values, even if a coarse mesh is employed. Shear and membrane locking are completely eliminated, shaping shear and membrane strains by means of suitable functions appropriately projected over Gauss integration points. A formulation has been developed which takes into account, in the elastic range, the possibility of working with a more general cross-section conceived as a set of layers. Numerical examples are also given to show the accuracy of the method and comparisons with previous models are made.

Coupled polynomial field approach for elimination of flexure and torsion locking phenomena in the Timoshenko and Euler–Bernoulli curved beam elements

The curvature related locking phenomena in the out-of-plane deformation of Timoshenko and Euler–Bernoulli curved beam elements are demonstrated and a novel approach is proposed to circumvent them. Both flexure and Torsion locking phenomena are noticed in Timoshenko beam and torsion locking phenomenon alone in Euler–Bernoulli beam. Two locking-free curved beam finite element models are developed using coupled polynomial displacement field interpolations to eliminate these locking effects. The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler–Bernoulli beam elements using the governing equations. The presence of penalty terms in the coupled displacement fields incorporates the flexure–torsion coupling and flexure–shear coupling effects in an approximate manner and produce no spurious constraints in the extreme geometric limits of flexure, torsion and shear stiffnesses. The proposed coupled polynomial finite element models, as special cases, reduce to the conventional Timoshenko beam element and Euler–Bernoulli beam element, respectively. These models are shown to perform consistently over a wide range of flexure-to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios and are inherently devoid of flexure, torsion and shear locking phenomena. The efficacy, accuracy and reliability of the proposed models to straight and curved beam applications are demonstrated through numerical examples.

Dynamic characteristics analysis of drillstring in the ultra-deep well with spatial curved beam finite element

Based on the spatial curved beam element, a new finite element model for drillstring dynamics was presented to understand the dynamic characteristics of the drillstring in a curved borehole. In view of the contact between drillstring and borehole, an iterative solution for the model was given to analyze drillstring dynamic characteristics, such as whirl orbit, whirl speed, lateral and axial accelerations. The results show that the proposed model is correct and reliable. Furthermore, the drillstring dynamic characteristics can be effectively calculated by using spatial curved beam finite element method. The proposed method can be used to improve the efficiency of dynamic analysis of drillstring with ultra-slenderness ratio in the ultra-deep well. It can be concluded that the proposed method could provide a new way and theoretical support for analyzing dynamic characteristics of the drillstring in the curved borehole.

A Finite Circular Arch Element Based on Trigonometric Shape Functions

The curved-beam finite element formulation by trigonometric function for curvature is presented. Instead of displacement function, trigonometric function is introduced for curvature to avoid the shear and membrane locking phenomena. Element formulation is carried out in polar coordinates. The element with three nodal parameters is chosen on curvature. Then, curvature field in the element is interpolated as the conventional trigonometric functions. Shape functions are obtained as usual by matrix operations. To consider the boundary conditions, a transformation matrix between nodal curvature and nodal displacement vectors is introduced. The equilibrium equation is written by minimizing the total potential energy in terms of the displacement components. In such equilibrium equation, the locking phenomenon is eliminated. The interesting point in this method is that for most problems, it is sufficient to use only one element to obtain the solution. Four examples are presented in order to verify the element formulation and to show the accuracy and efficiency of the method. The results are compared with those of other concepts.

General analysis and application to redundant arches under static loading

The aim of this paper is to define an exact formulation of a curved beam finite element for static analysis. The basic equations are combined in the coupled fundamental system in terms of radial displacement v, tangential displacement u and rotation ϕ. An original procedure for solving the fundamental system of equations is used. A finite element formulation based on shape functions that satisfy the homogeneous form of the fundamental system of differential equations is developed. The effects of bending moment, axial extension and transverse shear are taken into account. The exact elastic solution renders the element obtained free of shear and membrane locking. An efficient numerical procedure is presented for determining the pressure curve in the case of circular arches under static loading and arbitrary bonding conditions. The solution obtained is applicable to the analysis of both thin and thick curved beams. Several examples of arches with various loading and boundary conditions are investigated to illustrate the validity and the accuracy of the method. Finally, the effect of the arch rise on the structural response is pointed out.

ANALYSIS OF THREE-DIMENSIONAL LOCKING-FREE CURVED BEAM ELEMENT

Most existing curved beam elements suffer from poor convergence difficulties and a heavy computational burden while limit themselves to 2D problems. In this paper, we address and overcome these difficulties by developing a new three-noded locking-free 3D curved beam element. The element formulations, which employ coupled consistent polynomial displacement fields, satisfy the membrane locking-free requirement of being able to recover the inextensible bending mode of the curved beam. Quintic transverse displacement interpolation functions are used to represent the bending deformation of the beam, while the axial and torsional displacement fields are derived by integration of the presumably linear membrane and torsional shear strain fields, which are coupled with the transverse displacement fields. Numerical results of two- and three-dimensional applications are presented to demonstrate the superior accuracy and high convergence rate of the newly developed curved beam element compared with existing ones.

Behaviour of unpropped composite girders curved in plan under construction loading

Composite steel and concrete curved bridges are often used in highways, particularly for motorway interchanges where high speeds require smooth changes in direction. A composite steel and concrete curved bridge often consists of horizontally curved steel girders and concrete deck slabs. The use of steel I-section curved girders is commonplace because of their economy and ease of construction. During construction, the curved steel girder carries a uniformly distributed load, comprising its self-weight and the wet concrete deck. At this stage, each girder acts separately as an individual simply supported curved beam, with much of its load acting at the top flange level. It is often considered that linear analysis may be used to predict the structural behaviour of composite steel and concrete I-section girders curved in plan during construction. However, when an I-section girder curved in plan is subjected to a vertical uniformly distributed load, it experiences primary bending and non-uniform torsion actions. Because of this, the vertical deflections perpendicular to the plane of the girder are coupled with the twist rotations of the cross-section. The primary bending and torsion actions, vertical deflections and twist rotations couple together to produce second-order bending actions in and out of the plane of the curved girder. The interactions between these actions can grow rapidly, and produce early nonlinear behaviour and even early yielding. Hence, the predictions of a linear theory may be very misleading for the behaviour of composite steel and concrete I-section girders curved in plan during construction. This paper uses an efficient nonlinear inelastic curved beam finite element model developed by the authors to investigate the structural behaviour of composite steel and concrete I-section girders curved in plan during construction. It is found that a nonlinear analysis is needed to predict the structural behaviour of the girders under construction loading. The load carrying capacity of the steel curved girders during construction has to be ascertained and precautions may have to be taken to prevent the failure during construction of the composite curved girders.

Free vibration of arches using a curved beam element based on a coupled polynomial displacement field

The performance of a curved beam finite element with coupled polynomial distributions for normal displacement ( w) and tangential displacement ( u) is investigated for in-plane flexural vibration of arches. A quartic polynomial distribution for u is derived from an assumed cubic polynomial field for w using force–moment equilibrium equations. The coupling of these displacement fields makes it possible to express the strain field in terms of only six generalized degrees of freedom leading to a simple two-node element with three degrees of freedom per node. Numerical performance of the element is compared with that of the other curved beam elements based on independently assumed field polynomials. The formulation is shown to be free from any spurious constraints in the limit of inextensional flexural vibration modes and hence does not exhibit membrane locking. The resulting well-conditioned stiffness matrix with consistent mass matrix shows excellent convergence of natural frequencies even for very thin deep arches and higher vibrational modes. The accuracy of the element for extensional flexural motion is also demonstrated.

AN EFFICIENT CURVED BEAM FINITE ELEMENT

The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio. © 1997 John Wiley & Sons, Ltd.

Natural Vibrations of a Clamped-Clamped Arch With an Open Transverse Crack

The paper presents a finite element model of the arch with a transverse, one-edge crack. A part of the cracked arch is modelled by a curved beam finite element with the crack. Parts of the arch without the crack are modelled by noncracked curved beam finite elements. The crack occurring in the arch is nonpropagating and open. It is assumed that the crack changes only the stiffness of the arch, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a curved beam finite element with the crack is presented. The effects of the crack location and its length on the changes of the in-plane natural frequencies and mode shapes of the clamped-clamped arch are studied.

The development of an improved curvilinear thin-walled Vlasov element

The purpose of this study is to develop a more exact horizontally curved beam finite element in a cylindrical coordinate system. The variation method is used to formulate the stiffness matrix and the results of the solution are compared with another study using a closed form solution. The stiffness matrix for a curved beam element can be used in analyzing the behavior of horizontally curved bridges for either open or closed sections.

An effective curved composite beam finite element based on the hybrid-mixed formulation

A laminated curved-beam finite element with six displacement degrees of freedom and three stress parameters is derived and evaluated. Both thermal and hygrothermal effects are included. The element is based on the Hellinger-Reissner principle and the hybrid-mixed formulation. The Timoshenko beam theory and classical lamination theory are employed in the finite element description. Within an element linear displacement interpolation is used; the generalized stresses are interpolated by either stress functions based on the equilibrium equations ( P 1) or constant stress approximation ( P 2). The beam element stiffness is obtained explicitly and numerical results show very good displacement prediction compared to analytical solutions. Generalized stresses are predicted accurately at the mid-point of the finite element only for constant stress interpolation. The P 1-type element yields more accurate displacement and stress prediction.

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.