Displacement-based Finite Element Research Articles

Thickness-extensible Higher Order Plate Theory with Enforced C1 Continuity for the Analysis of PEEK Medical Implants.

Plate-like structures had been thoroughly studied in literature over years to reduce the computational space from 3D to 2D. Many of these theories suffer either from satisfying the free traction condition or thickness extensibility in addition to the consistency of transverse shear strain energy. This work presents a higher order shear deformation thickness-extensible plate theory (eHSDT) for the analysis of plates. The proposed eHSDT satisfies the condition of free traction as other theories do but it also satisfies the condition of consistency of transverse shear strain energy which is neglected by many theories in the area of plates and shells. The implementation of the proposed theory in displacement-based finite element procedure requires continuity of derivatives across elements. This necessary condition was achieved using the penalty enforcement method for derivative-based nodal degrees of freedom across the standard 9-nodes Lagrange element. The theory was tested for elastic bending deformation of Polyether-ether-ketone (PEEK) which is one of the basic materials for medical implants. The theory showed good accuracy compared to experimental data of the three-points bending test. The present eHSDT was also tested for different conditions with a wide range of aspects ratios (thin to thick plates) and different boundary conditions. The accuracy of the proposed eHSDT was verified against exact solutions for these conditions which showed the advantage over other approaches and commercial finite element packages.

A new finite element formulation for the dynamic analysis of beams under moving loads

Early studies highlighted the fact that moving loads on beams may induce significantly higher dynamic deflections and stresses than those observed in the quasi-static case. Since then, the dynamics of beams under moving loads has been an active research area, in particular for the past decades, with the rapidly increasing computer power allowing the development of highly robust and sophisticated computational and numerical methods in applied mechanics and engineering. This work introduces a novel finite element formulation for the dynamic analysis of Euler-Bernoulli beams subjected to moving loads. The formulation is consistent with a complementary form of the well known Hamilton's variational principle, and will be used to address some numerical tests in both modal and time domains. The effectiveness and accuracy of the formulation will be assessed and discussed by comparison between the obtained results and those rendered by the standard displacement-based finite element formulation. As it will be demonstrated, the proposed formulation not only renders continuous bending-moment and shear-force distributions, a desired feature in the structural design field, but also has a superior accuracy than that provided by the displacement formulation that uses the same number of nodal degrees-of-freedom.

Performance Evaluation of Nonlinear Viscoelastic Materials using Finite Element Method

This research paper applies the finite element method as a methodology to evaluate the structural performance of nonlinear viscoelastic solids. A finite element algorithm was built and developed to simulate the mathematical nonlinear viscoelastic material behavior based on incremental constitutive equations. The derived Equation of the incremental constitutive included the complete strain and stress histories. The Schapery’s nonlinear viscoelastic material model was integrated within the displacement-based finite element environment to perform the analysis. A modified Newton-Raphson technique was used to solve the nonlinear part in the resultant equations. In this work, the deviatoric and volumetric strain–stress relations were decoupled, and the hereditary strains were updated at the end of each time increment. It is worth mentioning that the developed algorithm can be effectively employed for all the permissible values of Poisson’s ratio by using a selective integration procedure. The algorithm was tested for a number of applications, and the results were compared with some previously published experimental results. A small percentage error of (1%) was observed comparing the published experimental results. The developed algorithm can be considered a promising numerical tool that overcomes convergence issues, enhancing equilibrium with high-accuracy results.

Ruled Shells of Conical Type on Elliptical Base

The information about main results on geometry of developable surfaces with an edge of regression which have a directrix ellipse in the base is gathered. These surfaces constitute a group called “Ruled surfaces of conical type on elliptical base”. This group includes elliptical cones, torses with two ellipses defined in the parallel planes, equal slope surfaces, and ruled surfaces with the main frame of three superellipses that are ellipses in one coordinate plane and broken straight lines in the other two coordinate planes. The paper presents a method for developing torses onto a plane, approximation of torses by folded surfaces, and parabolic ending of a thin sheet from elastic material into a torse shell. A brief review of the methods of stress-strain and buckling analysis of the considered ruled shells is given, including the displacement-based finite element method and variational energy method. It is shown that analytical methods can be used only in the case of applying the momentless shell theory for ruled thin shells of conical type. The analytical formulae for determining the normal and tangent internal forces in any momentless conic shell with a superellipse in the base are derived. References to forty four scientific articles of other authors, working or having worked on the subject of the paper are given. These references confirm the conclusions of the author and the perspectives of investigations of the considered ruled surfaces and shells.

A mixed finite element/finite volume formulation for granular bentonite mixtures

A mixed finite element/finite volume formulation is presented for characterising the macroscopic behaviour of granular bentonite mixtures. The formulation enables fitting the hydro-mechanical behaviour of a mixture parting from a double porosity numerical model. The granular bentonite mixture is abstracted as a finite set of homogeneous units made with bentonite grains (where microstructure and macrostructure coexist) embedded in megapores. The separate mass balances of macrostructural and microstructural water are solved for each unit using the finite volume method. The water mass balance in the megaporosity is calculated for the entire domain using the finite element method. To solve the equilibrium equation, a displacement-based finite element method is applied after defining a global stiffness matrix that integrates the effect of megaporosity. After implementing these approaches in a double porosity numerical model, satisfactory modelling results were obtained in agreement with the experimental tests. The results of the proposed model effectively fit those of a triple porosity bentonite unit model that uses a contact model for displacement compatibility of the units. This contributes to the qualification of the presented model’s ability to produce compatible strains of the units using the proposed global stiffness.

Two-layer spatial beam with inter-layer slip in longitudinal and transverse direction

The paper presents a new mathematical and numerical model for analysis of two-layered spatial beams with inter-layer slip taken into account. Each of the two layers can be made of different material, such as timber, concrete or steel. These layers, connected together, form a composite beam, for example concrete–steel, concrete–timber or timber–timber composite beam. There are several novelties in the presented mathematical and numerical model. The first novelty is that the model enables the analysis of the influence of shear deformations on the stiffness of two-layered spatial beams. The second novelty of the model is that the inter-layer slip can only happen in longitudinal and transverse direction, while inter-layer slip in perpendicular direction is not possible. Yet another important novelty of the model is a consistent separation of the basic equations of the model into two unrelated groups and introduction of appropriate constraining equations. The negative influence of poorly conditioned equations due to the constraint equations, which would otherwise occur during the solving of the equations of the model, is thus avoided. The equations of the presented numerical model are solved by applying the finite elements method. Thus, a new family of deformation based finite elements has been developed, where all the deformation quantities of the beam model are interpolated with Lagrange polynomials of arbitrary degree. By introducing a new deformation based finite element we are able to avoid all types of locking that is typical for displacement-based finite elements. Convergence and parametric studies, presented in the paper, show: (i) that the deformation-based finite elements are very accurate and therefore suitable for the analysis of two-layer spatial beams, (ii) that the inter-layer slip in both considered directions significantly affects the values of physical quantities and therefore has to be considered in the analysis and (iii) that shear deformations have smaller effect on the stiffness of two-layer spatial beams than they have on the stiffness of homogeneous spatial beams.

Displacement-based finite element approach on analysing flexible combined pile–raft foundation in layered soil

The present study proposes a new finite element methodology to analyse the behaviour of flexible combined pile–raft foundation (CPRF) situated in layered soil, in a displacement-based framework. The soil medium is idealised as an advanced elastic Pasternak medium and the piles and raft are modelled as bar and plate element, respectively. The two components of CPRF are analysed simultaneously and displacement compatibility is satisfied at the pile–raft junctions. A number of soil–structure interaction factors, which govern the behaviour of CPRF, are suitably subsumed in the analysis scheme. The proposed method is validated with available analytical and experimental studies. Further parametric studies, investigating the effects of soil layering and raft flexibility on the behaviour of CPRF, are explored. It is observed that the load sharing proportion between the components and the raft deformation pattern depend upon the thickness and position of the soft soil layer in a multilayered soil system. The thickness of the flexible raft plays a pivotal role in determining the behaviour of CPRF, founded in a multilayered soil profile. Thus, this research manifests notable advancement in understanding the behaviour of flexible CPRF in layered soil.

Unified three-dimensional finite elements for large strain analysis of compressible and nearly incompressible solids

This work proposes a displacement-based finite element model for large strain analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of the right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained by employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high-order hexahedral finite elements are derived by adopting a three-dimensional formalism based on the Carrera Unified Formulation. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of the present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells, and curved structures.

Melt flow-induced mechanical deformation of dendrites in alloy solidification: A coupled thermal fluid - solid mechanics approach

Melt flow during alloy solidification promotes more dendrite fragments or equiaxed grains due to dendrite fragmentation, while the complex interactions between melt flow and dendritic growth causing flow-induced dendrite deformation and dendrite fragmentations remain obscure. In this work, cellular automaton-finite volume approach and the displacement-based finite element method have been combined to simulate dendrite growth, fluid flow and flow-induced mechanical deformation in Al-4.5 wt%Cu alloy, and to reveal the stress evolution during dendritic growth under melt convection. It is found that dendrites can undergo visible mechanical bending under fluid flow. The stress increases with the enhancement of fluid flow. The primary dendritic trunk is the location of the dominant deformation under a parallel fluid flow. Though the secondary dendrite does not suffer the large stress, it significantly affects the stress concentration of the primary dendritic trunk. As flow velocity increases, the secondary dendrite arm spacing gradually decreases and bridging occurs due to the contact of tertiary dendrites. The bridging of secondary dendrites impedes the development of stress concentrations. Especially, as inflow velocity exceeds 0.05 m/s, the stress does not get larger than that under the velocity of 0.01 m/s under the complication interactions between the dendritic growth and flow patterns. Unlike previous understanding, the maximum stresses induced by melt flow are mainly located at the position where primary dendritic trunk intersects with unconnected secondary dendrites or the thin dendrite trunk.

Effect of interaction between polarization direction and inclined force on the dynamic stability of a pre-stressed piezoelectric plate

The current study deals with forced vibration analysis of a pre-stressed piezoelectric plate with finite lengths resting on a rigid foundation, exposed to an inclined time-harmonic force. The present investigation is modeled according to a new theoretical consideration in the context of the three-dimensional linearized theory of elasticity for solids under initial stress (TLTESIS). It is assumed that the plate exhibits a polarization effect in both vertical and horizontal directions to the top surface and there is an exact clamping condition at the interface between the plate and the rigid foundation, that is, no jumping in the quantities. For obtaining the governing and boundary condition equations of our problem, the virtual work principle is used according to Hamilton's principle and the piecewise homogeneous body model. The three-dimensional displacement-based finite element model is developed for solving the system of partial differential equations and analyzing the dynamic behavior of the plate. The reliability of the solution procedure is proved by convergence analysis and comparing the results to the current literature. The target of the paper for surveying the numerical results and findings is divided into two sections. The first one presents the influence of the inclination of the dynamic external force on the forced vibration characteristics of the current model. In the second one, the influence of inclination angle, thickness ratio, aspect ratio, and initial stress parameter on the frequency response of the harmonic force are investigated in detail. As a desirable result for the designer, while a decrease in the angle of the force with the vertical axis exceeds the resonant mode of the stress, increasing values of the initial stress prevents that mode.

New displacement-based finite element analysis method for predicting the surface residual stress generated by ultrasonic nanocrystal surface modification

Peening techniques can be applied to relieve high tensile residual stress, which is one of the main causes of primary water stress corrosion cracking (PWSCC). Ultrasonic nanocrystal surface modification (UNSM) is a peening technique that generates a high compressive residual stress on the surface of a structure. Finite element (FE) analysis can be utilized for quantitative surface stress prediction according to the UNSM in various materials and loading conditions. In this paper, a new method to predict the surface residual stress generated by UNSM process through FE analysis is proposed. In order to accurately simulate the UNSM process through FE analysis, it is necessary to define an equivalent displacement calculation method that can replace the complex load states of UNSM. For this, we proposed the new equivalent displacement calculation method based on the contact mechanics theory and the energy conservation law. The new equivalent displacement calculation method was validated by comparing the indentation depth obtained by single-path UNSM FE analyses and experiments under various loading conditions. The actual UNSM process is carried out in multiple paths over a large area of the structure. By applying the new equivalent displacement calculation method, the compressive residual stresses were calculated through multi-path UNSM FE analyses and compared with the results obtained from the multi-path UNSM experiments. The surface residual stress results of the experiments and analyses showed good agreement, and it was confirmed that the UNSM process can be simulated well if the new equivalent displacement calculation method is used for the FE analysis.

Free Vibration Analysis of Thin-Walled Beams Using Two-Phase Local–Nonlocal Constitutive Model

Abstract A mathematical model is developed based on the thin-walled beams theory for free vibration analysis of nano/micro scale beams having nonlocal properties and arbitrary cross sections. Constitutive relations are defined by using two-phase local–nonlocal constitutive formulation. Equations of motion are derived by use of Hamilton‘s principle. Both the local and nonlocal part of the model is solved by the displacement-based finite element method. Numerical results are obtained and examined for nonlocal box beams and collapsed carbon nanotubes. In general, it is observed that the natural frequency decreases by increasing the nonlocal parameter or the volume fraction of the nonlocal part.

New robust and efficient global iterations for large deformation finite element analysis of beams and shells with material nonlinearity

In large rotation analyses of beams and shells with displacement-based discretization, the iterative burden grows considerably with the membrane/flexural stiffness ratio. For linear elastic materials, it was shown that a stress–displacement iteration solves this issue and reduces the computational cost even of several times. Convergence problems occur also when large rotations are coupled to material nonlinearity, even if this case is not well addressed in the literature. The extension of the mixed iteration to large deformation problems with nonlinear constitutive laws is faced in this work, focusing on elastoplasticity. New iterative schemes are derived for both displacement-based finite elements (DFEs) and mixed finite elements (MFEs), each featuring further variables in the linearization along with displacements. For DFEs, it is shown that the most convenient approach is to impose the constitutive law for an independent integration point strain, with the strain–displacement compatibility solved together with the global equilibrium. For MFEs, as an extension of this strategy, the most performing scheme solves element compatibility and global equilibrium simultaneously, with the element state evaluated for independent strains work-conjugate of the stress DOFs. In both cases, global linear systems in displacements only are required and strain-driven material laws are easily considered. Numerous tests validate the proposed iterative schemes showing a significant reduction of the computational cost compared to standard approaches.

A Finite Element Solution to Normal Contact Forces of Viscoelastic Particles

In this paper, an approach to solve the normal contact forces of viscoelastic particles with the finite element method is presented. A viscoelastic constitutive model is deduced and established for the finite element solution, where the stress tensors of the Zener-type model expressed as partial differential equations are obtained by applying the generalized Hooke’s law, and the incremental equations are further deduced with the backward difference method. An iterative matrix of the viscoelastic constitutive model that depends on the current strain, current stress and strain increment is derived; a user material subroutine is programed based on the iterative matrix to implement the viscoelastic constitutive model in the displacement-based finite element modeling. The validity of the finite element solution to the normal contact forces of elastic particles is validated with the Hertz contact force model, and that of the solution to the normal contact forces of viscoelastic particles is verified by the experimental data. The results obtained by the proposed solution agree well with those predicted by the Prony series, and the computational efficiency of this solution is higher for different values of the shear viscosity.

About Influence of Choosing the Force Method Principal System on the Accuracy of Calculations in Solving a Stress-Based Plane Elasticity Problem

Introduction. It is known that the solution of a problem in elasticity theory is possible based on displacements, stresses, as well as mixed method. The most common is the displacement-based finite element method (FEM), however, it has its imperfections. The aim of this article is to study the features of numerical solution of a stress-based plane elasticity problem using the finite difference method (FDM) as a possible alternative to the FEM. The options for choosing the force method principal system are considered aiming to obtain the boundary conditions in calculating a rectangular beamwall by the grid method along with the influence thereof on the final result during stresses calculation.Materials and methods. The calculation of a rectangular beam-wall is made in Balka-Stenka software developed in DSTU. Comparison between a solution made by the finite difference method and a solution made by the displacementbased finite element method is made in the LIRA-SAPR software. The accuracy of calculations is determined by performing the static balance tests of the cut-off part of a structure. For the structure under consideration, the choice of five options of the force method principal system (FMРS) is analysed.Results. For the first time the conclusion was made that in contrast to the calculation of bar systems, the choice of FMРS does not significantly affect the final stress values in solving a plane elasticity problem.Discussion and conclusion. The comparison of calculation results made by the displacement-based finite element method and by the finite difference method showed some advantages of the latter. The finite difference method can act as an alternative to the displacement-based finite element method in solving a plane elasticity problem on condition of automation of the process of building the internal forces epures in a wall beam-wall contour.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

Scaled Boundary Perfectly Matched Layer (SBPML): A novel 3D time-domain artificial boundary method for wave problem in general-shaped and heterogeneous infinite domain

Artificial boundary method is widely used in the numerical modeling of unbounded wave problem. However, the accurate modeling of truncated infinite domain with general geometry and heterogeneous materials is still a challenging task, especially for direct time-domain analysis in three dimensions (3D). In this paper, a novel 3D time-domain artificial boundary method, called Scaled Boundary Perfectly Matched Layer (SBPML), is proposed. This method is a generalization of the Perfectly Matched Layer (PML) based on a scaled boundary coordinates transformation inspired by the Scaled Boundary Finite Element Method (SBFEM), which is capable of using artificial boundary of general geometry (not necessarily convex) and considering plane physical surfaces and interfaces extending to infinity in the truncated infinite domain. Local scaled boundary coordinates are firstly introduced on the element-level into the truncated infinite domain to describe general geometry properties of the infinite domain. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. The spatial discretization of the SBPML domain produces semi-discrete mixed displacement–stress​ unsplit-field formulations of third orders in time. The order of the obtained formulation can be reduced by one, enabling a seamless coupling with the standard displacement-based finite element formulation of the interior domain. The coupled system can be solved by an explicit time integration algorithm efficiently. The validation of the SBPML is demonstrated through several benchmark tests, including wave problems in unbounded domains with general geometries and heterogeneous material properties. Furthermore, the application of the SBPML in dynamic soil–structure interaction (SSI) is demonstrated using two engineering problems, including an impact analysis of soft rock-nuclear island system and a vibration analysis of soil-lined tunnel system.

Objectivity in quasi-brittle structural failure via adaptive formulation and mesh refinement

In this work, the combined implementation of Adaptive Formulation Refinement (AFR) and Adaptive Mesh Refinement (AMR) strategies is proposed for the efficient and mesh objective evaluation of quasi-brittle fracture.A critical issue when computing localized structural failure is the spurious mesh bias dependence of the crack trajectories and collapse mechanisms that ensues when adopting the standard displacement-based finite element (FE) formulation. This problem is resolved by the introduction of the mixed strain/displacement FE formulation, which allows for local convergence, producing mesh objective results.On the one hand, the proposed AFR approach allows to adaptively switch between the standard and mixed FE formulations during the numerical simulation depending on the level of accuracy required in different regions of the domain. Within this framework, the more accurate mixed formulation is activated only in the areas where the crack onset and evolution take place, while the standard formulation is adopted for the rest of the structure.On the other hand, the adopted octree-based AMR strategy allows to adaptively refine only certain areas of the domain, also in function of the level of accuracy required. Using this methodology, it is possible to start the simulation with an initially relatively coarse mesh and adaptively introduce several levels of mesh refinement only in the regions of interest, where cracks appear and propagate. This approach guarantees the necessary mesh resolution to capture the fracture phenomenon accurately while further increasing the computational efficiency of the simulation. The subsequent adaptive coarsening of the FE mesh is also allowed.Structural failure and fracture propagation are reproduced via a local isotropic damage model. For this reason, a refinement/coarsening criterion defined in function of the equivalent effective stress is conveniently adopted with both AFR and AMR strategies.To assess the capabilities of the proposed method, a set of numerical simulations featuring benchmark problems and experiments is included. Results show the mesh objectivity and cost-efficiency of the approach combining AFR and AMR in fracture problems. The proposed model is able to reproduce experimental crack paths, collapse mechanisms and force–displacement curves with accuracy and without spurious mesh bias dependence.The paper includes a comparative assessment of the performance of the proposed method and the standard FE formulation exclusively, showing the superior performance of the AFR methodology and its mesh objectivity. Convergence of the computed results when introducing different levels of AMR is also verified. Computational efficiency of the AFR and AMR techniques is specifically assessed. The combination of AFR and AMR approaches allows to provide accurate and mesh objective results in quasi-brittle fracture problems with very significant savings in computational cost.

Justification of fracture criteria for salt rocks

The study of salt rocks deformation and fracture processes is an essential part of mining parameters justification for mineral salt deposits. The results of uniaxial compression tests on large salt rock specimens are presented as a loading curve and diagrams of the transverse-longitudinal displacements at various distances from the side faces. Based on an isotropic elastoplastic model, a multivariant numerical simulation was performed. Its purpose was to select of fracture criteria that accurately describe the loading diagram of specimen and its transverse-longitudinal deformations. The following fracture criteria are considered: Tresca with the associated plastic flow rule, the associated and non-associated Mohr-Coulomb, the parabolic analogue of Mohr-Coulomb criterion and the volumetric fracture criterion. Numerical simulation was carried out by the displacement-based finite element method. Three-dimensional hexahedral eight-node isoparametric elements were used for discretization of the solution domain. It has been established that within the elastoplastic model of media the process of uniaxial compression of a large cubic salt rock specimen is adequately described by the linear Mohr-Coulomb fracture criterion with the non-associated plastic flow, as well as by the associated volumetric parabolic yield criterion with the linear isotropic hardening.

Generalised Beam Theory formulation for the buckling analysis of thin-walled members with circular axis

This paper presents an extension of Generalised Beam Theory (GBT) that enables performing buckling (bifurcation) analyses of thin-walled members with circular axis (and no pre-twist). The bifurcation eigenvalue problem is obtained by applying the linear stability analysis concept to the non-linear equilibrium equations, while incorporating the classic GBT kinematic assumptions, which are essential to obtain significant computational savings. A displacement-based finite element is proposed and used to assess the accuracy and efficiency of the developed GBT formulation in several illustrative numerical examples involving complex global-distortional-local buckling. It is shown that the proposed finite element leads to results that match accurately those obtained with refined shell finite element models (such refinement is essential to obtain correct solutions in curved members), but with much less DOFs. Moreover, it is also shown that the GBT modal features can provide an in-depth insight into the nature of the buckling modes in curved members.