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In this paper, the accurate free vibration characteristics of thin-walled box beam are discussed and the kinematic model is established by using fine shear deformation theory. The model adopts the in-plane and out-of-plane displacement fields including extension, torsion, warping and distortion, as well as the transverse shear due to bending and warping due to torsion. One-dimensional high-order beam theory is applied to the dynamic solution of thin-walled box beam, and the analytical solution of high free vibration modes of multi-deformation coupled modes under different boundary conditions is derived. The results show that warping, distortion and shear deformation play an important role in the free vibration characteristics of thin-walled box beam, and the validity of the model is verified. In addition, finite element software (ANSYS) is used for finite element simulation. The application of vibration mode in the structure design of thin-walled box beam is summarized, especially in the case of higher natural frequency. The calculation method is in good agreement with the finite element results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis of the pure torsion and distortion of straight box beams with trapezial cross-sections and hinged walls. A one-dimensional mechanical model for this kind of system subjected to anti-symmetric loads on the end cross-sections and no warping constraints is developed. The distortional stiffness of the system is provided by the torsional rigidity of the wall panels. The cross-sectional kinematic condition for which torsion and distortion are uncoupled has been determined. Novel explicit expressions of the internal and external distortional moments, the distortion constant, and the distortional warping pattern have been deduced; they can be directly translated to the classical distortion theory. Results of representative test cases with different section shapes and loads show excellent agreement with finite element models using shell elements. The model is a first step to analyse bridge decks with a distortionable central cell for wind engineering applications. Finally, an extension of the model, including the distortional stiffness provided by the frame bending stiffness of the cross-section walls, is presented. The extended model is applicable to assess the large-scale torsional–distortional effects in long beams with closed cross sections. [Display omitted] • 1D model for pure torsion and distortion of trapezial box beams with hinged walls. • The kinematic reference for uncoupling torsion and distortion has been determined. • Explicit formulas for distortional moment, distortion constant and warping pattern. [ABSTRACT FROM AUTHOR]

In this research effort, the generalized warping and distortional problem of straight or horizontally curved composite beams of arbitrary cross section, loading and boundary conditions is presented. An inclined plane of curvature is considered. Additionally, the stiffness of diaphragmatic plates has been introduced in the formulation in order to compare with the case where rigid diaphragms are assumed. Isogeometric tools (NURBS) are employed in order to obtain the results for the 1D formulation and 3D shell models are developed in FEM commercial software for composite cross sections with diaphragms. The number of intermediate diaphragms according to bridges design specifications is compared to the analyzed diaphragmatic arrangements in order to assess the overall structural behavior of bridges decks. For this purpose, examples of curved beam models with open or closed cross sections and various arrangements of diaphragms have been studied.

Viewing various stereo images under different viewing conditions has escalated the need for effective object-level remapping techniques. In this paper, we propose a new object spatial mapping scheme, which adjusts the depth and size of the selected object to match user preference and viewing conditions. Existing warping-based methods often distort the shape of important objects or cannot faithfully adjust the depth/size of the selected object due to improper warping such as local rotations. In this paper, by explicitly reducing the transformation freedom degree of warping, we propose an optimization model based on axis-aligned warping for object spatial remapping. The proposed axis-aligned warping based optimization model can simultaneously adjust the depths and sizes of selected objects to their target values without introducing severe shape distortions. Moreover, we propose object consistency constraints to ensure the size/shape of parts inside a selected object to be consistently adjusted. Such constraints improve the size/shape adjustment performance while remaining robust to some extent to incomplete object extraction. Experimental results demonstrate that the proposed method achieves high flexibility and effectiveness in adjusting the size and depth of objects compared with existing methods. [ABSTRACT FROM AUTHOR]

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.

Abstract In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method. Highlights • Axial warping and distortional effects, on top of shear, flexural, torsional ones, via sequential equilibrium scheme and BEM. • Higher-order beam theory taking into account Poisson's ratio - advantages over 3-D solid or shell solutions. • General cross-section and boundary conditions. • Axial mode is evaluated - essential for future buckling analysis (axial mechanism is the most important one in buckling). • Results as accurate as those of Solid FEM are obtained, with order of magnitude less number of degrees of freedom. [ABSTRACT FROM AUTHOR]

Highlights • The free vibration computation is performed using a (1D) refined beam theory (RBT). • RBT displacement model is free from classical beam assumptions and can deal with any cross-section. • RBT displacement model includes the main own sectional displacement modes of the cross-section. • RBT comes with a friendly Matlab numerical tool (CSB) to quickly analyze the beam dynamic behavior. • Results are compared to 3D-FEM computations for a significant set of beam configurations. Abstract Free vibration problem of an arbitrary cross-sectional homogeneous or composite beam is investigated using a refined 1D beam theory (RBT). This theory includes a set of 3D displacement modes of the cross-section (CS) which reflects its mechanical behavior: the main part of these sectional modes is extracted from the 3D Saint Venant's solution and another part is related to the CS dynamic behavior. These sectional modes, which are first derived from a CS analysis, lead to a consistent 1D beam model which really fits the section nature (shape and materials), and hence the beam problem. The numerical strategy to apply such general approach, is based on a first set of CS problems solved by 2D-FEM computations to get the sectional modes, and then the dynamic beam problem is solved by 1D-FEM computation according to RBT displacement model to provide (in fine) the first natural frequencies and 3D vibration mode shapes of the beam. To do so and in order to easily apply such method, a user friendly Matlab numerical tool named CSB (Cross-Section and Beam analysis) has been developed. To illustrate the capabilities and the accuracy of the method to catch the main 3D-effects, such as elastic/inertial coupling effects and 3D local/global mode shapes, a significant set of beam cross-section configurations with isotropic and anisotropic materials are analyzed. The first ten natural frequencies and 3D mode shapes are systematically compared to those obtained by full 3D-FEM computations, and some of them to literature. [ABSTRACT FROM AUTHOR]

Linear buckling of homogeneous or composite I-beams are investigated using a refined beam theory built on 3D Saint-Venant's solution. The kinematic model includes sectional Poisson's effects and out-of plane warpings extracted from 3D Saint-Venant's solution and distortions related to the free vibrational behaviour of the cross-section. Available for an arbitrary cross-section and free from the classical beam and shell-like assumptions, such displacement model leads to a consistent 1D beam theory which really matches the cross-section nature (shape and materials). Moreover, in order to easily apply this method, a user friendly numerical tool has been developed. To show the capabilities and the accuracy of the proposed model to deal with thin-to-thick walled beams, homogeneous and laminated I-beam configurations subjected to axial or transversal loads are performed. The numerical results are compared to literature and some of them to 3D-FEM solid model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A higher-order beam theory is proposed for the analysis of a thin-walled beam with a generally shaped cross section, which consists of straight cross-section edges and is non-uniform along the axial direction. To derive cross-sectional shape functions for the higher-order deformation modes, a new approach is introduced using a set of beam frame models. The distortions with inextensional cross-sectional walls are determined by solving an eigenvalue problem of a beam frame model under inextensional wall constraints. Subsequently, the distortions with extensional cross-sectional walls are evaluated by considering orthogonality with respect to the inextensional distortions. Moreover, the extensional distortions due to the Poisson effect, which is generated due to the uniform axial strain of the rigid-body cross-sectional deformations, are considered. Warpings induced by the inextensional and extensional distortions are consistently defined based on the orders of the tangential displacements of their corresponding distortions. To deal with the varying cross section, three-dimensional displacements at an arbitrary point are interpolated using those at the cross sections of the nodes, where the beam frame analyses are performed. The proposed method is validated by performing static and vibration analyses of beams with varying single- and multi-cell cross sections. [ABSTRACT FROM AUTHOR]

Practical design of bridges and other structures requires the use of quick and simple calculation methods, rather than the use of tridimensional models using shell or solid finite elements. These methods have to be used for a general loading state, taking into account the different structural mechanisms, and generating the results required to apply the verifications of the structural codes and to understand the structural behavior. In this work, a beam-type element is proposed to adress these objetives for the case of thin walled sections. This element has three nodes, with five-degrees of freedom per node, more than the six degrees of a conventional 3D beam, incorporates the effects of shear lag, torsion and distortion homogeneous and non homogeneous in the distribution of normal stress. Various examples were tested to verify the validity of the beam element according different calculation methods. [ABSTRACT FROM AUTHOR]

• A new modeling method for a welded multicell thin-walled beam was presented based on a higher-order beam theory. • Partial welding lines in the axial direction of a beam can be accurately simulated by adopting constraint equations among sectional modes. • The optimized locations of welding lines can be found by using a genetic algorithm (GA). • The large number of analyses required for GA is not problematic due to the fast analysis of the proposed beam model. • Optimized welding lines are found near beam ends and joint regions because of large reaction forces and end effects. • Optimized welding lines on the top and bottom surfaces of a multicell beam do not generally overlap by complementing each other. • The proposed method is very helpful in reducing development time for new vehicle structures consisting of many multicell thin-walled beams. Thin-walled multicell structures observed in vehicle frames are partially welded to lower the welding process cost while maintaining their overall structural stiffness comparable to those of fully-welded multicell structures. A fast and reliable beam-based model for analyzing such structures is required in the concept stage of the design process. In this investigation, an analysis method for a thin-walled beam of a multi-cell cross-section partially welded along its axial direction is newly established using a higher-order beam theory (HoBT). Our approach uses the Lagrange multipliers to impose the partial welding condition along the common edges of multiple thin-walled closed sections using the three-dimensional field derived from a higher-order beam theory. The developed method in this study can be an accurate and effective alternative without full finite element analysis employing shell and solid elements. To find the optimal welding locations, we introduced binary design variables parameterized through their corresponding welding constraints to identify the status of welding or non-welding along a candidate welding line. A genetic algorithm is adopted to solve optimization problems. After establishing the analysis method and optimization technique, their validity was checked using double-cell and triple-cell sectioned beams and beam frame structures, including a bus structure. The stiffnesses of the structures with optimized partial welding lines (20%) drop less than 6% from those of fully welded counterparts. [ABSTRACT FROM AUTHOR]

A higher order beam model for the buckling analysis of thin-walled structures is presented in this paper.The model considers an enrichment of the displacement field so as to accurately represent the three-dimensional behaviour of thin-walled structures. The definition of an uncoupled set of deformation modes allows a meaningful definition of hierarchical higher order solutions, which are useful for the linear buckling analysis of thin-walled structures. A criterion for the definition of local and global buckling modes, as well as possible interaction between modes is put forward. A comparison between the results obtained with the higher order beam model and results obtained from a shell finite element model implemented in Abaqus allows to conclude not only the efficiency of the beam model but also its simplicity of use. [ABSTRACT FROM AUTHOR]

A thin-walled beam finite element with a varying quadrilateral cross section is formulated based on a higher order beam theory. For the calculation of distortions, the beam frame approach, which models the cross section by using two-dimensional Euler beams, is used. Distortions induced by the Poisson’s effect and warpings are analytically derived. Three-dimensional displacements at an arbitrary point of a present beam element can be described by interpolating three-dimensional displacements at the end sections. Straight and curved thin-walled beams with varying cross sections are solved to show the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents a one-dimensional finite element analysis of tapered thin-walled box beams under out-of-plane loads and twisting moments by developing new C 0 -continuous tapered higher-order beam elements. While higher-order sectional deformations such as distortion were considered earlier, there was no theoretical or finite element analysis on the tapering effect of variable cross-sectioned box beams. Case studies considering tapered box beams with various height-to-width ratios and a system of joint structure with a tapered box beam show that the results by the developed method are comparable with the shell finite element results unlike the stepped Timoshenko beam element results. [ABSTRACT FROM AUTHOR]

Two finite-element formulations are developed for the general distortional analysis of beams with monosymmetric sections. In the first formulation, cubic and linear Hermitian polynomials are adopted to interpolate the nodal displacements; whereas in the second formulation, shape functions that exactly satisfy the governing field equations were used. Because the distortional lateral-torsional and the longitudinal-transverse responses are fully uncoupled, separate finite elements were developed for both types of behaviors. A comparison with other finite-element solutions and a recently developed distortional theory established the validity of the present formulations. A study was then performed on the stability and convergence characteristics of both elements. The new elements were then adopted to solve linearly static analysis of simple beams and beams with overhangs. The formulation is shown to reliably capture the difference in behavior between stiffened and unstiffened beams. [ABSTRACT FROM AUTHOR]

A distortional theory is developed for the analysis of doubly symmetric and mono-symmetric wide flange beams under general loading. The governing differential equations of equilibrium and associated boundary conditions are derived based on the principle of potential energy. The theory captures shear deformation effects in the web and local and global warping effects. In contrast to classical beam theories, the present study captures web distortion by accounting for its flexibility within the plane of the cross-section while considering the flanges as Euler-Bernoulli beams. The formulation yields two systems of coupled differential equations of equilibrium in seven displacements fields. The first system governs the longitudinal transverse response and involves three displacement fields, and the second system governs the lateral torsional response and involves four displacement fields. Closed form solutions are then developed for both coupled systems under general loading. Numerical solutions for practical problems are then provided to illustrate the applicability of the formulation. Comparisons to results based on 3D shell finite element solutions show the validity of the results. The theory preserves the relative simplicity of one dimensional beam theories while effectively capturing the three-dimensional distortional phenomena normally captured within computationally expensive 3D FEA. [ABSTRACT FROM AUTHOR]

Beam-type elements based on the theories of Euler–Bernoulli, Timoshenko, and Vlasov are widely used in civil engineering. However, shell and solid finite elements are often used when the effects on normal stresses of either shear deformation or distortion are considered important. Numerically validated in an earlier study with finite element models for shell-type structures, the same one-dimensional finite element model is further developed in this study with a low number of degrees of freedom per node that includes all the structural mechanisms without using 3D finite element models. Laboratory testing of an instrumented steel box girder is conducted, to improve validation of the goodness of fit of the finite element model with real structural behavior. • Calculation of the resistant modes of sections (shear, torsion and distortion). • Improvement of the description of the new beam element by applying energetic methods. • Laboratory testing of an instrumented steel box girder. • Validation of fitting with a one-dimensional finite element of the real behavior. [ABSTRACT FROM AUTHOR]

Shear lag effect and torsional and distortional warping can significantly affect the performance of thin-walled curved box girders used in modern bridge engineering. The structural behaviour of these bridges exhibits complexity due to coupled bending and torsion together with warping effects of non-uniform torsion, distortion and shear lag. A practical method of analysis, based on the symplectic approach, with the same perspective as the Higher Order Beam Theories, is presented for overcoming the difficulties of numerical approaches via the Finite Element Method. In this paper, the Hamiltonian Structural Analysis method implements the analysis of the shear lag effect together with non-uniform torsion and distortion, for curved box girder bridges. The examples given can allow engineers to evaluate the effects of the negative shear lag phenomenon and the distribution of axial stresses in box slabs and webs, due to different warping effects, which strongly influence thin-walled box sections subjected to point loads and restrained areas. • The analysis of thin-walled curved beams is carried out via symplectic approach. • The contribution to axial stress of torsion, distortion and shear lag is showed. • A method of analysis for curved bridges with box or T sections is developed. • Positive and negative shear lag effects are investigated. • Applications to straight and curved girders are presented. [ABSTRACT FROM AUTHOR]

Highlights: [•] A higher order model for thin-walled structures with a general and deformable section is proposed. [•] The model relies on deformation modes obtained by a quartic eigenvalue problem. [•] The set of non-null eigenvalues define the higher order deformation modes. [•] The cross-section warping and distortion modes are consistently derived. [•] An hierarchic selection of modes is defined. [ABSTRACT FROM AUTHOR]

Highlights: [•] A new beam element with an enriched kinematics is derived. [•] A full description for the method to determine warping and distortion modes is given. [•] An exact solution of the equilibrium equation is performed. [ABSTRACT FROM AUTHOR]

Abstract: 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. To handle arbitrary boundary conditions as well as the possibility of adding concentrated loads as nodal loads the formulation of a beam element is needed. This paper presents the formulation of such a generalized one-dimensional semi-discretized thin-walled beam element including distortional contributions. It should be noticed that we are only dealing with a basic generalized beam theory and not an extended finite element formulation of an approximate beam element, which allows the addition of special (transverse extension and shear lag) modes. Illustrative examples showing the validity and the accuracy of the developed distortional semi-discretized thin-walled beam element are given and it is shown how the novel approach provides accurate results making it a good alternative to the traditional and time consuming FE calculations. [Copyright &y& Elsevier]

This investigation presents one-dimensional static and eigenvalue analyses of thin-walled straight beams with generally shaped closed single-cell or multicell sections. For accurate beam analysis, sectional warping and distortional deformations should be considered in addition to the standard Timoshenko displacement field, but it is difficult to obtain the deformation functions analytically for arbitrarily shaped sections. Thus, a numerical method is proposed to obtain sectional deformations for any arbitrarily shaped sections. Once the deformations are identified, they can be integrated over a cross section to yield one-dimensional higher order beam equations. For the numerical determination, the cross section of a thin-walled beam is modeled as a beam frame, where the warping and distortional deformation functions of the section are identified as the eigenmodes of the frame model; the lowest few energy mode sets of in-planar and out-of-planar modes are selected as the distortional and warping deformation functions, respectively. The validity of this approach is checked by comparing the present results with shell finite-element results. For numerical tests, several thin-walled closed sections, including those with flanges or varying wall thicknesses, are considered. The effect of the number of selected warping and distortion sets on solution convergence is also investigated. [ABSTRACT FROM AUTHOR]

To take into account the flexibility resulting from sectional deformations of a thin-walled box beam, higher-order beam theories considering warping and distortional degrees of freedom (DOF) in addition to the Timoshenko kinematic degrees have been developed. The objective of this study is to derive the exact matching condition consistent with a 5-DOF higher-order beam theory at a joint of thin-walled box beams under out-of-plane bending and torsion. Here we use bending deflection, bending!shear rotation, torsional rotation, warping, and distortion as the kinematic variables. Because the theory involves warping and distortion that do not produce any force!moment resultant, the joint matching condition cannot be obtained just by using the typical three equilibrium conditions. This difficulty poses considerable challenges because all elements of the 5x5 transformation matrix relating the field variables of one beam to those in another beam should be determined. The main contributions of the investigation are to propose additional necessary conditions to determine the matrix and to derive it exactly. The validity of the derived joint matching transformation matrix is demonstrated by showing good agreement between the shell finite element results and those obtained by the present box beam analysis in various angle box beams. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents distorting buckling solutions for semi-discretized thin-walled columns using the coupled differential equations of a generalized beam theory (GBT). In two related papers recently published by the authors a novel semi-discretization approach to GBT has been presented. The cross section is discretized and analytical solutions are sought for the variation along the beam. With this new approach the general GBT equations for identification of a full set of deformation modes corresponding to both homogeneous and non-homogenous equations are formulated and solved. Thereby giving the (complex) deformation modes of GBT which decouple the state space equations corresponding to the reduced order differential equations. In this paper the developed semi-discretization approach to generalized beam theory (GBT) is extended to include the geometrical stiffness terms, which are needed for column buckling analysis and identification of buckling modes. The extension is based on an initial stress approach by addition of the related potential energy terms. The potential energy of a single deformation mode is formulated based on a discretization of the cross section. Through variations in the potential energy and the introduction of the constraints related to beam theory this leads to a modified set of coupled homogeneous differential equations of GBT with initial stress for identification of distortional displacement modes. In this paper we seek instability solutions using these GBT initial stress equations for simply supported columns with constrained transverse displacements at the end sections and a constant axial initial stress. Based on the known boundary conditions the reduced order differential equations are solved by using the trigonometric solution functions and solving the related eigenvalue problem. This gives the buckling mode shapes and the associated eigenvalues corresponding to the bifurcation load factors. Thus the buckling modes are found directly by the analytical solution of the coupled GBT-equations without modal decomposition. Illustrative examples showing global column buckling, distortional buckling and local buckling are given and it is shown how the novel approach may be used to develop signature curves and elastic buckling curves. In order to assess the accuracy of the method some of the results are compared to results found using the commercial FE program Abaqus as well as the conventional GBT and FSM methods using the software packages GBTUL and CUFSM. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Very large container ships are rather flexible due to the large deck openings. Therefore, hydroelastic stress analysis is required as a basis for a reliable structural design. In the early design stage, the coupling of the beam model with a 3D hydrodynamic model is rational and preferable. The calculation is performed using the modal superposition method, so natural hull modes have to be determined in a reliable way. Therefore, the advanced thin-walled girder theory, which takes the influence of shear on both bending and torsion into account, is applied for calculating the hull flexural and torsional stiffness properties. A characteristic of very large container ships is the quite short engine room, whose closed structure behaves as an open hold structure with increased torsional stiffness due to the deck effect. The paper deals with the calculation of its effective torsional stiffness parameters by utilizing the energy balance approach. Also, estimation of distortion of transverse bulkheads, as a result of torsion and warping, is given. The procedure is checked by the 3D FEM analysis of a ship-like pontoon. Such a modified beam model of the engine room structure can be included in the general beam model of a ship hull for the need of hydroelastic analysis, where only a few first natural frequencies and mode shapes are required. For practical use in the preliminary design stage of ship structures, the simplicity of the beam model presents an advantage over 3D FEM models. [ABSTRACT FROM AUTHOR]

Abstract: By using existing beam theories alone, it is not possible to correctly predict the structural behavior of thin-walled closed beams connected at an angled joint that is under in-plane bending. Sectional deformations occurring near a joint cannot be estimated by using the Euler/Timoshenko beam elements, but they significantly increase the overall flexibility of a beam structure. Following a recent development of a higher-order beam analysis for joined box beams at an arbitrary angle under out-of-plane bending and torsion, a higher-order analysis suitable for box beams connected at an angled joint under in-plane bending is newly developed and implemented in finite element formulation. For the analysis, two sectional deformation degrees of freedom accounting for bending distortion and warping are included in addition to the standard degrees of freedom of extension, bending deflection and bending/shear rotation. The employed sectional deformation function for bending distortion is the same as that for a curved box beam, but the function for bending warping is newly developed. Then, a procedure to match the five one-dimensional field variables from extension to bending warping at an angled joint is presented. The validity and effectiveness of the developed analysis are confirmed by comparing the present beam results with those obtained by shell finite elements. [Copyright &y& Elsevier]

Higher-order sectional modes of a thin-walled beam such as distortion and warping significantly affect structural stiffness levels. Because this higher-order effect becomes even greater near the joints of a beam frame structure, finding the correct connection conditions of sectional modes at beam joints is crucial for an accurate analysis. For conventional beam elements based on the Euler-Bernoulli/Timoshenko beam theory, the joint connection conditions are obtained by component-wise matching of the force and moment vectors at the joint node. However, this simple approach is no longer valid for the joints of higher-order beam elements because warping and distortion modes have zero force/moment resultants on the beam cross-section and therefore cannot be considered through the equilibrium condition of their resultants. In this investigation, three-dimensional displacements and rotation angles are set to be continuous at the connection points on a so-called joint section, which is defined as a virtual plane shared by joining beams. We propose to comprise the connection points using the vertices of the joint section and intersection points on the joint axis and impose the continuity conditions at these points using Lagrange multipliers. The proposed joint connection conditions can be applied to a beam frame structure with general section shapes without requiring any geometry-dependent conditions as done in earlier studies. The validity of the proposed method is demonstrated by conducting static and vibration analyses of two-beam joint structures, a T-joint structure, and a vehicle frame structure. [Display omitted] [ABSTRACT FROM AUTHOR]

If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with numerical examples. [ABSTRACT FROM AUTHOR]

• A new thin-walled beam-joint modeling method is proposed to deal with the joint flexibility effect of a car body. • A thin layer element is employed to absorb the strain energy at a beam-joint interface. • A surrogate model is built to approximate the absorption parameter of a thin layer element. • Sectional constants for torsion, bending and distortion are adopted for input parameters of a surrogate model. • The proposed beam-based model enables a fast structural optimization as well as a fast analysis at the concept design phase. A new joint modeling method for a beam-based analysis of a car body applied during the concept design phase is presented. The body-in-white (BIW) of a car mainly consists of thin-walled beams with closed cross-sections, the joints of which show a significant flexibility owing to higher-order deformation modes. The joint flexibility effect cannot be evaluated using conventional beam theories. Although a hybrid modeling method using shell elements for a joint region, condensed in the form of a superelement, can be employed, its increase in accuracy for the joint flexibility is marginal. This is because the interface of the shell-modeled joint and that of conventional beam elements is rigidly connected, and thus the flexibility of the shell elements of the joint disappears at the interface. In this investigation, a thin elastic layer is introduced to account for a flexible connection between the interfaces of the shell-modeled joint and those of the beam elements. By employing an absorption parameter, the stiffness of the thin layer is set such that the thin layer can represent not only the accurate flexibility of the shell-modeled joint, but also higher-order deformations such as distortion and warping of thin-walled beams that are connected to the joint. To determine the absorption parameter of a thin layer, a surrogate model is built with respect to the geometric parameters of the cross-section of a connected beam, joint angle, and wall thickness. The effectiveness of the proposed beam-joint modeling method is shown by solving two-beam joint problems with various cross-sections and the BIW of a passenger car. [ABSTRACT FROM AUTHOR]

• Develops a recursive analysis method to derive hierarchical sectional modes. • Analytically derives the sectional mode shapes in closed form. • Establishes explicit stress-generalized force relations using the recursive method. • Shows these relations are consistent with those by the Timoshenko beam theory. • Found that rapid stress variations are accurately predicted by the proposed theory. 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. [ABSTRACT FROM AUTHOR]

A new systematic approach is presented to derive the cross-section deformation modes of thin-walled beams with arbitrary sections within the framework of a higher-order beam theory (HoBT). New sets of higher-order modes, in this case warping and distortion, are derived hierarchically from the lowest mode set by considering the consistency between the strain field and the stress field generated by the modes in lower sets. Warping modes are derived by the shear stress of in-plane modes while distortion modes are induced by out-of-plane deformations via Poisson's effect. Higher-order modes are shown to be built as a linear combination of the integrated functions of lower-order modes, where the combination coefficients are determined by the orthogonality condition among the higher-order modes. Because the proposed method does not require any approximation when determining sectional mode shapes, no cross-section discretization, commonly used in existing studies, is needed. The effectiveness of the proposed mode derivation process is verified by comparing the static and modal analysis results of thin-walled beams with open, closed, and flanged cross-sections obtained by the proposed method, other HoBTs, and shell finite elements. • A method for deriving cross-section modes is presented based on a higher-order beam theory. • The cross-section analysis is conducted recursively and hierarchically. • No sectional discretization is used for the cross-section analysis. • The derived modes are decoupled from each other due to the orthogonality condition. • Rapidly changing responses by the end effect can be captured by the proposed beam theory. [ABSTRACT FROM AUTHOR]

• newly develops a higher-order beam theory for torsion consistent with the Vlasov torsion theory. • includes hierarchical N (N ≥ 1) sets of higher-order torsion modes together with the Vlasov modes. • establishes explicit F - U and σ - F relations (U : kinematic variables, F : generalized forces, σ : stresses). • shows that the explicit relations are fully consistent with those by the Vlasov torsion theory. • found that stresses calculated directly using the explicit σ - F relation are sufficiently accurate. 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. [Display omitted] [ABSTRACT FROM AUTHOR]

• Formulation for the generalized warping and distortional analysis of composite curved beams with intermediate diaphragms. • Spatial curved beams under general loading and boundary conditions. • Finite Element models with shell elements for bridge decks with composite open and closed cross sections usually employed in practice. • Influence of the diaphragms' plate thickness and their optimum positions. • Assessment of the design guidelines which specify the maximum spacing of intermediate. In this research effort, the generalized warping and distortional problem of straight or horizontally curved composite beams of arbitrary cross section, loading and boundary conditions is presented. An inclined plane of curvature is also considered (with respect to the horizontal plane) in order to account for a slope in the cross-section plane. Additionally, a finite stiffness of diaphragmatic plates has been introduced in the formulation in order to compare with the beam models where rigid diaphragms are considered, as usually is assumed in practice. The numerical method employed for the 1D beam formulation is based on Isogeometric tools (NURBS) while the 3D shell or solid models are developed in Finite Element commercial software for composite cross sections with diaphragms. The influence of friction on the contact interaction between the concrete and steel parts is also considered. The number of intermediate diaphragms is determined according to commonly used bridge design guidelines and the results are compared to the developed arrangements in order to assess the overall structural behavior of bridge decks. Straight or curved beam models with open or closed composite cross sections of full (1D proposed beam) or partial material-bonding (3D solid/shell models) and various arrangements of intermediate diaphragms and materials have been studied. [ABSTRACT FROM AUTHOR]