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Higher order beam theory for linear local buckling analysis
Abstract
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.
... 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. ...
... Nevertheless, all the aforementioned researches deal with the problem of local buckling employing assumptions of TTT and in some cases and the applicability of their methods depends on the cross-sectional shape. To overcome these disadvantages, Argyridi and Sapountzakis [82] developed a higher order beam theory for the buckling analysis of arbitrarily shaped beams where warping and distortional modes (axial additionally to flexural and torsional ones) are evaluated employing the concept of sequential equilibrium scheme. This higher order beam theory is employed in the present paper to examine the influence of distortion in the response of beams evaluated by higher order beam theories. ...
... In order to investigate the influence of distortional phenomenon in the response of beams evaluated by higher order beam theories a numerical example drawn from the literature [98,99] is examined dealing with linear buckling analysis of beams. Closed cross-section cantilever beams (Figure 1a) Figure 1: Boundary conditions and loading (a), cross section (b) and buckling loads of beams of numerical example as obtained from E/BBT [98], TBT [98], GWBT [98], Solid FEM with and without diaphragms [98], Wang and Li [99] and HOBT of [82]] for various Ndofs(c). ...
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.
... 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. ...
... with a b ¼ € u b , u b ¼ u xb u yb u zb ½ T and ρ b is the effective mass density for bending (15). ...
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.
... FSDT does not consider the in-plane and out-plane deformation. To overcome the disadvantages of FSDT, HSDT considers the shear lag and distortion effects by including higher order axial displacement along the thickness of the beam [2,18]. ...
This paper investigates the free vibration and buckling behaviors of functionally graded graphene platelets (FG-GPLs) reinforced porous beam under axially variable loads. The internal pores and GPLs are either uniformly or non-uniformly distributed along the thickness direction. Halpin–Tsai micromechanics model is used to calculate the effective elastic modulus. The variation of Poisson’s ratio along the thickness and the relation between mass density and porosity coefficients are determined using mechanical properties of closed-cell solid under the Gaussian random scheme. The equilibrium equations are derived by Hamilton’s principles, and critical buckling load and dimensionless natural frequency are determined by Ritz formulation. Results revealed that buckling and free vibration behavior of the porous FG-GPL beam are influenced by the GPLs grading pattern and the type of axially varying load. Furthermore, the grading pattern of porosity has more influence on the buckling behavior compared to the free vibration behavior. It is also observed that buckling mode and the fundamental vibration mode of the porous FG-GPL are influenced by the loading conditions and remain unaffected by the grading pattern of porosities and GPLs.
... Elastic stability has been widely studied in classical books [3][4][5][6]. Recently, more complicated studies regarding theory of stability considering higher order beam theories [7] and more rigorous mathematical solution involving theoretically exact derivation for some stability problems [8] have been carried out. On the contrary, a simple and comprehensible procedure to address the study of the elastic stability of compressed members is presented here. ...
There are three possible modes of buckling of thin-walled straight steel columns: flexural buckling, torsional buckling, and flexural-torsional buckling. These modes of buckling are considered in the specification for the design of steel structures, such as Eurocode 3, which includes the particular case of torsional-flexural buckling of centrically loaded members with monosymmetric cross-sections. The system of differential equations that governs the stability of centrically loaded weightless members was presented in the mid-twentieth century and has been widely addressed in both steel structures and instability books. In this work, a simpler way to obtain the differential equations of stability for both torsional and flexural-torsional buckling modes by using equivalent forces is presented. The presented idea is especially useful in the academic context of civil engineering. Students and faculty members will appreciate the deduction of the instability equations governing the equilibrium in a few simple steps.
The present study investigates the aeroelastic flutter characteristics of the graphene platelets (GPL) reinforced metal foam beam. Closed-cell metal foam beams having graded distribution of pores and functionally graded reinforcement of GPLs are considered in this study. The closed-cell metal foam model has been used for deriving the mechanical properties of the foam matrix, which makes provision for determining the relation between the co-efficient of porosity and the co-efficient of density. Modified Halpin-Tsai micromechanics is used to obtain the effective Young’s modulus of the GPLs reinforced composite beam, Density and Poisson’s ratio are calculated with the help of the rule of mixture. The Hamilton’s principle together with the Ritz method, employing the first-order piston theory gives the governing equations of motion for aeroelastic flutter characteristics of the beam for different end conditions. Juxtaposition of dimensionless natural frequencies with the results previously published by others is executed for validating the correctness of the approach followed in the present model. A study of various parameters has been executed, and the results in tables and graphs present the influence of porosity as well as GPLs reinforcement, different boundary conditions and thermal loading on aeroelastic flutter characteristics of the FG-GPL reinforced metal foam beam.
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.
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.
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.
Thin-walled box beams generally exhibit complex sectional deformations that are not significant in solid beams. Accordingly, a higher-order beam theory suitable for the analysis of thin-walled box beams should include degrees of freedom representing sectional deformations. In a recent study, a recursive analysis method to systematically derive sectional membrane deformations has been proposed to establish a consistent higher-order beam theory. In this study, another recursive analysis method is proposed that is suitable for the closed-form derivation of new sectional bending deformations representing the bending of edges (or walls) of the cross-section shown in a box beam under doubly symmetric loads. A consistent 1D higher-order beam theory appropriate to include these additional deformation modes as beam degrees of freedom is also established. The proposed theory provides explicit formulas that relate stresses to generalized forces including self-equilibrated forces such as bimoments. Furthermore, sectional modes are hierarchically derived so that the level of solution accuracy can be effectively and systematically controlled. Thus, the accuracy for static displacement/stress calculations and eigenfrequencies can be adjusted to be fully comparable with plate/shell results. When general doubly symmetric loads are applied to a box beam, the edge membrane modes derived in an earlier study can also be used as additional degrees of freedom besides the edge-bending modes derived in this study. The validity of the proposed beam approach is verified through the analyses of static displacements and stresses as well as eigenfrequencies for free vibration problems.
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 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.
This paper presents a general formulation for the distortional analysis of beams of arbitrary cross section under arbitrary external loading and general boundary conditions. The nonuniform distortional/warping distributions along the beam length are taken into account by employing independent parameters multiplying suitable deformation modes accounting for in-plane and out-of-plane cross-sectional deformation (distortional/warping functions). The paper proposes a novel procedure for cross-sectional analysis which results in the solution of separate boundary value problems for each resisting mechanism (flexure, torsion) on the cross-sectional domain instead of relying on eigenvalue analysis procedures encountered in the literature. These distortional and warping functions are computed employing a boundary element method (BEM) procedure. Subsequently, sixteen boundary value problems are formulated with respect to displacement and rotation components as well as to independent distortional/warping parameters along the beam length and solved using the analog equation method (AEM), a BEM-based technique. After the establishment of kinematical components, stress components on any arbitrary point of each cross section of the beam can be evaluated, yielding a prediction in good agreement with three-dimensional finite-element method (FEM) solutions, in contrast to conventional beam models.
In this paper, the influence of the variable axial force and of the Secondary Torsion-Moment Deformation-Effect (STMDE) on the deformations of beams due to torsional warping is investigated. The investigation is based on the second-order torsional warping theory of doubly symmetric beams with thin-walled open or closed cross-sections. The effect of the axial force on the torsional stiffness of thin-walled beams is considered according to the second-order torsional warping theory. The solutions of the underlying differential equations are used for setting up the relations, needed for application of the transfer matrix method. They are derived, considering both static and dynamic action. This enables stablishing the local element matrix of the twisted beam in the framework of the Finite Element Method (FEM). The numerical investigation comprises static and modal analyses of thin-walled beams with I cross-sections and rectangular hollow cross-sections. The results are compared with results obtained by the FEM, using solid and beam elements available in standard software.
In this paper, an accurate and computationally efficient Generalised Beam Theory (GBT) finite element is proposed, which makes it possible to calculate buckling (bifurcation) loads of steel–concrete composite beams subjected to negative (hogging) bending. Two types of buckling modes are considered, namely (i) local (plate-like) buckling of the web, possibly involving a torsional rotation of the lower flange, and (ii) distortional buckling, combining a lateral displacement/rotation of the lower flange with cross-section transverse bending. The determination of the buckling loads is performed in two stages: (i) a geometrically linear pre-buckling analysis is first carried out, accounting for shear lag and concrete cracking effects, and (ii) an eigenvalue buckling analysis is subsequently performed, using the calculated pre-buckling stresses and allowing for cross-section in-plane and out-of-plane (warping) deformation. The intrinsic versatility of the GBT approach, allowing the incorporation of a relatively wide range of assumptions, is used to obtain a finite element with a reasonably small number of DOFs and, in particular, able to comply with the principles of the “inverted U-frame” model prescribed in Eurocode 4 [1]. Several numerical examples are presented, to illustrate the application of the proposed GBT-based finite element and provide clear evidence of its capabilities and potential.
A method to determine the elastic buckling strength of imperfect closed polygon columns has been developed. A set of eight equilibrium equations was developed but was reduced to a set of four equations by eliminating the displacements at a typical midpoint node. Typical corner displacements were expanded by Fourier series and a 4 x 4 stability matrix was obtained. Matrix size was independent of the number of sides of the closed polygon. Design curves are given for representative imperfections and geometric parameters for several different polygon shapes.
This paper provides the basis for a very general approach to the determination of initial buckling stresses of long stiffened panels in uniform longitudinal compression. The panels are assumed to consist of a series of long flat strips, rigidly connected together at their edges, as in panels with top-hat or Z-section stringers, or in sandwich panels with corrugated cores. Whatever the buckling mode, the individual flats are subjected, just after buckling, to sinusoidally varying systems of both out-of-plane and in-plane edge forces and moments, superimposed on the basic state of uniform compression. The stiffness matrices corresponding to these sinusoidal edge loads are derived, taking account of the destabilising effect of the basic longitudinal compressive stress, not only in the out-of-plane but also in the in-plane deformations. For the latter purpose a non-linear theory of elasticity is used. The application of these stiffness matrices to specific panels is briefly described. All possible modes are incorporated within one determinantal equation. For panels with identical stiffeners spaced at equal intervals, the order of the determinant is independent of the number of stiffeners.
Material strength of steel columns with octagonal cross section, residual stress distribution in them and initial cross-sectional deformation were measured. Axial compression tests were carried out. Using finite element method the buckling strength of such columns was compared with that of the columns of polyhedral cross-section.
This paper presents a finite-element technique for the analysis of tubular member stability under combined external pressure and structural loads. A tube element is developed for the purposes of this research. While polynomial (quadratic) interpolation is used in the longitudinal direction, Fourier series expansions of the displacement components are adopted at the nodal cross sections. The formulation accounts for large inelastic deformation and recognizes initial imperfections and residual stresses. To trace unstable equilibrium paths, arc-length procedures are implemented. A simple, yet effective estimate of the contribution of external pressure to the tube element stiffness matrix, particularly significant in the analysis of slender tubes, is included. Preliminary results regarding the behavior of tubular members subjected to pressure and bending are reported and discussed. The effects of initial imperfections and residual stresses on the response to pressure along with bending are summarized. Finally, the influence of residual stresses on thrust-moment interaction in tubular beam-columns is examined briefly.
In a recent paper (Gonçalves and Camotim, 2013 [1]), the authors presented an investigation concerning the buckling (bifurcation) behaviour of uniformly compressed thin-walled tubes with regular polygonal cross-sections (RCPS). The present paper complements the previous work by addressing the local and distortional buckling behaviour of RCPS members subjected to bending or torsion and aims at providing a novel insight into these phenomena. In particular, the specialization of Generalized Beam Theory (GBT) for RCPS, as recently proposed in Gonçalves and Camotim (2013) [2], is employed to obtain closed-form analytical solutions and also to carry out parametric studies by means of numerical analyses which are both computationally efficient (due to the small number of d.o.f. involved) and clarifying (due to the modal decomposition features of GBT). For validation purposes, solutions taken from the literature and also standard shell finite element model results are employed.
This paper presents and discusses the results of a GBT-based numerical investigation concerning the local, distortional and global buckling behaviour of lipped channel and zed-section cold-formed steel purlins restrained by steel sheeting and subjected to an uplift loading. Strengthened (lapped) joints, commonly employed at internal supports to preclude the occurrence of local/distortional buckling phenomena, are also investigated and an illustrative application of the use of GBT to determine strengthening lengths is also presented. The sheeting restraint is modelled by means of elastic translational and rotational springs, located at the purlin upper flange, and the joint strengthening is modelled by doubling the cross-section wall thickness. For validation, the GBT-based results are compared with values yielded by ANSYS shell finite element analyses.
This paper investigates the elastic buckling (bifurcation) behaviour of uniformly compressed thin-walled tubular members with single-cell regular polygonal cross-sections (RCPS), such as those employed to build transmission line structures, towers, antennas and masts. A specialisation of Generalised Beam Theory (GBT) for RCPS, reported in a recent paper (Gonçalves and Camotim, 2013) [1], is used to obtain both analytical and numerical results concerning the most relevant buckling modes and provide novel and broad conclusions on the structural behaviour of this type of members. In particular, local, cross-section extensional, distortional and multi-mode (including global flexural) buckling phenomena are addressed. For validation purposes, the GBT-based results are compared with solutions taken from the literature and also with numerical values obtained from finite strip analyses.
In this paper, a boundary element method is developed for the general flexural-torsional linear buckling analysis of Timoshenko beams of arbitrarily shaped composite cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The beam is subjected to a compressive centrally applied load together with arbitrarily axial, transverse and/or torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the method can treat composite beams of both thin and thick walled cross-sections taking into account the warping along the thickness of the walls, while the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the analog equation method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency. The significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.
The local buckling behavior of regular polygonal, short length steel columns, fabricated by welding two half sections made of folded steel plates, is described. The polygonal sections are composed of five different section profiles with four to eight sides and each profile having component plates with one to four varied width-to-thickness ratios. A total of 15 specimens are used in the compression test, sustaining uniform compression stress in the fixed end condition. Accurate measurements of welding and cold-forming residual stresses and geometric imperfections were taken prior to testing and are presented in this paper. The test strengths are compared with the current plate buckling code in Japan and with the ECCS recommendations for unstiffened circular cylinders. The empirical design formula based on the test data is also presented to predict the local buckling strength of the polygonal section columns.
A finite strip method of inelastic analysis for local buckling moments of composite T beams with residual stresses is presented. Comparisons with test data show that the method gives reliable predictions, even when strain hardening is present. This method and an existing method for predicting distortional buckling resistances of non-compact unbraced and unstiffened continuous beams were used to analyze eleven fixed-ended composite bridge girders, with spans between 20 m and 40 m. From the results, modification to the design method of BS 5400: Part 3 were developed. These give resistances to hogging bending that are roughly double those given by Part 3.
A simple analytical model is proposed to describe the effects of local buckling of circular cross section on the maximum strength and behavior of tubular beam-columns. The behavior is presented in the form of load-deflection and load-shortening relationships. These relationships are developed on the basis of an assumed deflection method coupled with the moment-thrust-curvature relationship including the softening branch of the relationship due to the local cross-sectional distortion. The analytically obtained maximum strength interaction curves of beam-columns show a reasonably good agreement with the available experimental results. The trend of the analytical load-deflection available experimental results. The trend of the analytical load-deflectionand load-shortening curves is very similar to that of the available experimental results. It is found that the effects of the local buckling on the behavior and strength of tubular beam-columns become more severe with an increase in diameter-to-thickness ratio, and with a decrease in slenderness ratio
This paper summarizes results concerning the capacity of tubular members under combined loading due to external pressure and bending. A nonlinear finite-element technique, outlined in the companion paper is used. Three-dimensional analyses are performed towards an investigation of the effects of localized deformation. To verify the validity of the analytical procedure, computational results are obtained and compared with data from long-column and stub-column tests. Two-dimensional (cross-sectional) calculations furnish accurate estimates of the bending capacity of long, unstiffened tubes. However, three-dimensional effects must be taken into account when dealing with short tubes. Both experiments and computations show that short tubes undergo substantial inelastic deformation prior to buckling at a level of moment higher than the plastic moment. The presence of external pressure reduces the bending capacity and ductility of tubular members. Finally, in agreement with experimental data, the analytical results demonstrate the beneficial effects of capped-end compression on the pressure capacity of stub tubes.
For thin-walled sections, lateral and lateral-torsional buckling are often affected by distortion of the section and this can severely reduce the critical stress. A calculation method for the critical stress which takes account of distortional effects, based on Generalised Beam Theory (GBT), is presented in this paper. The limiting slenderness at which distortional effects begin to take effect is evaluated for channel- and hat-sections on the basis of parametric studies. Approximate formulae for the limiting slenderness are then given.
This paper reports the results of an investigation aimed at providing fresh insight on the mechanics underlying the local and global buckling behaviour of angle, T-section and cruciform thin-walled steel members (columns, beams and beam-columns). Due to the lack of primary warping resistance, members displaying these cross-section shapes possess a minute torsional stiffness and, therefore, are highly susceptible to buckling phenomena involving torsion – moreover, it is often hard to distinguish between torsion and local deformations. Almost all the numerical results presented are obtained by means of Generalised Beam Theory (GBT) analyses and, taking advantage of its unique modal features, it is possible to shed some new light on how to characterise and/or distinguish the local and global buckling modes of the above thin-walled members. Finally, some comments are made concerning the development of a rational and efficient (safe and economic) approach for their design.
The objective of this paper is to provide implementation details of, and practical examples for, modal decomposition of the cross-section stability modes of thin-walled members by constraining a traditional finite strip method (FSM) solution. The theoretical development of the proposed method is provided in a companion to this paper [Ádány S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-walled Structures, submitted for publication, companion to this paper.] The constraint matrix, which is directly applied to the elastic and geometric stiffness matrices of a traditional FSM solution in order to constrain the deformations, is provided along with all formulae necessary in its construction. In addition, a completely worked out numerical example is provided to aid in implementing the constrained FSM solution. The authors implemented the constrained FSM in the open source program CUFSM. This modified version of CUFSM is then used to provide a series of numerical examples that illustrate (i) the advantages of performing modal decomposition, (ii) the importance of understanding and defining the deformation fields related to a desired mode, and (iii) the behavior of constrained FSM stability solutions compared with classical analytical solutions, GBT, and unconstrained FSM. Decomposition of the cross-section buckling classes related to global and distortional modes is demonstrated. Further, the impact of how to select the deformation fields and perform modal decomposition for cross-section stability modes within a class, e.g., for the traditional three global modes (weak-axis flexure, strong-axis flexure and flexural-torsional buckling), is explored and the impact of the deformation field definitions demonstrated. Comparisons of the constrained FSM solutions with other available solutions demonstrate the importance of properly determining when beam theory and plate theory should apply to the cross-section stability of thin-walled members.
This paper provides the first detailed presentation of the derivation for a newly proposed method which can be used for the decomposition of the stability buckling modes of a single-branched, open cross-section, thin-walled member into pure buckling modes. Thin-walled members are generally thought to have three pure buckling modes (or types): global, distortional, and local. However, in an analysis the member may have hundreds or even thousands of buckling modes, as general purpose models employing shell or plate elements in a finite element or finite strip model require large numbers of degrees of freedom, and result in large numbers of buckling modes. Decomposition of these numerous buckling modes into the three buckling types is typically done by visual inspection of the mode shapes, an arbitrary and inefficient process at best. Classification into the buckling types is important, not only for better understanding the behavior of thin-walled members, but also for design, as the different buckling types have different post-buckling and collapse responses. The recently developed generalized beam theory provides an alternative method from general purpose finite element and finite strip analyses that includes a means to focus on buckling modes which are consistent with the commonly understood buckling types. In this paper, the fundamental mechanical assumptions of the generalized beam theory are identified and then used to constrain a general purpose finite strip analysis to specific buckling types, in this case global and distortional buckling. The constrained finite strip model provides a means to perform both modal identification relevant to the buckling types, and model reduction as the number of degrees of freedom required in the problem can be reduced extensively. Application and examples of the derivation presented here are provided in a companion paper.
The present paper focuses on the structural stability of long uniformly pressurized thin elastic tubular shells subjected to in-plane bending. Using a special-purpose non-linear finite element technique, bifurcation on the pre-buckling ovalization equilibrium path is detected, and the post-buckling path is traced. Furthermore, the influence of pressure (internal and/or external) as well as the effects of radius-to-thickness ratio, initial curvature and initial ovality on the bifurcation moment, curvature and the corresponding wavelength, are examined. The local character of buckling in the circumferential direction is also demonstrated, especially for thin-walled tubes. This observation motivates the development of a simplified analytical formulation for tube bifurcation, which considers the presence of pressure, initial curvature and ovality, and results in closed-form expressions of very good accuracy, for tubes with relatively small initial curvature. Finally, aspects of tube bifurcation are illustrated using a simple mechanical model, which considers the ovalized pre-buckling state and the effects of pressure.
A non-linear elastic Generalised Beam Theory (GBT) is formulated and used to investigate the buckling behaviour of aluminium and stainless steel thin-walled columns. The modifications that must be incorporated in the conventional GBT, in order to handle the material non-linearity, are addressed and particular attention is paid to the need to define the stability problem in terms of instantaneous elastic moduli. After validating the proposed GBT, by means of its application to compressed rectangular plates, the unique features and capabilities of the theory are illustrated through the presentation and discussion of results concerning C-section and RHS columns. Stress–strain laws of the Ramberg–Osgood type are used to model the uniaxial behaviour and both J2-flow and J2-deformation plasticity theories are implemented.
First-order generalized beam theory describes the behaviour of prismatic structures by ordinary uncoupled differential equations, using deformation functions for bending, torsion and distortion. In second-order theory, the differential equations are coupled by the effect of deviating forces. The basic equations for second-order generalized beam theory are outlined. Solutions for pin-ended supports are presented, demonstrating the coupling effect by modes and by loads. In the different ranges of length, the individual modes are sufficient approximations for the critical load. The application to a thin-walled bar with C-section under eccentric normal force demonstrates the quality of the single-mode compared to the exact solution.
The present paper examines instabilities of long thin elastic tubes. Both initially straight and initially bent tubes are analyzed under in-plane bending. Tube response, a combination of ovalization instability and bifurcation instability (buckling), is investigated using a nonlinear finite element (FE) technique, which employs polynomial functions in the longitudinal tube direction and trigonometric functions to describe cross-sectional deformation. It is demonstrated that the interaction between the two instability modes depends on the value and the sign of the initial tube curvature. The ovalization of initially bent tubes is examined in detail and, in particular, the case of opening moments. Furthermore, the paper emphasizes on bifurcation instability. It is shown that buckling may occur prior to or beyond the ovalization limit point, depending on the value of the initial curvature. Using the nonlinear FE formulation, the location of bifurcation on the primary path is detected, post-buckling equilibrium paths are traced, and the corresponding wavelengths of the buckled configurations are calculated. Moreover, results over a wide range of initial curvature values are presented, extending the findings of previous works. Finally, several analytical approaches, introduced in previous research works, are also employed to estimate the moments causing ovalization and bifurcation instability. These approaches are based on nonlinear flexible shell theory or simplified ring analysis. The efficiency and accuracy of those analytical methods with respect to the nonlinear FE formulation are examined.
Analysis of shear lag in box beams by the principle of minimum potential energy
- Reissner
Second-order generalised beam theory
- Davies