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Abstract

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.

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... 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. ...
... where E m is the effective modulus for membrane properties (14) and ν is the mean-value of Poisson's ratio. The offdiagonal terms in (23) account for the coupling of normal strains and normal stresses: Due to numerical experiments [1,4] we propose to use ν Ã ¼ ν here within the cross-sectional analysis, while ν Ã ¼ 0 will be used in the member analysis. ...
... Þdenotes the plane stress elasticity matrix from (23) with E b being the effective modulus for bending properties (14), while γ ¼ γ xz γ yz ½ is the vector of mean transverse shear strains. The third summand in (29) is proportional to the shear correction factor α s . ...
Article
Full-text available
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.
... The aforementioned stiffness matrix takes into account shear deformation, generalized warping due to both flexure and torsion (shear lag effects) and distortional effects. Nonuniform distortional/warping distributions are taken into account by employing 10 independent distortional/warping parameters (DOFs) multiplying corresponding distortional/warping functions which are obtained by solving corresponding boundary value problems at cross sectional analysis level, developed within the corresponding beam theories presented in [43,44] by the authors. Local stiffness matrix and local equivalent nodal load vector of the element are computed numerically applying the Analog Equation Method (AEM) [45] a Boundary Element Method (BEM) based technique. ...
... , , j P, S) introduced in [44]. The above warping/distortional functions are properties of the member cross section and are evaluated through a cross sectional analysis procedure based on a sequential equilibrium scheme as presented in [44]. ...
... , , j P, S) introduced in [44]. The above warping/distortional functions are properties of the member cross section and are evaluated through a cross sectional analysis procedure based on a sequential equilibrium scheme as presented in [44]. ...
Article
In this paper an advanced 32 × 32 stiffness matrix and the corresponding nodal load vector of a 3-D beam element of arbitrary cross section taking into account shear deformation, generalized warping (shear lag effects) and distortional effects due to both flexure and torsion is presented. Nonuniform distortional/warping distributions are taken into account by employing 10 additional degrees of freedom per node. Local stiffness matrix and local equivalent nodal load vector of the element are computed numerically applying the Analog Equation Method (AEM), a Boundary Element Method (BEM) based technique. Warping and distortional functions as well as geometric constants of the cross section are evaluated employing a 2-D BEM approach. The developed element is incorporated into a standard Direct Stiffness Method (DSM) algorithm employed for the solution of spatial frames. The problem of handling distortional/warping transmission conditions in non-aligned members is alleviated by using an approximate technique for a proper transformation of higher-order degrees of freedom at frame joints.
... Elastic stability of beams is one of the most important criteria in the design of structures. In this paper, a higher order beam theory is employed for local buckling analysis [3] of beams of arbitrarily shaped, homogeneous cross-section [4], taking into account warping and distortional phenomena due to shear, flexure and torsion. 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 analysis consists of two stages. In the first stage, where the Boundary Element Method (BEM) 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 [4]. In the second stage, where the Finite Element Method (FEM) 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 calculation of the , 1 are obtained after solving corresponding boundary value problems, formulated exploiting the local equilibrium equations and the corresponding boundary conditions according to a sequential equilibrium scheme as presented in [4] employing the Boundary Element Method. More specifically, in order to compute the involved warping and distortional functions, local equilibrium is sequentially fulfilled by introducing additional warping and distortional functions, so as to equilibrate non-equilibrated stress residuals [4]. ...
... Genoese, Genoese, Bilotta and Garcea (2014) in [29] developed a beam model with arbitrary cross section taking into account warping and distortion with their evaluation being based on the solution of the 3D elasticity problem for bodies loaded only on the terminal bases and a semi-analytic finite element formulation. Finally, Dikaros and Sapountzakis (2016) in [30] presented a general boundary element formulation for the analysis of composite beams of arbitrary cross section taking into account the influence of generalized cross sectional warping and distortion due to both flexure and torsion. In this proposal, distortional and warping functions are evaluated by the same eigenvalue problem and in order of importance. ...
... Genoese, Genoese, Bilotta and Garcea (2014) in [29] developed a beam model with arbitrary cross section taking into account warping and distortion with their evaluation being based on the solution of the 3D elasticity problem for bodies loaded only on the terminal bases and a semi-analytic finite element formulation. Finally, Dikaros and Sapountzakis (2016) in [30] presented a general boundary element formulation for the analysis of composite beams of arbitrary cross section taking into account the influence of generalized cross sectional warping and distortion due to both flexure and torsion. In this proposal, distortional and warping functions are evaluated by the same eigenvalue problem and in order of importance. ...
... The aim of this paper is to propose a new formulation by enriching the beam's kinematics both with out-of-and in-plane deformation modes and, thus, take into account both cross section's warping and distortion in the final 1D analysis of curved members, towards developing GBT further for curved geometries while employing independent warping parameters, which are commonly used in Higher Order Beam Theories (HOBT). The approximating methods and schemes proposed by Dikaros and Sapountzakis in [30,55] are employed and extended in this study. Adopting the concept of end-effects and their exponential decay away from the support [27], appropriate residual strains are added to those corresponding to rigid body movements. ...
Article
Towards improving conventional beam elements in order to include distortional effects in their analysis, independent parameters have been taken into account in this study. Curved beam’s behavior becomes more complex, even for dead loading, due to the coupling between axial force, bending moments and torque that curvature produces. Thus, the importance of simulating geometry exactly arises in order to approximate accurately the response of the curved beam. For this purpose, the isogeometric tools (b-splines and NURBS), either integrated in the Finite Element Method (FEM) or in a Boundary Element based Method (BEM) called Analog Equation Method (AEM), are employed in this contribution for the static analysis of horizontally curved beams of open or closed (box-shaped) cross sections. Responses of the stress resultants, stresses and displacements to static loading have been studied for various cross sections.
... For this purpose, the so-called higher order beam theories have been developed, taking into account shear lag (out-of-plane deformation) [1][2] and distortional (in-plane deformation) effects [3][4]. Higher order beam theories are of increased interest due to their important advantages over approaches such as 3-D solid or shell solutions as they [5][6] : a) require less modelling time, b) permit isolation of structural phenomena and results' interpretation (rotations, warping parameters, stress resultants etc. are also evaluated in addition to displacements and stress components), c) facilitate modelling of supports and application of external loading, d) require significantly less number of degrees of freedom (dofs) reducing computational time, e) facilitate parametric analyses without the construction of multiple models. ...
... 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. ...
Conference Paper
Full-text available
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.
... Warping and distortional fields are found simultaneously in Genoese et al (2014) where an eigenvalue cross-sectional problem is developed. Finally, Dikaros and Sapountzakis (2017); Sapountzakis (2018, 2019) 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 cross-sections and does not stand on any corresponding assumption. ...
Chapter
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.
... 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. ...
... 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] . ...
... 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. ...
Article
Full-text available
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.
... Additionally, a FEM procedure based on a mixed variational formulation for orthotropic beams by developing an eigenvalue cross-sectional problem yielding simultaneously distortional and warping functions of the arbitrarily-shaped cross-section was developed by Genoese et al. [72]. Finally, an advanced beam element in which the so-called sequential equilibrium scheme is employed for the computation of warping and distortional functions [73]. The advantages of this approach over the eigenvalue analysis one are that distortional and warping functions are evaluated by the same problem and in order of significance, while contrary to eigenvalue analysis, it permits the separate evaluation of axial, flexural and torsional mode groups. ...
Article
Full-text available
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.
Article
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.
Article
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A simple and efficient method is proposed for the analysis of twist of rectangular box-girder bridges, which undergo distortion of the cross section. The model is developed in the framework of the Generalized Beam Theory and oriented towards semi-analytical solutions. Accordingly, only two modes are accounted for: (i) the torsional mode, in which the box-girder behaves as a Vlasov beam under nonuniform torsion, and, (ii) a distortional mode, in which the cross section behaves as a planar frame experiencing skew-symmetric displacements. By following a variational approach, two coupled, fourth-order differential equations in the modulating amplitudes are obtained. The order of magnitude of the different terms is analyzed, and further reduced models are proposed. A sample system, taken from the literature, is considered, for which generalized displacement and stress fields are evaluated. Both a Fourier solution for the coupled problem and a closed-form solution for the uncoupled problem are carried out, and the results are compared. Finally, the model is validated against finite element analyses.
Article
In higher-order beam theories, cross-sectional deformations causing complex responses of thin-walled beams are considered as additional degrees of freedom. To fully capture their bending responses, enriched sectional modes departing from Vlasov’s assumptions have been utilized in recent studies. However, due to these bending-related modes, no available higher-order beam bending theory has established explicit stress-generalized force relations that are fully consistent with those by the classical beam theories and earlier studies based on Vlasov’s assumptions. If they are available, physical significance of the bending-related generalized forces can be readily understood. In addition, equilibrium conditions at a joint of multiple thin-walled beams can be explicitly derived. Here, we newly propose a higher-order beam bending theory that not only includes as many bending-related sectional modes as desired, but also provides the desired explicit stress-generalized force relations. To this end, we establish a recursive analysis method that derives hierarchical bending-related sectional modes. We show that this method can yield certain relations among the sectional mode shapes, which are critical in establishing the desired explicit relations. The validity of the present theory is confirmed by calculating the static, free vibration, and buckling responses of several thin-walled rectangular hollow section beams.
Article
Non-negligible sectional deformations, such as warping and distortion, occur in thin-walled beams under a twisting moment. For accurate analysis, these deformations need to be considered as additional kinematic degrees besides the degrees of freedom used in the classical St. Venant torsion theory. Vlasov pioneered to develop a higher-order beam theory for torsion that incorporates warping and distortion, but more sectional deformation modes than those considered in the Vlasov theory are needed to improve solution accuracy. Several theories were developed towards this direction, but no higher-order beam theory for torsion appears to allow explicit F-U and σ-F relations (U: kinematic variables, F: generalized forces, σ: stresses) as established by the Vlasov theory. In that the explicit relations are useful to interpret the physical significance of the generalized forces and can be critical in deriving explicit equilibrium conditions among the generalized forces at a joint of multiple thin-walled beams, a theory allowing the explicit relations needs to be developed. In this study, we newly propose a higher-order Vlasov torsion theory that not only includes as many torsion-related modes as desired but also provides the explicit F-U and σ-F relations that are fully consistent with those by the Vlasov theory. Towards this direction, we show that expressing the σ-U relation only with sectional mode shapes orthogonal to each other is critical in establishing explicit F-U and σ-F relations. We then establish new recursive relations that can be used to express each of derivatives for the sectional mode shapes involved in the σ-U relation as a linear combination of other orthogonal sectional mode shapes. In the developed theory, even stresses at off-centerline positions of the beam cross-section are explicitly related to F. The validity and accuracy of the proposed theory are confirmed by examining displacements, stresses, and eigenfrequencies for several torsion problems. The numerical results by the proposed theory are in good agreement with those by the shell analysis.
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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.
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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.
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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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In this work, we present a new formulation of a 3D beam element, with a new method to describe the transversal deformation of the beam cross section and its warping. With this new method we use an enriched kinematics, allowing us to overcome the classical assumptions in beam theory, which states that the plane section remains plane after deformation and the cross section is infinitely rigid in its own plane. The transversal deformation modes are determined by decomposing the cross section into 1D elements for thin walled profiles and triangular elements for arbitrary sections, and assembling its rigidity matrix from which we extracts the Eigen-pairs. For each transversal deformation mode, we determine the corresponding warping modes by using an iterative equilibrium scheme. The additional degree of freedom in the enriched kinematics will give rise to new equilibrium equations, these have the same form as for a gyroscopic system in an unstable state, these equations will be solved exactly, leading to the formulation of a mesh free element. The results obtained from this new beam finite element are compared with the ones obtained with a shell model of the beam.
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In this paper an effective finite element of constant stiffness is developed for torsion with warping of both open- and closed-shape cross-sections. The secondary torsion moment deformation effect has been taken into account. The local element stiffness matrix that describes torsion with warping of both open- and closed-shape cross-sections has been derived using an analogy between torsion with warping (including the secondary torsion moment deformation effect) and the 2nd order beam theory (including the shear force deformation effect). The deformation effect of the secondary torsion moment must be taken into account first of all when dealing with closed-shape cross-sections. The warping part of the first derivative of the twist angle has been considered as an additional degree of freedom in each node at the element ends. This can be regarded as part of the twist angle curvature caused by the warping moment. Numerical results are presented to demonstrate the efficiency and accuracy of this new beam finite element. The necessity of including the deformation effect of the secondary torsion moment into the solution is demonstrated on the torsion of closed cross-sections. The results are compared with those obtained using other special commercial codes.
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An analytical model is presented for determining the displacement and stress distributions of the Saint-Venant extension, bending, torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy. The determination of the complete displacement field requires solving a coupled two-dimensional boundary value problem for the local in-plane deformations and warping out of the section plane. The principle of minimum potential energy is applied to a discretized representation of the cross-section (Ritz method) to calculate solutions to this problem. The behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions, where definitions are presented for the shear center, center of twist, torsion constant and a new geometric parameter: the line of extension bending centers. Two sets of numerical results are presented to illustrate how section geometry, beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam.
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This work presents exact solutions of the boundary element integral coefficients for the and influence matrices for off- and on-element boundary integrations based on the direct boundary element method. The interior integral expressions for potential and fluxes are also presented. The exact solutions obtained in this investigation are for the two-dimensional Laplace equation using the constant, linear and quadratic elements. The element geometry in all three cases is considered straight and a formulation is considered for the linear and quadratic elements so that boundary solutions for potential and flux may be discontinuous, partially discontinuous and continuous. The exact expressions presented were verified using numerical integration methods of both Gauss and Romberg. Various geometric cases were also considered and included positioning the source point on and off the element and repositioning the Cartesian coordinate origin. Furthermore, two benchmark problems were also considered to verify the exact integrations presented. Mathcad and Mathematica were used to develop the analytical relationships and verify these relationships.
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This paper compares two distinct approaches for obtaining the cross-section deformation modes of thin-walled members with deformable cross-section, namely the method of Generalized Eigenvectors (GE) and the Generalized Beam Theory (GBT). First, both approaches are reviewed, emphasizing their differences and similarities, as well as their resulting semi-analytical solutions. Then, the GE/GBT deformation modes for four selected cross-sections are calculated and examined in detail. Subsequently, attention is turned to the efficiency and accuracy of the GE/GBT mode sets in typical benchmark problems, namely the calculation of the global–local–distortional first-order and buckling (bifurcation) behaviors of bars with the previously analyzed cross-sections. It is concluded that GE and GBT, both based on the method of separation of variables, yield accurate results although they use different structural models and mode selection strategies. Therefore they offer complementary advantages, which are put forward in the paper.
Article
In this paper a general formulation for the nonuniform shear warping analysis of composite beams of arbitrary simply or multiply connected cross section with at least one axis of symmetry, under general boundary conditions is presented. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading passing through the shear center of the cross section, as well as to bending and warping moments. The nonuniform shear warping distribution is taken into account by employing one independent warping parameter multiplying a shear warping function, which is obtained by solving a corresponding boundary value problem, formulated exploiting the longitudinal local equilibrium equation. By taking into account the aforementioned nonuniform shear warping distribution, the developed formulation is capable of capturing “shear lag” effect on beams under flexure. Three boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameter and solved using the Analog Equation Method, a Boundary Element Method based technique. The warping function and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach.
Article
This paper addresses the development and illustrates the application of a generalised beam theory (GBT) formulation intended to perform first-order elastic-plastic analyses of thin-walled members made of isotropic non-linear materials exhibiting strain-hardening. After presenting an overview of the main concepts and procedures involved in the above GBT formulation, its application is illustrated through the analysis of (i) simply supported Z-section beams and (ii) fixed-ended lipped channel beams. In both cases, a bilinear elastic-plastic material model is adopted, which exhibits three strain-hardening levels, namely Esh = 0 (perfectly plastic model), Esh = E/100 and Esh = E/50. The results presented and discussed consist of equilibrium paths, modal participation diagrams, displacement profiles, beam deformed configurations and stress diagrams and contours. For validation purposes, most of the GBT results are compared with values obtained from shell finite element analyses − with a few relatively minor exceptions, a very good correlation is always found. Finally, the paper closes with some remarks concerning the influence of the strain-hardening slope on the structural behaviour of thin-walled steel beams.
Article
In this two-paper contribution, a general formulation for the nonuniform warping analysis of composite beams of arbitrary simply or multiply connected cross sections, under arbitrary external loading and general boundary conditions, is presented. Part I is devoted to the theoretical development and numerical implementation of the solution method, whereas Part II discusses the examined numerical applications illustrating the efficiency, accuracy, and range of applications of the proposed method as well as the effects arising because of nonuniform warping. The nonuniform warping distributions are taken into account by using four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, respectively, which are obtained by solving the corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. A shear stress correction is also performed to improve the stress field arising from the kinematical considerations used. Ten boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the analog equation method, a boundary element method (BEM)-based technique. The warping functions and the geometric constants including the additional ones due to warping are evaluated with a pure BEM approach.
Article
In this paper an “extended” nonuniform torsion theory of bars taking into account the secondary torsional moment deformation effect (STMDE) is formulated without adhering to assumptions employed in Thin Tube Theory (TTT). This theory is extensively applied to bars of arbitrarily shaped doubly symmetric steel cross section, subjected to arbitrarily distributed or concentrated torsional loading and to the most general torsional boundary conditions. In order to satisfy local equilibrium considerations to this “extended” theory, a torsional shear correction factor (determined through an energy approach) is required at the global level to correct the secondary torsion constant along with a suitable warping shear stress distribution at the local level. Three boundary value problems with respect to (i) the primary warping function, (ii) the secondary warping function and (iii) the total angle of twist coupled with its primary part per unit length are formulated and numerically solved employing the Boundary Element Method (BEM). The torsional geometrical parameters of a large number of either open- or closed-shaped steel sections are determined, verifying wherever possible the accuracy of the employed method and quantifying the errors induced by the TTT. Moreover, for bars of the aforementioned cross sections of various lengths, the influence of the STMDE is systematically quantified, while the accuracy of the results is verified by FEM solutions employing solid or shell elements. The limitations of TTT in computing stress components of closed-shaped section bars are also discussed, while the EC3 – part 1.1 guideline that permits as a simplification the ignorance of torsional warping effects in closed hollow section bars is assessed. Finally, the necessity of employing a torsional shear correction factor to the problem at hand is demonstrated.
Article
A higher order model for the analysis of linear, prismatic thin-walled structures that considers the cross-section warping together with the cross-section in-plane flexural deformation is presented in this paper. The use of a one-dimentional model for the analysis of thin-walled structures, which have an inherent complex three-dimensional (3D) behaviour, can only be successful and competitive when compared with shell finite element models if it fulfills a twofold objective: (i) an enrichment of the model in order to as accurately as possible reproduce its 3D elasticity equations and (ii) the definition of a consistent criterion for uncoupling the beam equations, allowing to identify structural deformation modes.
Article
A linear model for beams with compact or thin-walled sections and heterogeneous anisotropic materials is presented. It is obtained by means of a Ritz–Galerkin approximation using independent descriptions of the stress and displacement fields. These are evaluated by a preliminary semi-analytic solution based on a finite element description of the cross section. A coherent definition of the deformations and stresses is obtained which includes both the generalized Saint Venànt solution for generic materials and some significant additional effects, due to out-of-plane warping and section distortions. The so-built 1-D model maintains the richness of the 3-D solution using a small number of variables.
Article
For thin-walled beams, the classic theory for flexural and torsional analysis of open and closed cross-sections can be generalized by including distortional displacements. In a companion paper it is shown that using a novel semi-discretization process, it is possible to determine specific distortional displacement fields which decouple the reduced order differential equations. In this process the cross section is discretized into finite cross-section elements, and the natural distortional modes as well as the related axial variations are found as solutions to the established coupled fourth order homogeneous differential equations of GBT.In this paper the non-homogeneous distortional differential equations of GBT are formulated using this novel semi-discretization process. Transforming these non-homogeneous distortional differential equations into the natural eigenmode space by using the distortional modal matrix found for the homogeneous system, we get the uncoupled set of differential equations including the distributed loads. This uncoupling is very important in GBT, since the shear stiffness contribution from St. Venant torsional shear stress as well as “Bredt's shear flow” cannot be neglected nor approximated by the combination of axial stiffness and transverse stiffness, especially for closed cross sections. The full analytical solutions of these linear non-homogeneous differential equations are given, including four illustrative examples, which illustrate the strength of this novel approach to GBT. This new approach is a considerable theoretical achievement, since it without approximation gives the full analytical solution for a given discretization of the cross section including distributed loading. The boundary conditions considered in the examples of this paper are restricted to built in ends, which are needed for future displacement formulation of an exact first-order distortional beam element.
Article
In this paper, a new beam finite element is presented, with an accurate representation of normal stresses caused by “shear lag” or restrained torsion. This is achieved using an enriched kinematics, representing cross-section warping as the superposition of “warping modes”. Detailed definitions and computational methods are given for these associated “warping functions”. The exact solution of the equilibrium equations is given for a user-defined number of warping modes, though elastic results are totally mesh-independent.
Article
In this paper, a boundary element method is developed for the nonuniform torsion of simply or multiply connected bars of doubly symmetrical arbitrary constant cross section, taking into account secondary torsional moment deformation effect. The bar is subjected to arbitrarily distributed or concentrated twisting and warping moments, while its edges are restrained by the most general torsional boundary conditions. To account for secondary shear deformations, the concept of shear deformation coefficient is used leading to a secondary torsion constant. Four boundary value problems with respect to the variable along the bar primary and secondary angles of twist and to the primary and secondary warping functions are formulated and solved employing a pure BEM approach, that is only boundary discretization is used. The warping and the primary torsion constants are evaluated employing the aforementioned primary and secondary warping functions using only boundary integration, while the secondary torsion constant is computed employing an effective automatic domain integration. Numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. The influence of the secondary torsional moment deformation effect of closed shaped cross sections is verified, while the accuracy of the proposed numerical procedure for the calculation of the secondary torsion constant compared with a FEM one is noteworthy.
Article
An application of the recently developed thin-walled box beam element to the analysis of multibox bridges which arises in practical design, is presented. The thin-walled box beam element, which also takes account of warping distortional effects, when combined with traditional beam elements into a grillage model may adequately represent the three-dimensional behaviour of multibox superstructure. Equivalent sectional properties for the transverse grillage members across individual boxes are computed from a frame analysis. A numerical iterative procedure is introduced to take account of the interaction due to distortion.Comparisons with other numerical methods and model experiments demonstrate the accuracy and economy of the proposed method.
Article
This paper considers the problem of distortion in thin-walled structural members with closed cross-sections such as might be used as single-spined, single or multi-cell box beams in bridge decks. The problem of distortion has been formulated in a more general way than in previous attempts. The distortion of a cross-section has been characterised by a single representative parameter and appropriate functions of this parameter have been used as the degrees of freedom in a finite element representation.Consideration is first given to the cross-sectional deformation due to torsion and then a general distortional moment is defined. The paper continues with derivations of expressions to obtain the distortional normal and shear stresses in a thin-walled section. Consideration is then given to the transverse resistance of a cross-section to distortion. A discussion follows on the interaction between bending, torsion and distortion and the way in which the formulation can be incorporated into the finite element method. Finally, examples are solved for straight and curved box beams.
Article
A study of rectangular cross-section thin-walled beams under torsional, distortional and bimomental loads is presented. The assumed displacement field describes both torsional and distorsional behaviour, the shearing strain in the walls being taken into account. By a variational principle the equilibrium and boundary equations are derived and a physical interpretation of the natural conditions is given.A closed form solution is given with reference to the matrix form of the governing differential system together with a discussion of a sixth-order ‘uncoupled’ differential equation generalizing the BEF analogy.The present formulation shows that, in general, torsion and distortion are pair of coupled problems and a study of distortion without considering torsion would therefore not be legitimate.The comparison of the results predicted by the theory is in good agreement with experimental evidence.
Article
A previous paper presented a method of analysis for any open unbranched thin-walled section considering both rigid body movement and cross-section distortion (including local buckling). It described briefly the calculation procedures required in order to obtain section properties related to each of a number of cross-section distortion modes. These properties were generally calculated using first-order theory.This paper describes in full how second-order theory can be used to calculate the section properties for all modes, including each of the four rigid body modes. In the process of developing the second-order generalized beam theory, additional section properties evolve which enable the first-order equilibrium equation to be modified to consider second-order elastic critical load problems.
Article
This paper considers the relationship between torsional warping and distortional warping for the analysis of thin-walled beams. The paper explains why warping displacement can be defined by a single warping function in special cases, but in more general cases this description is insufficient.The paper also considers the relationships between warping functions and the torsional and distortional angles in thin-walled beam analysis.Finally, in order to demonstrate the essential features of the paper, a numerical example is solved for a thin-walled beam. Both open and closed cross-section profiles are investigated.
Article
The purpose of this paper is to present a formulation of a curved thin-walled box beam finite element having a variable cross-section with at least one vertical axis of symmetry. To allow for the effects of warping and distortion, three degrees of freedom have been included in the formulation. These degrees of freedom have been designated as the rate of change of twisting angle, the distortional angle of the cross-section, and the rate of change of distortional angle. The effect of shear lag has also been included. The element may be used for the elastic analysis of a variety of thin-walled structures and in particular for the preliminary analysis of box bridge decks where a three-dimensional analysis may be unnecessary. The accuracy of the element has been demonstrated by comparison of the results obtained with known results from other methods for some examples.
Article
Diaphragms in box girder bridges are implemented primarily to prevent premature excessive distortional deformation under torsional loading condition. Distortional warping and transverse bending stresses, which are the major stress components resulting from distortion, should be appropriately limited to a specific level for efficient use of the cross-section by installing adequate intermediate diaphragms. The objectives of the present study are to develop a thin-walled box beam finite element and to propose tentative design charts for adequate spacing of intermediate diaphragms. The developed beam element possesses nine degrees of freedom per node and the validity was intensively verified from a series of comparative studies using a conventional shell element. Also, performed herein are extensive parametric studies for continuous box girder bridges of doubly symmetric steel box section. The design parameters taken into account were the desired ratio of the distortional warping normal stress to the bending normal stress, the number of spans, the span length, the aspect ratio of the box section, and the spacing of the intermediate diaphragms. The results were summarized into tentative design charts indicating efficient spacing of intermediate diaphragms for the various stress ratios.
Article
Free rigid body modes in Neumann problems are typically eliminated by suitably restraining the body. An alternative approach, here called “regularization”, involves first computing the singular stiffness matrix and then suitably modifying it using ideas from linear algebra. This idea has been suggested by Vêrchery (1990) for symmetric matrices. This paper is concerned with regularization of nonsymmetric stiffness matrices that arise from the boundary element method (BEM) for linear elasticity. Existence and uniqueness issues, as well as properties of the displacement field, for elasticity problems with tractions prescribed at every point on the boundary, are discussed in this paper.
Article
This paper addresses the formulation and validation of GBT-based beam finite elements, intended to analyse the physically non-linear (plastic zone) behaviour of thin-walled metal members. Both stress-based and stress resultant-based elastoplastic formulations are developed. The stress-based formulation is generally more accurate, but the stress resultant-based formulation, which employs the Ilyushin yield function, leads to significant computational savings, namely (i) numeric integration in the through-thickness direction is not required and (ii) constraints to the stress resultant and work-conjugate strain field, typical of linear elastic GBT-type formulations, are straightforwardly enforced. The choice of interpolation functions and the cross-section discretization procedure are also discussed. In order to illustrate the application, provide validation and demonstrate the capabilities of the proposed finite elements, several numerical results are presented and discussed. These results are compared with those obtained with standard 2D-solid and shell finite element analyses.
Article
The paper addresses the elastic post-buckling behaviours of cold-formed steel lipped channel simply supported columns affected by mode interaction phenomena involving distortional buckling, namely local/distortional, distortional/global (flexural-torsional) and local/distortional/global mode interaction. The results presented were obtained by means of Abaqus shell finite element analyses adopting column discretisations into fine 4-node element meshes. In order to enable a thorough assessment of all possible mode interaction effects, the column lengths and cross-section dimensions were carefully selected to ensure similar local, distortional and/or global buckling loads. One analyses otherwise identical (elastic) columns having initial geometrical imperfections (i) with various configurations (combinations of the competing critical buckling mode shapes) and (ii) sharing the same overall amplitude.
Article
This two-part-paper proposes a higher order composite beam theory that can be viewed as an extension of Saint-Venant’s theory. Saint-Venant’s solution, is known to represent the exact 3D solution in the interior part of a beam, far from the end-sections where the boundary conditions are applied. The difference between these solutions contains the end-effects. The objective of the proposed theory is to capture a significant part of these end-effects in order to predict the 3D-stresses in a larger interior-part of the beam, and therefore better describe its structural behavior. Based on a kinematics built from the exact form of Saint-Venant displacement, the present theory is rigorously derived for the case of symmetric cross section made of orthotropic materials. Closed-form expressions are obtained for the stiffness matrix of the structural behavior and for the 3D-stresses. Easy to compare to those of Saint-Venant, these results highlight the contribution of this approach. Part-I is devoted to the theoretical developments and part-II illustrates the predictive capability of this theory through the analysis of tip loaded cantilever beams, focusing the built-in effects influence on the structural behavior of the beam.
Article
The equations of conventional thin plate theory are used to formulate an eigenvalue problem for effects of self-equilibrating end loads in thin-walled rectangular cross section tubes, or box beams. The problem is analyzed by a perturbation procedure, which is based on a small parameter proportional to the square root of the ratio of wall thickness to cross section width. Solution of the unperturbed problem yields a family of membrane and inextensional end-effect eigenfunctions which are found to have decay distances on the order of the beam width or shorter. The perturbation procedure is carried out to obtain closed form asymptotically valid solutions for warping and distortional effects which decay much more slowly.
Article
This Note proposes an extension to composite section of the non-uniform (out-of-plane) warping beam theory recently established for homogeneous and isotropic beam by R. El Fatmi (C. R. Mecanique 335 (2007) 467–474). For the present work, which constitutes a first step of this extension, the cross-section is assumed to be symmetric and made by orthotropic materials; however, Poisson's effects (called here in-plane warping) are also taken into account. Closed form results are given for the structural behavior of the composite beam and for the expressions of the 3D stresses; these ones, easy to compare with 3D Saint Venant stresses, make clear the additional contribution of the new internal forces induced by the non-uniformity of the (in and out) warpings. As first numerical applications, results on torsion and shear-bending of a cantilever sandwich beam are presented.
Article
This paper introduces the second-order terms associated with geometric nonlinearity into the basic equation of Generalised Beam Theory. This gives rise to simple explicit equations for the load to cause buckling in individual modes under either axial load or uniform bending moment. It is then shown how the explicit procedure can be extended to consider the interaction between local, distortional and global buckling modes. More general load cases require the use of numerical methods of analysis and the finite difference method offers a suitable procedure. The success of Generalised Beam Theory for a wide range of situations is demonstrated by comparing the results obtained using it with both test results and other analyses. It is shown that it offers particular advantages in the analysis of buckling problems in cold-formed sections.
Article
A new linear model for beams with compact or thin-walled section is presented. The formulation is based on the Hellinger–Reissner principle with independent descriptions of the stress and displacement fields. The kinematics is constituted by a rigid section motion and non uniform out-of-plane warpings related to shear and torsion. The stress field is built on the basis of the Saint-Venànt (SV) solution and with a new part to describe the variable warping.The formulation of a finite element with exact shape functions made possible to validate the beam model avoiding discretization errors.
Article
Due to the increased consumption of thin-walled structural elements there has been increasing focus and need for more detailed calculations as well as development of new approaches. In this paper a thin-walled beam element including distortion of the cross section is formulated. The formulation is based on a generalized beam theory (GBT), in which the classic Vlasov beam theory for analysis of open and closed thin-walled cross sections is generalized by including distortional displacements. The beam element formulation utilizes a semi-discretization approach in which the cross section is discretized into wall elements and the analytical solutions of the related GBT beam equations are used as displacement functions in the axial direction. Thus the beam element contains the semi-analytical solutions. In three related papers the authors have recently presented the semi-discretization approach and the analytical solution of the beam equations of GBT. In this approach a full set of deformation modes corresponding to the homogeneous GBT equations are found. The deformation modes of which some are complex decouple the state space equations corresponding to the reduced order differential equations of GBT and allow the determination of the analytical solutions. Solutions of the non-homogeneous GBT differential equations and the distortional buckling equations have also been found and analyzed. Thus, these related papers are not dealing with an element but dealing with analytical solutions to the coupled differential equations.
Article
A new approach is illustrated for the cross-sectional analysis to be performed in the context of the Generalised Beam Theory (GBT). The novelty relies in formulating the problem in the spirit of Kantorovich’s semi-variational method, namely using the dynamic modes of an unconstrained planar frame as in-plane deformation modes. Warping is then evaluated from the post-processing of these in-plane modes, thus reversing the strategy of the classical GBT procedure. The new procedure does not require several steps of the classical algorithm for the determination of the conventional modes, in which bending, shear and local modes are evaluated separately, and is applicable indifferently to open, partially-closed and closed sections. The efficiency and ease of use of the method are outlined by means of two examples, aimed to describe the linear–elastic behaviour of thin-walled members.Highlights► New approach for the derivation of the conventional deformation modes for GBT analysis. ► Novelty relies in using the dynamic modes of an unconstrained planar frame. ► Procedure applicable to open, partially-closed and closed sections.
Article
This paper addresses the distortional kinematics and mechanics of thin-walled sections and provides clear definitions of cross-section properties that characterise the distortional deformation, as it is usually done for conventional modes (axial, bending and torsion). In particular, a procedure to build the distortional displacement field of a given thin-walled section is described. The first part of the paper describes the essentials of distortion in comparison with the conventional modes of classical beam theories. It is shown that primary warping is the key factor that controls the distortion of thin-walled sections. Then, an analytical procedure to determine the distortional warping displacement distribution of a given cross-section is described, on the basis of orthogonality conditions existing between the distortional and conventional modes. Next, an overview of the kinematical assumptions underlying the distortional deformation is presented and a simple procedure to build distortional displacement fields of thin-walled sections is provided. This procedure is then applied to obtain the distortional displacement field of C-sections and general expressions of distortional cross-section properties are given. Finally, a simple example is presented to illustrate how the distortional displacement field of a C-section is built, on the basis of simple kinematics principles. The distortional critical stress and half wavelength are determined and good agreement with exact numerical estimates is found.
Article
A new displacement-based finite element is developed for thin-walled box beams. Unlike the existing elements, dealing with either static problems alone or dynamic problems only with the additional consideration of warping, the present element is useful for both static and dynamic analyses with the consideration of coupled deformation of torsion, warping and distortion. We propose to use a statically admissible in-plane displacement field for the element stiffness matrix and a kinematically compatible displacement field for the mass matrix so that the present element is useful for a wide range of beam width-to-height ratios. The axial variation of cross-sectional deformation measures is approximated by C0 continuous interpolation functions. Numerical examples are considered to confirm the validity of the present element. Copyright © 1999 John Wiley & Sons, Ltd.
Article
This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by distortional/global (flexural–torsional) buckling mode interaction. The results presented and discussed were obtained by means of analyses performed using the finite element code Abaqus and adopting column discretisations into fine 4-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections that may warp freely), (ii) have cross-section dimensions and lengths that ensure equal distortional and global (flexural–torsional) critical buckling loads, thus maximising the distortional/global mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combinations of the two “competing” critical buckling modes. After briefly addressing the lipped channel column “pure” distortional and global post-buckling behaviours, one presents and discusses in great detail a fair number of numerical results concerning the post-buckling behaviour and strength of similar columns experiencing strong distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (ii) curves or figures providing the evolution of the deformed configurations of several columns (expressed as linear combination of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii1) the location and growth of the plastic strains and (iii2) the characteristics of the failure mechanisms more often detected in the course of this research work.
Article
The classical Vlasov theory for torsional analysis of thin-walled beams with open and closed cross-sections can be generalized by including distortional displacement fields. We show that the determination of adequate distortional displacement fields for generalized beam theory (GBT) can be found as part of a semi-discretization process. In this process the cross-section is discretized into finite cross-section elements and the axial variation of the displacement functions are solutions to the established coupled fourth order differential equations of GBT. We use a novel finite-element-based displacement approach in combination with a weak formulation of the shear constraints and constrained wall widths. The weak formulation of the shear constraints enables analysis of both open and closed cell cross-sections by allowing constant shear flow. We use variational analysis to establish and clearly identify the homogeneous differential equations, the eigenmodes, and the related homogeneous solutions. The distortional equations are solved by reduction of order and solution of the related eigenvalue problem of double size as in non-proportionally damped structural dynamic analysis. The full homogeneous solution is given as well as transformations between different degree of freedom spaces. This new approach is a considerable theoretical improvement, since the obtained GBT equations found by discretization of the cross-section are now solved analytically and the formulation is valid without special attention also for closed single or multi-cell cross-sections. Further more the found eigenvalues have clear mechanical meaning, since they represent the attenuation of the distortional eigenmodes and may be used in the automatic meshing of approximate distortional beam elements. The magnitude of the eigenvalues thus also gives the natural ordering of the modes.
Article
 In this paper a solution to the problem of plates reinforced with beams is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems which are nonlinearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate–beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. The resulting deflections are considerably smaller than those obtained by other models.
Article
A formulation of generalised beam theory (GBT) developed to analyse the elastic buckling behaviour of circular hollow section (CHS) members (cylinders and tubes) is presented in this paper. The main concepts involved in the available GBT are adapted to account for the specific aspects related to cross-section geometry. Taking into consideration the kinematic relations used in the theory of thin shells, the variation of the strain energy is evaluated and the terms are physically interpreted, i.e., they are associated with the geometric properties of the CHS. Besides the set of shell-type deformation modes, the formulation also includes axisymmetric and torsion deformation modes. In order to illustrate the application and capabilities of the formulated GBT, the local and global buckling behaviour of CHS members subjected to (i) compression (columns), (ii) bending (beams), (iii) compression and bending (beam-columns) and (iv) torsion (shafts), is analysed. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed.
Article
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.
Article
This paper presents an expanded method for exact distortional behavior of multicell box girders subjected to an eccentric loading. This method decomposes the eccentric loading into flexural, torsional and distortional forces by using the force equilibrium. From the force decomposition, the complex behavior of the multicell box girders can be decomposed and in turn the distortional behavior can be considered independently. Based on the method, a thin-walled box beam finite element, which can be applied to practical distortional analysis of straight multicell box girder bridges, is also developed in this study. The present box beam element possesses nine degrees of freedom per node to consider each separate behavior of multicell box girders. The validation of the present box beam element is demonstrated through a series of comparative studies using a conventional shell element and a box beam element proposed by other researchers. From the independent consideration of exact distortional behavior, the present box beam element unlike shell element will be useful for practicing engineers to easily determine distortional bimoments and stresses of multicell box girders.
Article
Previous papers (Refs 1–4) have presented a method of analysis for any open unbranched thin walled section considering both rigid body movement and cross section distortion (including local buckling). Reference 1 described how the Generalized Beam Theory (GBT) can be used to calculate generalized section properties for all modes, including each of the four rigid body modes and the distortional modes. The additional section properties evolved from GBT were then used in Ref. 2 to consider second order elastic critical buckling problems.This paper compares the critical buckling predictions of GBT with the results obtained in two series of tests carried out on lipped and unlipped channels subject to a major axis bending moment. These predictions are then combined with the yield criteria of EC3 to allow a comparison with the analysis of these tests carried out by Lindner and Aschinger (Ref. 5). The paper concludes that the Generalized Theory is a powerful and effective analysis tool for the solution of interactive buckling problems where both local and overall buckling can occur.
Article
This paper presents the formulation of a Generalised Beam Theory (GBT) developed to analyse the structural behaviour of composite thin-walled members made of laminated plates and displaying arbitrary orthotropy. The main concepts and procedures involved in the available isotropic first-order GBT are revisited and adapted/modified to account for the specific aspects related to the member orthotropy. In particular, the orthotropic GBT fundamental equilibrium equations and corresponding boundary conditions are derived and their terms are physically interpreted, i.e., associated with the member mechanical properties. Moreover, different laminated plate material behaviours are dealt with and their influence on the GBT equations is investigated. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, a thin-walled orthotropic beam is analysed and the results obtained are thoroughly discussed.
Article
Torsional warping and distortion of the cross-section are important features of thin-walled beams and must be considered fully in the design of box girder bridges. A method of elastic analysis is developed based on the stiffness approach which includes the effects of warping torsion and distortion in addition to the more familiar actions of bending moment and torsion. The method, which is applicable to straight single cell box girders with at least one axis of symmetry, is demonstrated here in the analysis of three different box girder models for which experimental or analytical results were already available. The method is shown to be easy and economical to use and provides a physical insight into the structural response of thin-walled box girder bridges under general loading conditions.
Article
This paper presents the formulation and illustrates the application of a non-linear Generalised Beam Theory (GBT), which makes it possible to calculate bifurcation load factors of thin-walled members made of non-linear elastic–plastic materials and subjected to general loading conditions. This formulation is an extension of the non-linear GBT recently derived by the authors, which was valid for uniformly compressed members only. In order to illustrate the application of the proposed non-linear GBT, a beam finite element is formulated and some numerical results are presented and discussed. These results concern rectangular plates and thin-walled hat-section beams. The material instantaneous elastic moduli are determined on the basis of both the well known J2-flow and J2-deformation small strain plasticity theories.
Article
This paper presents a new formulation for thin-walled beams that includes cross-section deformation. The kinematic description of the beam emanates from the geometrically exact Reissner–Simo beam theory and is enriched with arbitrary cross-section deformation modes complying with Kirchhoff’s assumption. The inclusion of these deformation modes makes it possible to capture the cross-section in-plane distortion, wall (plate) transverse bending and out-of-plane (warping), which leads to a computationally efficient numerical implementation. Several illustrative numerical examples are presented and discussed, showing that the resulting beam finite element leads to solutions that are in very good agreement with those obtained with standard shell finite elements, albeit involving much less degrees-of-freedom.
Article
A geometrically exact and completely consistent finite element theory for curved and twisted beams is proposed. It is based on the kinematical hypothesis generally formulated for large deformation and accounts for all kinds of deformation of a three-dimensional solid: translational and rotational displacements of the cross-sections, warping of their plane and distortion of their contours. The principle of virtual work is applied in a straightforward manner to all non-zero six components of the strain and stress tensors. Expressions are given for tangent matrices of elastic, inertia and external forces and specific techniques for discretization and updating are developed for the analysis of beams in inertial and non-inertial frames. Finally, the numerical properties of the finite element models are demonstrated through examples.
Article
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.