Thin-walled Bars Research Articles

Elastic Critical Resistance of the Simple Beam Grillage Resulting from the Lateral Torsional Buckling Condition: FEM Modelling and Analytical Considerations.

Transversely loaded beam grillages are quite often used in industrial construction. In order to produce a safe design of such structures, it is necessary to account for the lateral torsional buckling phenomenon, which reduces load-bearing capacity. To be able to calculate the relevant reduction factor, the elastic critical load must be determined. As regards the existing design practice for such structures, simplified conditions are assumed for the mutual restraint of the component beams. However, this approach does not correspond to reality. This study discusses the results of numerical investigations and analytical calculations concerning the effect of the elastic action of simple beam grillage (SBG) joints on the critical load, which results from the lateral torsional buckling (LTB) condition. The SBG was defined as a flat system of interconnected beams, unstiffened laterally and loaded perpendicularly to the grillage plane. The analysis covered H-shaped grillages with different span ratios of component beams, in which the main (coupling) beam was decisive for instability. The effectiveness of the use of closed-section stiffeners at the grillage joints was also investigated. The grillage elastic critical resistances (ECR) were determined for two variants of joint stiffening. The computations were performed by means of FEM numerical simulations. The spatial models were discretised with the following elements: (1) solid ones in Abaqus, (2) shell ones in ConSteel, and (3) thin-walled bars in ConSteel. The LTB critical moments of the weakest beam (critical beam), elastically restrained against warping and against lateral rotation (in the LTB plane), were computed using the analytical methods developed by the authors. To this end, the methods were proposed to determine the indexes of the critical beam elastic restraint in the adjacent stiffening beams. In the study, it was demonstrated that (1) taking into account the conditions of mutual elastic restraint and interaction of the component beams provides a more accurate assessment of the grillage ECR, (2) the use of closed-section stiffeners in the grillage joints increase the ECR compared with classic flat stiffeners, (3) the grillage ECR can be estimated based on the critical moment Mcr of the weakest beam (critical beam) when the conditions of its elastic restraint in joints are accounted for.

Influence of adjacent segments on the torsional load-bearing behaviour of assembled half-shell towers

This article deals with the influence of adjacent segments on the torsional load bearing behaviour of assembled half-shell towers for wind energy turbines. The main objective is to detect and understand the load transfer in the modular tower structure considering the effects of adjacent segments. The engineering approach already developed for two assembled half-shells – based on the warping theory of thin-walled bars – is investigated for its applicability to the entire tower structure. Numerical comparative studies show that an adjacent half-shell pair level at the top and bottom is sufficient to take the influence of the entire tower structure into account. The adjacent segments affect the quantitative determination of the load-bearing components from shear force and torsion. Furthermore, the coupling conditions in the horizontal joint depend on the stress consideration and include only one (normal stresses via linear superposition) or both half-shells (shear stresses via Bredt shear flow). Therefore, an analytically and numerically based approach is presented that can be applied to the use case of segmental towers for wind turbines and the constant component heights of the precast segments. In this case, the methods take the decrease of the bending shear force component and the simultaneous increase of the torsion component in the system axis into account. The boundary conditions of the frame structure remain unaffected. Based on the further developed engineering model, the complex torsional load-bearing behaviour and the stress state in the horizontal joint of modular tower structures can now be analytically determined in the compressed and static friction condition based on the warping theory of thin-walled bars.

Prediction of Mechanical Properties of Thin-Walled Bar with Open Cross-Section under Restrained Torsion

Thin-walled bars with an open cross-section are widely used in mechanical structures where weight and size control are particularly required. Thus, this paper attempts to propose a theoretical model for predicting the mechanical properties of a thin-walled bar with an open cross-section under restrained torsion. Firstly, a theoretical model with predictions of shear stress, buckling normal stress, and secondary shear stress of the thin-walled bar with open cross-section under the condition of restrained torsion was developed based on torsion theory. Then, physical test and finite element modeling data were employed to validate the theoretical predictions. The results indicate that the theoretical predictions show good agreements with data of finite element modeling and experiments. Therefore, the proposed theoretical model could be used for the prediction of the mechanical response of a thin-walled bar with an open annular section under restrained torsion.

Saint-Venant torsion of thin-walled nonhomogeneous elliptical tube

This paper deals with the uniform torsion of thin-walled elliptical tube. The material of the tube is nonhomogeneous and it depends on one of the curvilinear coordinates which defines the cross section of thin-walled bar with closed profile. The approximate solution for the stresses, torsion function and torsional rigidity are obtained by the application of two extreme value theorems of linearized elasticity.

Algorithm of correcting bimoments in calculations of thin-walled bar systems

Algorithm of correcting bimoments in calculations of thin-walled bar systems

Generalized beam theory-based advanced beam finite elements for linear buckling analyses of perforated thin-walled members

This paper presents a numerically efficient tool for linear buckling predictions of perforated thin-walled bars within the framework of generalized beam theory (GBT). GBT-based beam finite element methods (GBT-FEM) have been well developed for eigenvalue buckling analyses of non-perforated thin-walled bars. The novelty of this paper consists in an extension of the standard GBT to the scope of thin-walled members with arbitrarily shaped and placed holes. This is achieved by combining the standard GBT and the extended finite element method (X-FEM). More specifically, insert a set of locally supported enrichment functions, accounting for the discontinuities on displacement fields arising from the cross-section cut-outs/holes, into the GBT-based finite element approximations of the member configuration spaces, using the partition of unity method (PUM), where a set of level-set functions are used to describe the geometric profiles of hole edges, i.e., with the hole edges being the zero level sets, and also used to construct the enrichment functions. The proposed approach makes it possible to calculate the deformation mode participations for any perforated thin-walled members as the classic GBT for non-perforated ones. Finally, the proposed approach is calibrated against the shell/solid finite element analysis with four illustrative examples. It can be found that the presented approach is of higher computation efficiency than the shell model.

A geometrically exact beam finite element for curved thin-walled bars with deformable cross-section

This paper presents a geometrically exact beam finite element for curved and twisted thin-walled bars undergoing large displacements and finite rotations, combined with cross-section in-plane and out-of-plane (warping) deformation. The underlying formulation is based on a previous work by the authors Gonçalves et al. (2010), which is now improved and extended for the pre-curved/twisted case. Cross-section deformation is allowed for by adding kinematic DOFs, the so-called “cross-section deformation modes”, which are obtained using Generalised Beam Theory. All expressions required to implement the proposed finite element are derived. For validation purposes, several numerical examples are presented and discussed. In each case, the results obtained with the proposed element and refined shell finite element models are compared, and an excellent agreement is found. In particular, it is demonstrated that the proposed element predicts very accurately the spatial behaviour of curved beams undergoing complex coupling between large displacements, finite rotations and cross-section deformation.

ВЛИЯНИЕ НАКЛОННЫХ РЕБЕР ЖЕСТКОСТИ В УЗЛАХ СОЕДИНЕНИЯ ДВУТАВРОВ НА ПЕРЕДАЧУ БИМОМЕНТОВ

Shear-free theory of V.Z. Vlasov remains one of the most reasonable approaches to calculating thin-walled bars taking into account constrained torsion. At the same time, the use of this theory for the analysis of deformations of frame structures still requires research in terms of the conditions for transferring forces in bar connections. As noted in some scientific papers, the balance of bimoments can be significantly broken at the joints of thin-walled bars of an open profile in some designs. This paper deals with this phenomenon for steel I-beam profiles, associated with the presence of inclined stiffeners in joint units. Using shell finite element models, the influence of inclined stiffeners on the appearance of bimoment jumps at the pairwise connection of bars is shown. A dependence is derived that makes it possible to take into account the stiffness of the inclined edge in the bar models from the point of view of the restraint of cross-section warping. On the basis of numerical experiments, it was determined that the introduction of such stiffness into the bar finite element schemes of frame structures allows to reflect the condition of bars interaction in the transferring of bimoments with a sufficiently high accuracy for engineering practice.

Local Buckling and Resistance of Continuous Steel Beams with Thin-Walled I-Shaped Cross-Sections

In modern steel construction, thin-walled elements with Class 4 cross-sections are commonly used. For the sake of the computation of such elements according to European Eurocode 3 (EC3), simplified computational models are applied. These models do not account for important parameters that affect the behavior of a structure susceptible to local stability loss. This study discussed the effect of local buckling on the design ultimate resistance of a continuous beam with a thin-walled Class 4 I-shaped cross-section. In the investigations, a more accurate computational model was employed. A new calculation model was proposed, based on the analysis of local buckling separately for the span segment and the support segment of the first span, which are characterized by different distributions of bending moments. Critical stress was determined using the critical plate method (CPM), taking into account the effect of the mutual elastic restraint of the cross-section walls. The stability analysis also accounted for the effect of longitudinal stress variation resulting from the varied distribution of bending moments along the continuous beam length. The results of the calculations were compared with the numerical simulations using the finite element method. The obtained results showed very good congruence. The phenomena mentioned above are not taken into consideration in the computational model provided in EC3. Based on the critical stress calculated as above, “local” critical moments were determined. These constitute a limit on the validity of the Vlasov theory of thin-walled bars. Design ultimate resistance of the I-shaped cross-section was determined from the plastic yield condition of the most compressed edge under the assumptions specified in the study. Detailed calculations were performed for I-sections welded from thin metal sheets, and for sections made from two cold-formed channels (2C). The impact of the following factors on the critical resistance and design ultimate resistance of the midspan and support cross-sections was analyzed: (1) longitudinal stress variation, (2) relative plate slenderness of the flange, and (3) span length of the continuous beam. The results were compared with the outcomes obtained for box sections with the same contour dimensions, and also with those produced acc. EC3. It was shown that compared with calculations acc. EC3, those performed in accordance with the CPM described much more accurately the behavior of the uniformly loaded continuous beam with a thin-walled section. This could lead to a more effective design of structures of this class.

Model tests of bending and torsional deformations of thin-walled profiles stiffened with elements made in 3D printing technology

The paper presents the benefits of using 3D printing technology as an effective way to increase the rigidity of thin-walled bars subjected to non-free torsion. The first part of the article concerns bending and torsion testing of C thin-walled unstiffened and longitudinally stiffened profiles. The proprietary solution made with 3D printing technology was used as the longitudinal stiffening. The impact of a stiffener used on bending – torsional displacements in selected profile sections was assessed. The results of experimental research and numerical analysis made with the finite element method were compared. Based on numerical simulations and experimental tests carried out, it can be stated that the stiffener that was used increases the flexural and torsional stiffness of thin-walled profiles. In the second part of the work, in order to present the legitimacy of the discussed topic, the impact of the disassembly of the outer plating in the storey of a single-story structure of a single-family building made of light steel framework technology on its stability is shown. For this purpose, three various calculation were assumed for the adopted object, whose models were subjected to numerical analysis by the finite element method. The analysis took into account possible charges to occur in accordance with the regulations in force in Poland. Due to the analysis of the building model without stiffening plating on the ground floor, it was confirmed that the displacement limits specified in the standards were exceeded. The study showed that the self-wedging assembly insert developed in 3D printing technology developed by the authors is an effective way to increase the rigidity of non-free torsion of C-type steel thin-walled profiles and is used in renovation and repair works when there is a need to stiffen a building block following the removal of wall sheathing.

Analysis of Thin-Walled Bars Stress State with an Open Section

The analysis of the thin-walled structural problem is still a topic of interest and research by many scientists with many different solutions and theories. This paper presents the occurrence of stress components on the thin-walled bars with open sections. Vlasov’s theory has been applied to analyze the stress state on thin rods. At the same time, using the Midas civil structure calculation program based on finite element principle for structural analysis with taking into account sectional warping. From the results obtained by the finite element method, it is necessary to select and design carefully for thin-walled bars.

Influence of Imperfection on Buckling Resistance of Perforated Thin-Walled Bar with Very Low Slenderness – FEM Analysis and Comparison with Experimental Results

In this paper, the results of modelling of compression test for the perforated thin-walled bars of very low slenderness are shown and confronted with the experimental results. Three sets of imperfections as well as a model without imperfections were analysed and compared. Application of initial imperfections improved the results in most cases, but not always in the same extent, there were also some cases, where they worsened the compliance with experiment instead.

Searching for shear forces flows in arbitrary cross-sections of thin-walled bars: numerical algorithm and software implementation

Development of a general computer program for the design and verification of thin-walled bar structural members remains an actual task. Despite the prevailing influence of normal stresses on the stress-strain state of thin-walled bars design and verification of thin-walled structural members should be performed taking into account not only normal stresses, but also shear stresses.Therefore, in the paper a thin-walled bar of an arbitrary cross-section which is undergone to the general load case is considered as investigated object. The main research question is development of mathematical support and knoware for numerical solution for the shear stresses problem with orientation on software implementation in a computer-aided design system for thin-walled bar structures.The problem of shear stresses outside longitudinal edges of an arbitrary cross-section (including open-closed multi-contour cross-sections) of a thin-walled bar subjected to the general load case has been considered in the paper. The formulated problem has been reduced to the searching problem for unknown shear forces flows that have the least value of the Castigliano’s functional. Besides, constraints-equalities of shear forces flows equilibrium formulated for cross-section branch points, as well as equilibrium equation formulated for the whole cross-section relating to longitudinal axes of the thin-walled bar have been taken into account.A detailed numerical algorithm intended to solve the formulated problem has been proposed by the paper. The algorithm is oriented on software implementation in systems of computer-aided design of thin-walled bar structures. Developed algorithm has been implemented in SCAD Office environment by the program TONUS. Numerical examples for calculation of thin-walled bars with open and open-closed multi-contour cross-sections have been considered in order to validate developed algorithm and verify calculation accuracy for sectorial cross-section geometrical properties and shear stresses caused by warping torque and shear forces. Validity of the calculation results obtained using developed software has been proven by considered examples.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Abstract This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

SEARCHING SHEAR FORCES FLOWS FOR AN ARBITRARY CROSS-SECTION OF A THIN-WALLED BAR: DEVELOPMENT OF NUMERICAL ALGORITHM BASED ON THE GRAPH THEORY

Searching problem of shear stresses on outside longitudinal edges of an arbitrary cross-section (including open-closed multi-contour cross-sections) of a thin-walled bar subjected to the general load case has been considered in the paper. Detail numerical algorithm intended to solve the formulated problem using mathematical apparatus of the graph theory has been proposed by the paper. The algorithm is oriented on software implementation in systems of computer-aided design of thin-walled bar structures.

The semi-shear theory of V. I. Slivker for the stability problems of thin-walled bars

The semi-shear theory of V. I. Slivker for the stability problems of thin-walled bars

The stress-strain state of thin-walled bar of variable cross section with different variants of fastening of the ends that is used in the energy construction

The authors proposed equations for the calculation of the stability and strength of thin-walled beams with swivel bearing ends can be generalized to the problem with boundary conditions corresponding to the elastic interaction with neighboring elements of the lattice pylons.

Vibration behavior of thin-walled steel members subjected to uniform bending

Abstract This article reports the results of an investigation on the effects of internal moments on the vibration behavior of thin-walled steel members. The analyses are based on the Generalized Beam Theory (GBT), a thin-walled bar theory accounting for cross-section in-plane deformations - its main distinctive feature is the representation of the member deformed configuration by means of a linear combination of cross-section deformation modes, multiplied by their longitudinal amplitude functions. The study concerns a simply supported T-section (with unequal flanges) members exhibiting a wide range of lengths and subjected to uniform internal moment diagrams - their magnitudes are specified as percentages of the corresponding critical buckling values. After providing a brief overview of the main concepts and procedures involved in performing a GBT-based structural analysis, the vibration behavior of load-free and loaded T-section members is addressed - the influence of the applied loadings is assessed in terms of (i) the fundamental frequency difference and (ii) the change in the corresponding vibration mode shape. For validation purposes, some GBT results are compared with values yielded by shell finite element analysis performed in the code ABAQUS (Simulia, 2008).

Refinement of the accounting methodology of bi-moments transfer at the junctions of the I-section bars

Modern construction standards require an accounting of bi-moments when determining normal stresses in the cross-sections of the bars. This factor may first be essential to provide the required safety and reliability of thin-walled bars of an open profile. The bars restrained torsion theory was sufficiently considered in literature, however, in recent years the classic concepts about interaction of such bars in frame systems have been challenged by some authors. Within the framework of the finite element method such problem may be solved on the basis of a combined approach, when the bar finite elements are imposed out of the area of joined nodes and strains at the joints are described with the help of plate finite elements. At the same time, calculation models become more complicated in comparison to using only the bar finite elements. In this paper the detailed studies of interaction of the bars with I-sections are performed on the basis of mathematical modeling of their strains with the help of plate finite elements. The methodology of these studies contains definition of bi-moments in several bars cross-sections, including the sections adjacent to the joints. Different variants of relative positions of the bars are considered. It has been determined that in some instances the condition of bi-moments balance is provided in assembly joints with a high degree of accuracy. At the same time, in some schemes of the bars connection such a balance is being largely damaged. The paper gives the explanation for the phenomenon based on the features of the transfer of the internal forces factors in the connection joints of the bars. It also represents a way of obtaining imbalance on bi-moments in accordance with the real work condition of these thin-walled constructions. The obtained results will make it possible to increase the accuracy of calculations of the thin-walled constructions formed by the bars with I-sections within the bars calculation models. The considered approach to the analysis of the interaction of the thin-walled bars may be the basis for development of the calculation algorithms of thin-walled bar systems with other forms of cross-sections.

Free Vibrations of Thin-Walled Box-Section Bars Allowing for Shear Strains

Purpose. The purpose is to study free vibrations of thin-walled bar with combined bisymmetrical section allowing for the effects of shear strains.Methods. We used the main hypothesis of V.Z. Vlasov concerning thin-walled bars. Solving motion equations analytically, we used the method of initial parameters and the known methods for differential equations solution. In the natural experiment we applied Autodesk Inventor system. In the course of the natural experiment on shaker, we used the method of smooth change in the frequency of sinusoidal vibrations.Results. We provide the analytical solutions of the shapes and the natural frequencies of thin-walled bar with combined bisymmetrical section allowing for shears caused by bends and constrained torsion. We compared the results of the calculations conducted in CAE system and the results of the full-scale experiment to determine the frequencies of vibrations of box beam.Conclusions. The results showed that the analytical calculations are in good compliance with the results of numerical and physical experiments and can be used for dynamic calculations of thin-walled structural elements, in particular, for the prevention of the destruction of structures in the event of resonance phenomena.