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Advanced analysis of arbitrarily shaped axially loaded beams including axial warping and distortion
Abstract
In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method.
... -The curvilinear coordinates' system attached to the beam will be denoted by x 1 -For the classical derivative, the following notations are used: a ,i := ∂a ∂x i , a ,ij := ∂ 2 a ∂x i ∂x j . ...
... The unknowns of this system are the stress and strain tensors σ (0) and ε (0) , and the displacement vector u (1) . The loading terms appearing in (56d) and (56e) are in terms of the derivative along x 3 of the last order displacement vector u (0) . ...
... The Saint-Venant Solution of a 3D Tapered Beam δ int σ (0) , u (1) ...
In this article, an extension of the well-known Saint-Venant solution for prismatic beams to tapered beams, i.e., straight beams with linearly varying cross-sections, is presented. The geometry of the tapered beam will be mapped to a reference constant beam, where the 3D equilibrium equations will be expressed, using a differential geometry framework. Then, the asymptotic expansion method will be used to solve these equations, leading to a series of hierarchical problems to solve. The final solution hence obtained is written as a combination of displacement and stress modes, which can be employed as a stress-recovery method for the tapered beam. It is proven in this work that the classical Saint-Venant solution is still valid for the tapered beam, but with the addition of correction modes, representing the effect of the taper.
... In this paper, the higher order beam theory developed in [7] is employed for linear analysis of beams of arbitrarily shaped, homogeneous cross-section, including warping and distortional phenomena due to axial [7], shear, flexural, and torsional behavior [6]. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. ...
... In this paper, the higher order beam theory developed in [7] is employed for linear analysis of beams of arbitrarily shaped, homogeneous cross-section, including warping and distortional phenomena due to axial [7], shear, flexural, and torsional behavior [6]. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. ...
... are derived employing the local equilibrium equations exploiting the aforementioned strain and stress components. The whole procedure is described in detail in [7]. ...
Both Euler-Bernoulli and Timoshenko beam theories maintain the assumptions that neither out-of-plane (warping) nor in-plane (distortion) deformation contribute to beams response. To account for shear lag effects, the inclusion of non-uniform warping is necessary, relaxing the assumption of plane cross section. The shear flow associated with non-uniform warping leads also to in-plane deformation of the cross-section, relaxing the no-distortion assumption. For this purpose, the so-called higher order beam theories have been developed taking into account shear lag and distortional effects. In this paper, the higher order beam theory developed in a previous work of the authors is employed for linear analysis of beams of arbitrarily shaped, homogeneous cross-section, including warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to general load and boundary conditions. The analysis consists of two stages. The first stage is a cross-sectional analysis, establishing the possible distortional and warping deformation patterns (axial, flexural and torsional modes by means of the sequential equilibrium scheme and the Boundary Element Method). The second stage is a longitudinal analysis where the four rigid body displacements along with the extracted deformation patterns multiplied by respective independent parameters expressing their contribution to the beam deformation are included in the beam analysis (Finite Element Method). Axial warping and distortional modes are examined. In addition, numerical examples with practical interest are presented in order to highlight the importance of axial modes in linear analysis of beams-cross-sections of class 4 (EN1993-1-1), i.e., vulnerable to axial load are analyzed.
... Moreover, although the cross-section can be thin-or thick-walled and assumptions of thin walled beam theory are not adopted in [72], axial modes are calculated through the solution of an eigenvalue problem. Finally, Argyridi and Sapountzakis [81] developed a higher order beam theory for generally loaded beams of arbitrary cross-section where axial warping and distortional modes are evaluated employing the concept of sequential equilibrium scheme. In order to exemplify the axial warping and distortional modes of [81] and taking into account that in [81] these modes of a hollow rectangular cross section (closed crosssection) are illustrated, in table 1 warping and distortional modes of W250x45 cross-section (open cross-section) are presented according to the sequential equilibrium scheme [81]. ...
... Finally, Argyridi and Sapountzakis [81] developed a higher order beam theory for generally loaded beams of arbitrary cross-section where axial warping and distortional modes are evaluated employing the concept of sequential equilibrium scheme. In order to exemplify the axial warping and distortional modes of [81] and taking into account that in [81] these modes of a hollow rectangular cross section (closed crosssection) are illustrated, in table 1 warping and distortional modes of W250x45 cross-section (open cross-section) are presented according to the sequential equilibrium scheme [81]. In contrast, in the numerical example of Section 4, the linear buckling of a hollow rectangular cross section, which hasn't been examined in [82], is examined as a comparison with literature. ...
... Finally, Argyridi and Sapountzakis [81] developed a higher order beam theory for generally loaded beams of arbitrary cross-section where axial warping and distortional modes are evaluated employing the concept of sequential equilibrium scheme. In order to exemplify the axial warping and distortional modes of [81] and taking into account that in [81] these modes of a hollow rectangular cross section (closed crosssection) are illustrated, in table 1 warping and distortional modes of W250x45 cross-section (open cross-section) are presented according to the sequential equilibrium scheme [81]. In contrast, in the numerical example of Section 4, the linear buckling of a hollow rectangular cross section, which hasn't been examined in [82], is examined as a comparison with literature. ...
Comparing Euler-Bernoulli or Tismoshenko beam theory to higher order beam theories, an essential difference can be depicted: the additional degrees of freedom accounting for out-of plane (warping) and in-plane (distortional) phenomena leading to the appearance of respective higher order geometric constants. In this paper, after briefly overviewing literature of the major beam theories taking account warping and distortional deformation, the influence of distortion in the response of beams evaluated by higher order beam theories is examined via a numerical example of buckling drawn from the literature.
... Ватина [10], И.Л. Кузнецова [11], S. Back [12], R. Pavazza [13][14][15], N. Rizzi [16], G. Jang [17], R. Vieira [18], M. Brunetti [19], P. Dey [20], A. Argyridi [21]. ...
... Panovko [6], B.N. Gorbunov [7], A.R. Tusnin [8], A.G. Belyy [9], N.I. Vatin [10], I.L. Kuznetsov [11], S. Back [12], R. Pavazza [13][14][15], N. Rizzi [16], G. Jang [17], R. Vieira [18], M. Brunetti [19], P. Dey [20] and A. Argyridi [21]. ...
Introduction. Today thin-walled structures are widely used in the construction industry. The analysis of their rigidity, strength and stability is a relevant task which is of particular practical interest. The article addresses a method for the numerical analysis of stability of an axially-compressed i-beam rod subjected to the axial force and the bimoment. An axially compressed i-beam rod is the subject of the study.
Materials and methods. Femap with NX Nastran were chosen as the analysis toolkit. Axially compressed cantilever steel rods having i-beam profiles and different flexibility values were analyzed under the action of the bimoment. The steel class is C245. Analytical data were applied within the framework of the Euler method and the standard method of analysis pursuant to Construction Regulations 16.13330 to determine the numerical analysis method.
Results. The results of numerical calculations are presented in geometrically and physically nonlinear settings. The results of numerical calculations of thin-walled open-section rods, exposed to the axial force and the bimoment, are compared with the results of analytical calculations.
Conclusions. Given the results of numerical calculations, obtained in geometrically and physically nonlinear settings, recommendations for the choice of a variable density FEM model are provided. The convergence of results is estimated for different diagrams describing the steel behavior. The bearing capacity of compressed cantilever rods, exposed to the bimoment, is estimated for the studied flexibility values beyond the elastic limit. A simplified diagram, describing the steel behaviour pursuant to Construction regulations 16.13330, governing the design of steel structures, is recommended to ensure the due regard for the elastoplastic behaviour of steel. The numerical analysis method, developed for axially-compressed rods, is to be applied to axially-compressed thin-walled open-section rods. National Research Moscow State University is planning to conduct a series of experiments to test the behaviour of axially-compressed i-beams exposed to the bimoment and the axial force. Cantilever i-beams 10B1 will be used in experimental testing.
... For this purpose the so-called higher order beam theories have been developed taking into account shear lag [1] and distortional (in-plane deformation) effects [2]. In this paper, a higher order beam theory is employed for linear static analysis of beams of arbitrarily shaped, homogeneous cross-section, taking into account warping and distortional phenomena due to axial [3], shear, flexural, and torsional behavior [4]. The beam is subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. ...
... The finite element equations are formulated with respect to the displacement and the independent warping and distortional parameters. In the present paper, the higher order beam theory developed in [3] is employed in order to examine axial warping and distortional modes. In addition, numerical examples with practical interest are presented in order to highlight the importance of axial modes in linear static analysis of beams. ...
... An eigenvalue-type analysis [21] is proposed to study warping and distortion of cross-section exploiting the exponentially decaying character of end effects. Recently, a generalized warping analysis is proposed [22,23] with less computational effort and complexity, and can be incorporated with different numerical tools, such as BEM [24] and isogeometric analysis [25] . ...
... and23 , is well below 1% and those of the other stress components are all within 5%, which is good enough from an engineering perspective. To investigate the range of applicability of the Saint-Venant solution, the two main stress components 33 and 23 , located at point A ...
... Warping and distortional fields are found simultaneously in Genoese et al. [13], where an eigenvalue cross-sectional problem is developed. Finally, Dikaros and Sapountzakis [14] and Argyridi and Sapountzakis [15,16] developed a very advanced beam formulation based on a so-called sequential equilibrium scheme where within the cross-sectional analysis the boundary element method is employed. This formulation is not restricted to thin-walled crosssections and does not stand on any corresponding assumption. ...
This paper proposes an efficient generalized beam theory (GBT) formulation, which accounts for cross-sectional deformations in slender prismatic structures. It was shown by the authors in a recent publication [1] that in-plane distortional deformations and accompanied out of plane warping deformations of the cross-section influence the accuracy of results in beam dynamics especially if thin-walled cross-sections are applied. The GBT formulation proposed in [1] overcomes the inaccuracies of classical beam mechanics, however, requires a two-dimensional plane discretization of the cross-section. The computational complexity can be reduced vastly, if thin-walled cross-sections can be discretized with one-dimensional elements. Consequently, this paper discusses an approach with a line mesh discretizing the cross-section and having six degrees of freedom at each node. The membrane part consists of massless micro-polar rotations (drilling rotations) and can be derived independently from the bending part, where a shear elastic formulation is selected.
... Consequently, less cross-section subdomains or elements were needed to model an arbitrary shaped beam cross-section. Boundary element method was also used to analyze the beam cross-section by line or parabolic elements on only the boundaries of the cross-section instead of two-dimensional (2D) elements [14,15]. Moreover, a cross-sectional analysis based on Rayleigh-Ritz method proposed [16,17], where simple polynomial functions or B-spline series were used to eliminate the cumbersome 2D mesh generation process on cross-section. ...
This paper is intended to investigate the static behavior of pre-twisted composite beams using an isogeometric-based cross-sectional analysis. The three-dimensional pre-twisted beam problem is decomposed into a two-dimensional cross-sectional analysis and a one-dimensional beam model. The cross-sectional analysis obtains stiffness constants by considering three-dimensional warping deformation effects. The influence of pre-twist ratio is investigated on the stiffness constants and one-dimensional deformations of isotropic beams, composite strips and box-beams. Inconsistencies in results of pre-twisted composite beam theories in the literature are addressed and discussed. The present method eliminates the costly use of three-dimensional finite element analysis in the initial design steps.
... Besides, toward improving conventional beam elements in order to include distortional effects, independent parameters have been taken into account in beam analysis. The isogeometric tools (B-splines and NURBS), either integrated in FEM or in boundary element method (BEM) called analog equation method (AEM), are employed in the contribution for the static and dynamic analysis of straight beams (Argyridi and Sapountzakis, 2019;Dikaros and Sapountzakis, 2017;Tsiptsis and Sapountzakis, 2018) and horizontally curved beams Sapountzakis, 2017a, 2017b) of open or closed crosssections. Design guidelines for intermediate diaphragms have been applied for box girders and assessed as an indirect way to prevent distortional effects, which specified the maximum spacing of intermediate diaphragms for the case where the distortional effects can be ignored. ...
Toward estimating accurately the distortional response of box girders, in this article, distortion of steel box girders strengthened with intermediate solid diaphragms under eccentric loads is analyzed by employing the so-called initial parameter method. A new model of high-order statically indeterminate structure was established with three orthogonal redundant forces acting at the junction between the girder and diaphragms. Emphasis is put onto the interaction between the girder and diaphragms, where a hypothetical bi-moment B pi indicating all longitudinal redundant force components for diaphragm was proposed besides the moment M pi for in-plane shear component. Simplified initial parameter method solutions for distortional angle and distortional warping stresses and displacements were derived based on the in-plane and out-of-plane compatibilities between the girder and diaphragms. Taking box girders with three and five intermediate diaphragms as an example, the proposed initial parameter method solutions have good agreement with the finite element analysis ones. Finally, distortional behavior under moving eccentric loads is investigated, resulting in a bowl-shaped curve for moment M pi and an approximate trigonometric function for bi-moment B pi . Results show that diaphragms have a stronger resistance on in-plane distortional shear for the loads in midspan than on ends. Plus, the thick diaphragm holds a stronger restraint on distortional warping deformations and stresses than the thin one.
The warping effects may predominate in geometrically nonlinear analysis of open cross-section members. The formulation of conventional beam-column elements incorporating the warping effects is cumbersome due to the method considering the inconsistency between the shear center and centroid. To develop a concise warping element formulation, this paper presents a transformation matrix to integrate the inconsistent effects into the element stiffness matrix. The co-rotational (CR) method used to establish the element equilibrium conditions in the geometrically nonlinear analysis is adopted to simplify the element formulation and improve the efficiency of nonlinear analysis. A new beam-column element explicitly considering the warping deformation and the Wagner effects is derived based on the CR method and the Euler–Bernoulli beam theory. A detailed kinematic description is provided for considering large deflections and rigid body motions. Based on the mechanical characteristic, the coordinate and the rigid body motion transformation matrices are given. The secant relationship is developed to evaluate the element internal forces accurately and effectively in each iteration. Several verification examples are provided to validate the proposed method’s reliability and robustness. The verifications demonstrate that the proposed element leads to considerable computational advantages. The results of this paper are useful for future upgrading of frame analysis software with warping degrees-of-freedom (DOFs).
The present study develops a finite element formulation for the distortional buckling of I- beams. The formulation characterizes the distribution of the lateral displacement along the web height by superposing (a) two linear modes intended to capture the classical non-distortional lateral-torsional behaviour and (b) any number of user-specified Fourier terms intended to capture additional web distortion. All displacement fields characterizing the lateral displacements are taken to follow a cubic distribution in the longitudinal direction. The separation of variables is effectively achieved by exploiting the properties of the matrix Kronecker product. The finite element solution developed is shown to replicate accurately (a) the classical non-distortional lateral-torsional buckling solutions (b) previously developed distortional buckling solutions based on cubic interpolation of the lateral displacement, while (c) providing a basis to assess the effect to commonly omitted higher distortional modes on the predicted critical moments and buckling modes. The solution is then used to conduct a systematic parametric study of over 3900 cases to quantify the reduction in critical moments due to web distortion relative to the classical non-distortional predictions in the case of simply supported beams under uniform loads, point loads, and linear moment gradients, cantilevers, and beams with overhang.
This contribution extends the author’s previous research results concerning effect of spatially varying material properties on warping torsion of beams with open cross-section made of Functionally Graded Material (FGM). The author’s FGM WT beam finite element is used in the calculations of the primary quantities. The secondary deformations due to the angle of twist is considered. The warping part of the first derivative of the twist angle, caused by the bimoment, is accounted as an additional degree of freedom at the beam nodes. The author’s Reference Beam Method, (RBM), is used for homogenization of the spatially varying material properties in the real beam onto effective constant or longitudinally varying stiffnesses for the homogenized beam. Enhanced equations for calculation of the normal and shear stresses with the influence of the deformation effect caused by the primary and secondary torsion moment are established. These equations also contain effect of the warping ordinate function and its gradients that depend on spatially varying material properties. The focus of the numerical investigation is on non-uniform torsional analysis of straight FGM cantilever beams with I- cross-section. An effect of the warping ordinate function and its gradients on the normal and shear stresses is evaluated and discussed. A significant effect of the spatial variability of material properties on the deformation as well as on the stress state in the FGM beams with I-cross-section, has been found. The real beams with spatially varying material properties are modelled with only a single FGM WT beam finite element. Obtained results for the primary and secondary variables are compared with the ones calculated by a very fine mesh of standard 3D-solid finite elements. A very good agreement of all the results has been achieved.
A finite segment element including axial balance is formulated to describe shear lag in thin-walled box beams having constant or variable cross sections made from steel or other materials. The axial balance neglected in the conventional finite segment element model (CFSM) is enforced by adding the nodal longitudinal displacements, while shear lag and shear deformation are incorporated using the nodal shear lag functions and rotations, respectively. The homogeneous solutions deduced by the analytical method are utilized for constructing the element shape functions. By invoking the minimum potential energy theorem, the stiffness matrix and the equivalent nodal force vector are then derived for the element. The precision of the proposed finite segment model (PFSM) is verified against the results yielded from the solid finite element model (SFEM), the finite strip model (FSTM), and the experiments. A continuous box beam having varying cross sections is chosen for comparing the neutral axis depth to the centroidal axis depth. Subsequently, the influence of the axial balance on the mechanical behavior is evaluated. Moreover, the effects of three major geometric parameters are discussed for stress analysis. The results reveal that the proposed finite segment model is capable of reproducing the mechanical behavior of box beams having constant or varying cross sections, and that the stress analysis concerning the continuous box beam with variable cross sections is substantially affected by the axial balance condition.
In the analysis of thin-walled members, some types of boundary conditions met in practice cannot be correctly represented by conventional 1D models. Of course, both shell models or 1D generalised beam theory (GBT) can easily tackle this type of problems - at the cost, however, of the simple interpretation of results provided by conventional 1D models. This paper investigates the problem of combined bending and twisting of a doubly symmetric I-beam having only one flange supported, which precludes the application of the conventional Timoshenko's model, with its built-in assumption that the cross-sections do not distort in their own plane. A simple 1D model for this particular problem is developed, which accounts for the in-plane distortion of the web and for the relative rotation of the flanges. A mixed 3-field finite element was derived for the numerical implementation of this model and applied to three illustrative examples. Its principal merit is the ability to achieve high accuracy on very coarse meshes, particularly when it comes to the approximation of generalized stresses, which are of particular interest to designers.
This paper presents a numerically efficient tool for linear buckling predictions of perforated thin-walled bars within the framework of generalized beam theory (GBT). GBT-based beam finite element methods (GBT-FEM) have been well developed for eigenvalue buckling analyses of non-perforated thin-walled bars. The novelty of this paper consists in an extension of the standard GBT to the scope of thin-walled members with arbitrarily shaped and placed holes. This is achieved by combining the standard GBT and the extended finite element method (X-FEM). More specifically, insert a set of locally supported enrichment functions, accounting for the discontinuities on displacement fields arising from the cross-section cut-outs/holes, into the GBT-based finite element approximations of the member configuration spaces, using the partition of unity method (PUM), where a set of level-set functions are used to describe the geometric profiles of hole edges, i.e., with the hole edges being the zero level sets, and also used to construct the enrichment functions. The proposed approach makes it possible to calculate the deformation mode participations for any perforated thin-walled members as the classic GBT for non-perforated ones. Finally, the proposed approach is calibrated against the shell/solid finite element analysis with four illustrative examples. It can be found that the presented approach is of higher computation efficiency than the shell model.
In this contribution, which is an extension of author’s research [1,2], an influence of the in two and three directions varying material properties on non-uniform torsion of the Functionally Graded Material (FGM) thin-walled beams is originally investigated. Based on the semi-analytical solution of the fourth order differential equation for non-uniform torsion, the local finite element equations of the twisted FGM beam are presented, considering the non-uniform torsion with effect of warping and secondary deformations due to the angle of twist. The warping part of the first derivative of the twist angle caused by the bimoment is considered as an additional degree of freedom at the beam nodes. The Multi-Layers Method (MLM) [3], and the Reference Beam Method (RBM) [4] are extended for homogenization of the spatially varying material properties and relevant stiffnesses in the real beam onto effective longitudinally varying ones in the homogenized beam for the load case of non-uniform torsion.
The focus of the numerical investigation, with consideration of the warping and Deformation Effect due to the Secondary Torsional Moment (STMDE), is on elastostatic analysis of straight FGM beams with rectangular hollow cross-sections. The influence of the variation of the material properties with respect to two or three directions on the angle of twist and the bimoment normal and torsional shear stresses is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid finite elements. This FGM warping non-uniform torsion beam finite element can be used not only in torsional analysis of general thin-walled beams but also in the design of FGM sensors and actuators in mechatronics.
Mechanical properties of slender, prismatic structures are typically analyzed based on classical beam mechanics (Timoshenko’s shear force bending, Vlasov’s theory of warping torsion, …). There it is assumed that the cross-section remains rigid in its projection plane and in-plane distortional deformations of the cross-section are neglected. Such a model is predictive in case of static gradually distributed loading, and solid cross-sections, however, in case of thin-walled crosssections and dynamic loading severe deviations might occur. Therefore, a generalized beam theory is proposed, where warping fields and accompanied distortional fields of the cross-section are axially distributed each based on one generalized degree of freedom. The evaluation of pairs ofwarping and distortional fields in ascending order of importance is performed using a specific reference beam problem (RBP), where three-dimensional elasticity theory is applied in connection with semi-analytical finite elements (SAFE). Convergence of the resulting formulation is ensured by increasing the number of contributing pairs of warping and distortional fields. The resulting formulation yields significantly better results compared to classical beam mechanics especially in the dynamic regime.
We propose a consistent higher-order beam theory in which cross-sectional deformations defining degrees of freedom are derived in the framework consistent with the mechanics of the proposed one-dimensional beam theory. This approach contrasts with earlier methods in which the procedure used to derive sectional deformations and the final beam theory are based on models of different levels. An advantage of the proposed consistent approach is that the generalized force-stress relation even for self-equilibrated forces such as bimoments can now be explicitly written. Also, sectional deformations can be systematically derived in closed form by the recursive and hierarchical approach. Accordingly, the accuracy in both displacement and stress can be adjusted so that obtained results are fully comparable with plate/shell results. We mainly conduct analysis of membrane deformations occurring in thin-walled box beams subjected to doubly symmetric loads such as axially-loaded forces. This case is elaborately chosen to better explain the fundamental concepts of our newly proposed approach. A brief description is also provided to show that these concepts are applicable to other types of loads such as bending and torsion. We confirm the accuracy of the theory proposed here by calculating stress and displacement in several examples.
In this paper, a higher order beam theory is employed for linear local buckling analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to arbitrary concentrated or distributed loading, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross-sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the buckling analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters constitute the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. The buckling load is calculated and is compared with beam and 3d solid finite elements analysis results in order to validate the method and demonstrate its efficiency and accuracy.
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