Bending collapse theory of thin-walled twelve right-angle section beams

The present paper focuses on the bending collapse behavior of the twelve right-angle section (TTRS) beams. This paper presents the theoretical bending collapse mechanism of the TTRS beams around two axes based on the kinematic approach, and derives the expressions of the bending moments. The accuracy of the theoretical calculations are validated respectively by performing 36 groups finite element simulations including three kinds of materials, three different section dimensions and four different thicknesses. The results show that for 17.5<c/h<48.6, the theoretical bending collapse mechanism of the TTRS beams presented in this paper can describe the collapse process accurately, and the moment–rotation curves calculated by theory show a good consistency with simulation results.     

Lateral post-buckling analysis of thin-walled open section beams

Thin-walled beams with open sections are studied using a nonlinear model. This model is developed in the context of large displacements and small deformations, by accounting for bending-bending and bending-torsion couplings. The warping and shortening effects are considered in the torsion equilibrium equation. The governing coupled equilibrium equations obtained from Galerkin’s method are solved by a Newton–Raphson iterative process. It is established that the buckling loads are highly dependent on the pre-buckling deformations of the beam. The bifurcated branches are unstable and strongly influenced by shortening effects. Some comparisons are presented with the solutions commonly used in linear stability, like in the standard European steel code (Eurocode 3). The regular solutions appear to be very conservative, especially for I sections with large flanges.     

Nonlinear shortening and bending effect under pure torque of thin-walled open beams

Assuming a constant rate of twist under pure torque, general expressions are developed for the axial shortening and transverse displacements of the shear centre for monosymmetric sections. The effect of the shear stress distribution dur to non-uniform bending on the buckling of a monosymmetric beam is discussed.     

Quadruply coupled linear free vibrations of thin-walled beams with a generic open section

The coupled vibration of thin-walled beams with a generic open section induced by the boundary conditions is investigated using the finite element method. If the axial displacement of the pin end is restrained at another point rather than the centroid of the asymmetric cross section, the axial vibration, two bending vibrations, and torsional vibration may be all coupled. The element developed here has two nodes with seven degrees of freedom per node. The shear center axis is chosen to be the reference axis and the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element. Different sets of element nodal degrees of freedom corresponding to different pin ends are considered here. The relation between element matrices referred to different sets of element nodal degrees of freedom is derived.

Numerical examples are presented to demonstrate the accuracy of the proposed method and to investigate the effects of different pin ends on the coupled vibrations of the thin-wall beam.     

Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section

A finite element formulation is present for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies are derived by the virtual principle. The energy functional includes the effect of flexural–torsional coupling, the torsion warping and the shear center location. For finite element analysis, cubic polynomials are utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem has been solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.     

Plastic collapse analysis of thin-walled circular tubes subjected to bending

Circular tubes have been widely used as structural members in many engineering applications. Therefore, its collapse behavior has been studied for many decades, focusing on its energy absorption characteristics and collapse mechanism. In order to predict the collapse behavior of members, one could rely on the use of finite element codes or experiments. These tools are helpful and have high accuracy but are costly and require extensive running time. Therefore, an approximate model of tubes collapse mechanism is an alternative especially for the early step of design. This paper is also aimed to develop a closed-form solution to predict the moment–rotation response of circular tube subjected to pure bending. The model was derived based on the principle of energy rate conservation. The collapse mechanism was divided into three phases. New analytical model of ovalisation plateau in phase 2 was derived to determine the ultimate moment. In phase 3, the Elchalakani et al. model [Int. J. Mech. Sci. 2002; 44:1117–1143] was developed to include the rate of energy dissipation on rolling hinge in the circumferential direction. The 3-D geometrical collapse mechanism was analyzed by adding the oblique hinge lines along the longitudinal tube within the length of the plastically deformed zone. Then, the rates of internal energy dissipation were calculated for each of the hinge lines which were defined in terms of velocity field. Inextensional deformation and perfect plastic material behavior were assumed in the derivation of deformation energy rate. In order to compare, the experiment was conducted with a number of tubes having various D/t ratios. Good agreement was found between the theoretical prediction and experimental results.     

Influence of local postbuckling behaviour on bending of thin-walled beams

The influence of the local postbuckling behaviour on the bending of thinwalled beams is studied. The analysis of the postbuckling behaviour of a prismatic column-beam subject to compression and bending is employed in order to determine the overall flexural stiffness of a beam after local buckling. The asymptotic Koiter method is used in the second order approximation. The determination of post-buckling coefficients enables one to find the beam's flexural stiffness without the necessity of using hypotheses on effective widths of eccentrically compressed plates. It is shown that the post-buckling equilibrium branch in a bending moment-global curvature coordinate system for a perfectly elastic thin-walled beam is rectilinear (in the second order approximation of' the asymptotic method). Simple analytical relations are presented between the slope of the post-buckling equilibrium branch of global bending and post-buckling coefficients for local buckling mode. The results obtained are compared with data reported by other authors.     

Nonlinear theory of non-uniform torsion of thin-walled open beams

A nonlinear theory of non-uniform torsion based on finite displacements is developed. Expressions for the finite nonlinear strains in Lagrangian coordinates and the Kirchhoff stresses for thin-walled open beams are presented. Using the principle of stationary total potential, the dual forms of the beam equilibrium equations are derived. For conservatively loaded thin-walled open beams a static stability criterion, based on the positive definiteness of the second variation of the total potential, is presented. The criterion developed takes into account the effects of changes in beam geometry such as initial bending curvature, prior to instability.     

Numerical analysis of elastic-plastic bending of curved beams with thin-walled cross-section

It is well known that, when a curved beam is subject to bending, the shape of its cross-section is flattened and collapse occurs as the rigidity is reduced. It is important for design to determine the nonlinear behavior of such beams. This paper describes the elastic-plastic large deformation analysis of a curved beam with a thin-walled cross-section by the finite element method. The analytical formulation is developed by extending the kinematic work hardening model proposed by Ziegler. Several representative cases are computed.     

Behavior of thin-walled circular hollow section tubes subjected to bending

Thin-walled steel circular hollow sections (CHSs) are widely used in wind turbine towers. The tower tubes are mainly subjected to bending. There have been a few experimental studies on the bending behavior of thin-walled CHS steel tubes. This paper describes a series of bending tests to examine the influence of section slenderness on the inelastic and elastic bending properties of thin-walled CHS. In addition, the influence of stiffeners welded in the steel tube is considered. Sixteen bending tests were performed up to failure on different sizes of CHS with diameter-to-thickness ratio (D/t) varying from 75 to 300. The experimental results showed that the specimens with small diameter-to-thickness ratios failed by extensive plastification on the central part of the tube. With the increase of diameter-to-thickness ratio, the local buckling phenomena became more pronounced. The stiffeners in the steel tubes increased the load carrying capacity and improved the ductility of the specimens. The experimental results were compared with current design guidelines on thin-walled steel members in AISC-LRFD, AS4100 and European Specification. It was found that the test results agreed well with the results based on AS4100 design code.     

Numerical simulation of the axial collapse of thin-walled polygonal section tubes

This paper deals with the post-buckling deformation characteristics of aluminum alloy extruded polygonal section tubes subjected to dynamic axial impacts. The explicit finite element code LS-DYNA is the primary analytical tool used in this investigation. The study focuses on investigating a post-buckling deformation phenomenon that is primarily manifested by an axial crumpling action that generates material folds as the impact energy is dissipated. The research is conducted in two phases. The first phase consists of validating the LS-DYNA model parameters and numerical results pertaining to thin-walled aluminum extruded square tubes with actual published experimental data. The post-buckling deformation characteristics of the specimens such as the overall final configuration and the various folding deformation modes (extensional, symmetric and asymmetric) resulting from the axial collapse of the member is also investigated in a subsequent phase. Based on the numerical simulation results, it is apparent that the increase in the number of walls (flanges) has a direct impact on the mean axial crushing force and permanent displacement parameters. In particular, the adoption of a hexagonal tube section as an axially loaded energy absorbing column yields an average increase of 11% in the mean axial crushing force and an average decrease of 10% in the permanent displacement. The greatest benefits are obtained in the specimens with the thinnest nominal wall thickness, where the upper bound results show an average increase of 27% in the mean axial crushing force and average decrease of 20% in the permanent displacement.     

Calculation of the deep bending collapse response for complex thin-walled columns: I. Pre-collapse and collapse phases

The increasing complexity of transport vehicles means that more powerful finite element models are needed to simulate their crash behaviour. As existing models' calculation times are long and cannot effectively optimize structures in terms of peak moment and energy management, they should only be used as a final verifying tool. Distinct analytical models have been developed to determine the resistance to collapse of thin-walled structures subjected to a bending load. Part I of this paper concerns the theoretical prediction of bending strength in the pre-failure range for thin-walled structures of relatively complex geometry. Two types of buckling are considered: elastic and plastic.     

Thin walled beams with sinusoidal embedded patterns under deep bending collapse

Adding complex sinusoidal patterns on wall surfaces has been recently suggested in literature as an effective innovation for improving energy dissipation of thin walled box beams during axial collapses. This paper studies the energy absorption efficiency and performance of patterned beams under pure bending. A numerical investigation is carried out using commercially available Marc Mentat^™ solver and the investigation is also verified using existing analytical models. Five different types of patterns are tested. The result of adding patterns on individual and combination of walls and the effect of progressive triggering proposed previously in literature on the maximum bending moment of the beam is also studied. Results indicate that the beams with sinusoidal patterns are significantly better in energy absorption in deep bending collapse; an increase of about 53.49% at 0.6 rad is noted in the optimal case. The pattern types which had higher energy absorption in axial impact reflect the same trend in bending collapses. This research may further endorse the viability of using sinusoidal patterned beams in mainstream practical applications.     

Effective shaping of cold-formed thin-walled channel beams with double-box flanges in pure bending

The paper is devoted to cold-formed thin-walled channel beams with double-box flanges. Geometric properties of the C-section are described in terms of dimensionless parameters. The warping function and the warping inertia moment are analytically determined. The optimization criterion and the dimensionless objective functions are defined as a quality measure. The space of feasible solutions is constrained by the strength, global, local buckling, and geometric conditions. Analytical solutions of the problems of global and local buckling for thin-walled beams are presented. Results of the numerical calculations of the optimal shaping problem are presented in tables and figures.     

Distortional buckling of thin-walled inclined lipped channel beams bending about the minor axis

Based on the Generalised Beam Theory (GBT), two computing models are presented to analyse the distortional critical stress of cold-formed thin-walled inclined lipped channel beams bending about the minor axis. The computing model of rigid-body motion ignores the transverse bending deformation of the flange. However, the bending deformation of the flange is accounted for in the transverse bending computing model. Based on the transverse bending computing model, this paper puts up a simple method to take into account the in-plane bending of the flange. The results given by the rigid-body motion computing model does not correlate as well as those given by the transverse bending computing model with the results available in the literature. The accuracy of the transverse bending computing model is verified through comparison of its results with the known results. The comparisons demonstrate the importance of the bending deformation of the flange on the distortional buckling of cold-formed thin-walled channel beams bending about the minor axis.     

Distortional buckling of thin-walled inclined lipped channel beams bending about the minor axis

Based on the Generalised Beam Theory (GBT), two computing models are presented to analyse the distortional critical stress of cold-formed thin-walled inclined lipped channel beams bending about the minor axis. The computing model of rigid-body motion ignores the transverse bending deformation of the flange. However, the bending deformation of the flange is accounted for in the transverse bending computing model. Based on the transverse bending computing model, this paper puts up a simple method to take into account the in-plane bending of the flange. The results given by the rigid-body motion computing model does not correlate as well as those given by the transverse bending computing model with the results available in the literature. The accuracy of the transverse bending computing model is verified through comparison of its results with the known results. The comparisons demonstrate the importance of the bending deformation of the flange on the distortional buckling of cold-formed thin-walled channel beams bending about the minor axis.     

Load-capacity estimation and collapse analysis of thin-walled beams and columns—recent advances

The present paper is devoted to the recent results of research in the area of load-capacity and post-failure behaviour of thin-walled beams and columns (among them thin-walled cold-formed profiles). It deals with ultimate load and collapse of box-section girders (tubes) of different cross-sections under bending, as well as of lipped and plain channel-section beam-columns. The paper contains the presentation of theoretical analysis and experimental investigation of plastic mechanisms of failure and collapse behaviour of these thin-walled sections. The short review of results obtained in recent years in general precedes those obtained by the Department of Strength of Materials and Structures, TUL. The problem of post-failure behaviour is solved using the rigid-plastic theory adopted and modified for the purposes of the solution taking into consideration strain-hardening of the member’s material. On the basis of experimental investigations theoretical models of plastic mechanisms of failure are produced for different sections. Theoretical analysis is based on the principle of virtual velocities. The problem is solved in an analytical–numerical way. The particular attention has been paid to the influence of the strain-hardening of the material after yielding upon the collapse structural behaviour and also to the influence of cross-section shape and dimensions on the character of collapse. The upper-bound estimation of the load-carrying capacity of analysed thin-walled sections by combining results of non-linear, elastic post-buckling analysis with the results of plastic mechanism analysis is carried out. Results are presented in diagrams showing post-failure curves as well as curves representing structural behaviour in the whole range of loading up to and beyond the ultimate load. Some results are compared with experimental results and those obtained from FE analysis. A comparison of lower- and upper-bound estimation of the load-carrying capacity is discussed and illustrated in diagrams. Conclusions dealing with the influence of strain-hardening phenomenon displayed by the material upon the load-carrying capacity and collapse behaviour of examined sections are derived. Also conclusions concerning different upper- and lower-bound estimations of the load-carrying capacity of analysed sections are presented.     

Theoretical shape optimization of cold-formed thin-walled channel beams with drop flanges in pure bending

The paper is devoted to cold-formed thin-walled channel beams with open or closed profile of drop flanges. Geometric properties of two C-sections are described with consideration of warping functions and warping inertia moments. Optimization criterion and the dimensionless objective function as a quality measure are defined. Constraints of feasible solutions are strength, global and local buckling conditions and also geometric condition. Analytical solutions of the problems of global and local buckling for thin-walled beams are presented. Results of numerical investigation of optimization problem are compared and presented in tables and figures.     

不同截面形式高强冷弯薄壁槽钢构件受弯承载力试验研究

对36个屈服强度为550 MPa的高强冷弯薄壁槽钢受弯试件进行静力试验研究,考虑了加劲形式和卷边形式对试件受弯性能的影响,其中加劲形式分为无加劲、翼缘中间V形加劲和翼缘及腹板中间V形加劲3种,卷边形式分为直卷边、斜卷边和复杂卷边3种。试验结果表明：加劲形式和卷边形式是影响试件受弯承载力和屈曲模式的重要因素;与无加劲形式相比,采用板件中间V形加劲有效减小了板件宽厚比,试件受弯承载力提高了30%~70%;同种加劲形式下,短（直、斜）卷边试件受弯承载力提高幅度最大,复杂卷边试件提高幅度次之,长（直、斜）卷边试件提高幅度最小;试验过程中试件发生了局部屈曲、畸变屈曲与局部和畸变相关屈曲。对试验进行了有限元模拟,模拟结果与试验结果吻合较好。     

Distortion of thin-walled beams

First order Generalized Beam Theory (GBT) describes the behaviour of prismatic thin-walled structural members by using system of ordinary differential equations. Solution can lead to separation of the load components and then to subsequent combination of the stresses and deformations. Application of GBT to a steel, cold-formed, thin-walled cantilever beam with complex non-symmetrical open cross section is presented. Theoretical values are compared with experimental ones.