Distortional Mode Research Articles

On the out-of-plane buckling of RHS X-joints – Analytical and numerical approaches

Out-of-plane buckling (OPB) is not widely known failure mechanism in tubular X-joints with equal brace and chord widths (β = b1/b0 = 1.0) but it can occur in the chord member of tubular X-joints axially compressed by the brace members. The OPB consists of distortional and torsional deformations in the chord member, but only the distortional mode is involved in elastic energy of the system. In this work, an analytical formulation for the critical OPB capacity is established by the deformed shape in a beam on elastic foundation (BEF). A theoretical model for the ideal-elastic buckling capacity is computed by considering the in-plane bending of the chord faces and elastic bending of the cross-section as the frame structure. The obtained critical buckling loads are compared with numerical results obtained using linear buckling analyses by finite element (FE) models. The theoretical model provided a reasonable agreement with the numerically obtained eigenvalues with a tendency to slightly underestimate load-carrying capacity. The FE results showed that the OPB of X-joint can be the critical ideal-elastic failure mode in the wide range of various joint geometries and thus be regarded as a competitive failure mechanism for the widely known chord sidewall buckling. However, the results were obtained for the ideal-elastic buckling modes and, eventually, the actual failure mode is highly depended on the existing geometrical imperfections. Nevertheless, experimental tests have also indicated an existence of the OPB mode, and structural designers and analysts of tubular connections should be aware of this potential failure mode.

CFS lipped channel beams buckling in distortional modes at elevated temperatures: Behaviour, failure and DSM design

This work reports the findings of a comprehensive numerical investigation on the post-buckling behaviour, failure and Direct Strength Method (DSM) design of cold-formed steel (CFS) single-span simply supported lipped channel beams buckling in distortional modes at elevated temperatures (up to 800 °C) due to fire conditions. It extends the scope of a previous investigation carried out by Landesmann and Camotim [1], by analysing a substantially larger lipped channel beam set, exhibiting various cross-section dimensions and yield stresses, selected to cover wider distortional slenderness ranges. As done before, (i) the beams analysed display two end support conditions, (ii) the Eurocode 3 (part 1.2) model to describe the temperature-dependence of the CFS material properties is adopted and (iii) the results are obtained by means of Abaqus shell finite element GMNIA. After presenting and discussing the main features of the beam distortional post-buckling behaviour, extensive beam failure moment sets are gathered and used to develop and validate DSM-based design approaches. The methodology followed consists of modifying the most performant available DSM-based design curves (developed for beams at ambient temperature [2]), which naturally involves the temperature-dependant reduction factors of the CFS model. A merit assessment procedure shows that the modified DSM-based strength curves predict the lipped channel beam distortional failure moments with remarkable accuracy and reliability, thus constituting an excellent starting point to search for a DSM-based design approach capable of handling arbitrary CFS beams failing in distortional modes at elevated temperatures.

Inelastic buckling analysis of cold-formed steel members with residual stresses based on CFSM

Residual stresses in cold-formed steel members are generally composed of a membrane component and a bending component, which may result in the inelastic buckling of cold-formed members. The complex finite strip method (CFSM) has been widely used to analyze the buckling of thin-walled members and it can handle shear effects easily. As for inelastic buckling, the available CFSM mainly focuses on the local buckling of plates and cannot be applied to the buckling of cold-formed members that can buckle in a distortional or global mode. Besides, the bending component may be hard to be incorporated into the finite strip method. In the present study, a modified CFSM is proposed to analyze the inelastic buckling of cold-formed steel members with residual stresses. The effects of the membrane and bending components on inelastic buckling are considered through the applied stresses and modifying material properties, respectively. The results calculated by the proposed model are compared to the available solutions of several examples in previous studies, and good agreements are observed. Based on the proposed model, the effects of residual stresses on the inelastic buckling of plates and cold-formed members are investigated. The results show that the inelastic buckling stress is significantly lower than the elastic buckling stress when the elastic buckling stress exceeds the yield stress. The membrane component has a significant influence on the inelastic buckling stress of plates. The bending component mainly affects the inelastic local buckling of members under compression and the inelastic global buckling of members under compression as well as bending and shear.

Experimental investigation of the sectional buckling of built-up cold-formed steel columns

This paper presents an experimental program investigating the local and/or distortional (sectional) buckling of built-up cold-formed steel sections subjected to pure compression. The experiments were designed to study the influence of key parameters, including buckling mode, cross-section geometry and fastener spacing, on the ultimate sectional capacity of built-up columns. Three singly- or doubly-symmetric built-up sections were considered with two sets of component lipped channel sections, one buckling predominantly in the distortional mode and one buckling in the local mode. The component sections were connected using self-drilling screws at three different spacings. Material properties, geometric imperfections and residual stresses of the specimens were measured prior to testing. A total of 36 specimens with a length of 1 m were tested under concentric compression between fixed ends, which was achieved through a special end plate connection. The specimens failed in either distortional or the interaction of local and distortional buckling modes. Compared to the sum of the capacity of component sections, the ultimate capacity in built-up columns with distortional buckling mode increased significantly with the reduction of screw spacing, whereas the increase was minor for specimens that failed through local-distortional interaction. The ultimate strength of the specimens was predicted using the current Direct Strength Method, where either conservative or overly unconservative predictions were obtained depending on the considered composite action level in the elastic buckling analysis.

Local–distortional buckling mode of steel cold-formed columns: Generalized direct strength design approach

Cold-Formed Steel (CFS) members are highly susceptible to several structural instability phenomena, involving either individual (Local — L, Distortional — D and Global — G) or interaction (LD, LG, DG, and LDG) buckling modes. The buckling mode interaction of CFS columns and beams may lead to strength erosion compared to isolated modes and must be considered in structural design. The North American, Australian/New Zealand and Brazilian standards incorporated the Direct Strength Method (DSM) that provides accurate design approach for columns and beams affected by L, D, G, and LG buckling modes. However, the coupled LD, DG, and LDG buckling modes are not currently considered in the DSM and many recent research projects have been focused on these topics. The procedure adopted by DSM estimates the CFS member strength through experimentally calibrated design strength equations and curves, based on the relationship between the member elastic stability and the steel yielding property, respectively the elastic critical buckling load and the plastic load, represented by the slenderness factor. The more accurate the buckling load, the more improved will be the result of the column strength by the DSM equations. In this context, the present paper presents the main results of the investigation, based on both experimental and finite element method numerical results, that conducted to calibrated strength curves for CFS unstiffened sections columns without holes covering LD interaction buckling mode. To take into account the effect of the LD buckling, the proposed procedure for design purposes incorporates the slenderness factor ratio RλDL=λD/λL as the main variable, and λL, λD, λG as the local, distortional and global slenderness factors, respectively.

Distortional buckling behavior of axially loaded cold-formed steel built-up I-section columns

This paper presents a comprehensive experimental program in which the distortional buckling behavior of cold-formed steel built-up I-section columns with two different cross-sectional dimensions and three different screw spacings was investigated. In total, the results from 24 experimental tests, including 18 built-up I-section columns and 6 lipped channel section columns, were tested between fix-ended conditions. Material properties, geometrical cross-sectional dimensions, and initial imperfections for all test specimens were measured during tests. The experiments aim to quantify buckling behavior, failure deformation, assembly effect, and ultimate strength for built-up I-section columns. Nonlinear finite element models were established and the accuracy of the model was verified by experiments. Additionally, a theoretical model for the distortional buckling of built-up I-section columns was proposed, and the calculation formula for the distortional buckling critical stress was given in this paper. The validated finite element model was used to carry out extensive parametric research, revealing the influence of basic parameters such as boundary condition, cross-sectional dimension, and self-drilling screw arrangement on the distortional buckling. Finally, a calculation method for the distortional buckling bearing capacity that can consider the post-buckling strength was established by substituting the calculation formula of the distortional buckling critical stress into Schafer's bearing capacity curve. The effectiveness of the proposed method was verified by comparing the computational results with the experimental results and supplementary finite element results. The comparison results show that the proposed method can be successfully used to predict the bearing capacity of built-up I-section columns affected by pure distortional buckling mode.

Local/distortional and overall interactive buckling of thin-walled cold-formed steel columns with open cross-section

The difference between local and distortional buckling of thin-walled cold-formed steel sections with stiffened elements, of members in both compression and bending are taken into consideration applying relevant design formulas available in the codes. However, for unstiffened elements, or for those unstiffened enough, the difference between distortional and local buckling modes, in some cases, and particularly for U-, Z- and L-sections, might be unclear. When these sections are edge stiffened, in the case of lipped flanges, it is not always easy to identify which of the two are dominant, while in the case of interaction with an overall buckling mode, the situation might be even more difficult. Local buckling modes involve the buckling of cross-sectional walls, while distortional buckling, also known as “stiffener buckling” or “local-torsional buckling”, is characterised by the rotation of the flange at the flange/web junction in members with edge stiffened elements. Both types of buckling are also known as “sectional” buckling modes. For interactive local/distortional buckling with the overall one, the correct evaluation of the “short” member capacity might play a significant role when checking the design capacity of a slender thin-walled member prone to interactive sectional-overall buckling mode. In the case of a stiffened flange, the effectiveness of the stiffeners can make the difference between local and distortion buckling. A boundary between local and distortional bucking of a given open section can be established to be considered in the design. To correctly apply the Ayrton–Perry formula, in order to get the ultimate capacity of a thin-walled cold-formed steel slender member, prone to interactive local or distortional buckling with the overall one, the strength of the short member has to be identified. This paper summarises the study of this problem, including the main results and conclusions, emphasising that it is difficult to separate the local and distortional modes in the case of cold-formed steel open sections used in practice, and there is some interaction between these two modes.

Extension of modal decomposition approach for buckling analysis of thin-walled lipped channel columns: Local, distortional, flexural and torsional modes

This paper presents a new version of the modal identification method previously proposed by the authors. The method is aimed at experimental tests of lipped channel columns but may also be applied to numerical columns. The first version of the method considered only three main deformation modes: local, symmetric distortional and minor axis flexural global modes. In this new version, anti-symmetric distortional and flexural–torsional modes are included in the formulation and original expressions for local mode are improved. The inclusion of new modes does not alter the procedure to perform the modal identification of the first version. In short, simple displacement functions are assumed for each buckling mode and through an error minimization process the participation of each mode in a generic deformed shape of the column is assessed. The new version of the method is validated against numerical results from Generalized Beam Theory (GBT) for columns undergoing D–G interaction and applied to experimental results of lipped channel columns.

Constrained finite element method with displacement mapping

Understanding and analyzing the complex behavior of a structural member is often achieved by decomposing it into simpler phenomena and interpreting it as a superposition of these simpler phenomena. Accordingly, the deformations of thin-walled members are commonly decomposed into global, distortional, local-plate, and other modes. One well-known practical technique is the constraining of shell models; the idea first appeared in the constrained finite strip method, then later in the constrained finite element method, or cFEM in short. In spite of cFEM’s remarkable ability to handle a wide range of problems, one of its disadvantages is that it relies on a specific rectangular shell finite element. The research presented here seeks to overcome the limits set by this shell element. It is proposed to use two discretizations and two corresponding basis function systems. One is based on virtually arbitrary, ordinary shell finite elements, whereas the other is based on cFEM shell elements. Constrained spaces are defined with the basis functions of the cFEM and then mapped to the ordinary shell model. The modal analyses are finally completed in the ordinary shell model, but by using the constrained spaces obtained from the cFEM basis functions. In this paper we describe the proposed method and demonstrate its application with numerical studies.

Energy‐based Equations for Local and Anti‐symmetric Distortional Modes of Lipped Channel Columns

AbstractThis paper presents an energy‐based approach to obtain the critical length and stress of local (L) and anti‐symmetric distortional (D2) modes of simply supported lipped channel columns. For each mode, deformed shapes are assumed and the Rayleigh Quotient Method is applied to obtain analytical expressions for the buckling stress. The shape functions adopted for local buckling ensure compatibility of rotation and bending moments at wall junctions and at the stiffener's free end. For the D2 mode, equilibrium of bending and twisting moments are considered leading to an analytical derivation of the mode's warping stress distribution. While for L mode it is not feasible to obtain a closed expression for the critical length (but a simple length testing procedure is sufficient), for D2 mode an expression dependent on geometrical properties is derived. The developed equations for L and D2 modes are validated against CUFSM's pure local modes and GBTul's pure D2 mode (mode 6 in GBTul convention) respectively. For sections with flange‐to‐web width ratio, bf/bw, ranging from 0.2 to 1.0 and lip‐to‐web width ratio, bs/bw, from 0.1 to 0.4 the average ratios obtained for critical stresses (σcr,reference/σcr,proposed) are 1.014±0.019 and 1.011 ± 0.014 for L and D2 modes respectively.

New transverse extension modes for buckling analysis of thin-walled members by using the constrained finite strip method

In this paper a new way to define transverse extension modes is introduced in the context of modal analysis of thin-walled members. The new transverse extension modes are inspired by the strain distributions of global and distortional modes. The new modes are employed here in the constrained finite strip method. The new modes require some change in the interpolation functions of the semi-analytical finite strip method, and this change is briefly discussed. The practical advantage of the newly proposed transverse extension modes is that the artificial stiffness increase, which is characteristic to pure global and/or distortional deformations in constrained analyses, can be compensated by the consideration of the relevant new transverse extension mode(s), as illustrated by numerous proof-of-concept examples in the paper.

Distortional buckling of cold-formed steel rack section under uniform and non-uniform bending—Numerical analysis and DSM

Due to the wide variety of cross-sections and their good mass/strength ratio, cold-formed steel (CFS) components are gaining prominence among steel structures, although this material is more susceptible to local, distortional, and global buckling. The design procedure based on the direct strength method (DSM) presented in some codes such as the Brazilian (NBR), the American Iron and Steel Institute (AISI), and the Australia/New Zealand (AS/NZS), has been well accepted for estimating simply and safely the ultimate strength of beams subject to distortional buckling for many situations. However, more recent studies show that DSM design can lead to unsafe ultimate strength for beams with a high slenderness factor of distortional buckling. This study analyzes the behavior from 768 models developed using the finite element analyses (FEA) with the ABAQUS software. Beams were analyzed with 2 support conditions (SCA and SCB), under uniform major axis bending alone. The selection of the specimens in which the distortional buckling mode is predominant (modal participation analysis) was performed through a linear stability analysis using the GBTUL software. The finite element models used in nonlinear elastoplastic analyses included geometrical initial imperfections and material nonlinearity. A parametric study was developed to investigate the influence of the slenderness factor of distortional buckling (λd) on CFS rack beams’ ultimate strength. The FEA results were compared with DSM results to verify the accuracy of this method to predict distortional ultimate strength. It is shown that, for CFS rack beams subject to uniform and non-uniform bending and distortional buckling with λd > 1.0–1.5, the DSM overestimates the ultimate strength. Modifications were proposed in the DSM to improve the accuracy in predicting the ultimate strength of the analyzed conditions.

Optimization of cold-formed lipped C-section under bending using prediction equations as objective functions

The use of cold-formed sections (CFS) in construction is growing quickly in spite of the difficulty facing the designer in choosing the optimum sections’ proportions of any CFS in order to acquire the most economical section. In this paper, the optimization of CFS C-lipped channel beam will be carried out with respect to their flexural strength calculated using the direct strength method (DSM) given by the American specifications (AISI). The optimization will be carried out by developing an accurate and effective optimization code using particle swarm optimization (PSO). The mathematical model and the optimization technique used in this paper will help in finding the most optimum ratios between the different elements of the C-lipped channel beam that are studied in three groups. Nonlinear regression analysis is performed to obtain proposed equations that predict the flexural capacity of the cross section in two modes of failure: the local buckling mode and the distortional buckling mode. These proposed equations are used as objective functions in the optimization process as they showed that they are capable of predicting the moment capacity of the C-lipped channel effectively. It was found that the most suitable group of the three studied groups is the ratios between different elements of the cross section lies between 15 and 50%.

Investigations of horizontal rotation at the support of bending beams as proof of global distortional buckling mode

Presented research aims to prove the appearance of lateral-distortional buckling mode by observing the beams' deflection characteristics. Eight-layer channel and lip-channel cross-section glass fibre reinforced polymer (GFRP) beams subjected to pure bending in the web plane were discussed. The Finite Element Method and the Semi-Analytical Method were used to perform all calculations. Deflection angles at the beams' ends show that the lateral deflection sharply increases from load close to critical one and in the post-buckling range. It was observed that for simply supported beams subjected to bending, the lateral-distortional global buckling mode is accompanied by rotation at support in the flange plane.In the FEM approach, the lowest buckling mode is usually considered in beams subjected to bending, and the angle of rotation in the load plane corresponding to the critical bending moment is only analysed. However, in the interactive buckling analysis, the horizontal rotation at the supports is accompanied by the global mode, and its impact is often neglected due to its high bifurcation value compared to the lowest mode. Moreover, in the paper, the influence of this mode on the equilibrium paths, the critical buckling load, the load-carrying capacity, distributions of internal membrane forces, and vertical and horizontal angle of rotation was investigated.

Finite element formulation for distortional lateral buckling of I-beams

The present study develops a finite element formulation for the distortional buckling of I- beams. The formulation characterizes the distribution of the lateral displacement along the web height by superposing (a) two linear modes intended to capture the classical non-distortional lateral-torsional behaviour and (b) any number of user-specified Fourier terms intended to capture additional web distortion. All displacement fields characterizing the lateral displacements are taken to follow a cubic distribution in the longitudinal direction. The separation of variables is effectively achieved by exploiting the properties of the matrix Kronecker product. The finite element solution developed is shown to replicate accurately (a) the classical non-distortional lateral-torsional buckling solutions (b) previously developed distortional buckling solutions based on cubic interpolation of the lateral displacement, while (c) providing a basis to assess the effect to commonly omitted higher distortional modes on the predicted critical moments and buckling modes. The solution is then used to conduct a systematic parametric study of over 3900 cases to quantify the reduction in critical moments due to web distortion relative to the classical non-distortional predictions in the case of simply supported beams under uniform loads, point loads, and linear moment gradients, cantilevers, and beams with overhang.

Explicit equations for anti-symmetric distortional buckling of thin-walled lipped channel columns

This paper presents an energy-based method to obtain expressions for the critical length and buckling stress of anti-symmetric distortional mode for lipped channel columns using the Rayleigh quotient method. Displacement functions are assumed for each wall of the cross-section and, by applying compatibility and equilibrium conditions, a close representation of the stability phenomenon is reached with explicit calculation of its warping profile. The derived expressions for the critical length and buckling stress are compared to results obtained with GBTUL in a parametric study providing accurate predictions (proposed/GBTUL stress ratio = 1.011 ± 0.014).

Cold‐Formed Steel Profiles with Multiple Stiffeners Effective Section Calculation Using Alternative Methodology of the EN1993‐1‐3 Standard

SummaryIn this article, the use of paragraph 5.5.1 (7) is applied, only for stiffeners, and the use of the numerical results of linear stability analyses (LSA) obtained by the GBT (Generalized Beam Theory) methods is tested for the section, namely the distortional modes, relating them to the formulas currently present in EC3‐1‐3. For comparison and analysis of results, studies present in the bibliography were used that show experimental and numerical results, relevant and representative, of sections with stiffeners. In the calculation of effective sections, the Class4Section application, developed by the authors, is used to run several tests combining EC3‐1‐3 specifications and LSA results.

A modal decomposition approach for experimental buckling analysis of thin-walled lipped channel columns

This paper presents a method to perform the decomposition of a lipped channel column’s general deformed shape into the three main buckling modes: local, distortional and flexural global modes. Simple shape functions for each buckling mode are assumed and through an error minimization process the participation of each mode in the deformed shape of the column can be assessed. In a first step, the method is validated against numerical results using generalized beam theory (GBT) modal decomposition for columns undergoing L-D interaction. It is shown that the proposed technique provides good estimates for each buckling modes that contributes to the deformed shape. Lastly, an experimental column previously tested by the authors is analyzed using the proposed modal decomposition method. Although simpler than GBT and the constrained finite strip method (cFSM), the obtained results show that the proposed method can be a valuable and simple tool for a quick assessment of columns’ behavior, especially experimental specimens.

Joint modelling in advanced thin-walled beam models

A methodology for thin-walled frame analysis using displacement modes of beams at joint interfaces has recently been proposed. This paper introduces an implementation that uses a specific advanced beam element based on semi-analytical beam mode solutions of a generalised beam theory. The end cross-sections of the beam elements consist of wall elements with displacement degrees of freedom that can be joined to conventional finite shell element models of the joints. However, before joining the beam element and joint model at the interface, both beam and joint model are transformed into cross-section displacement mode degrees of freedom. This transformation enables the use of a reduced mode space at the interface as well as in the beam. The quadratic nature of the beam mode equations of the beam element leads to a non-unique number of beam displacement modes. However, this paper introduces an exact mode selection technique that leads to a unique set of modes. Necessary details of the beam element formulation are included in order to make a self-contained description of the mode selection technique. Mode reduction through a length scale dependent choice of the number of exponentially decaying distortional modes is investigated and discussed for three types of joint design in four examples. The three joint design types investigated are based on in-plane membrane action, flexural out-of-plane action and a combination of both membrane and flexural action.

Axial Compressive Strength of steel members made of cold formed angle sections

AbstractCold formed sections often fail in three basic modes which are local, distortional, and global. Local and distortional modes are characterized by entire deformations, whether translation or bending of the plates forming the section. Current design codes determine the ultimate strength of short to intermediate cold formed angle columns based on the local‐global interactive failure modes. However, these cold formed angles sections fail in flexural‐torsional mode. Thus, the predicated ultimate loads are either unsafe or excessively safe depending on boundary conditions. Therefore, in this study the ultimate axial strength of pined‐pined columns composed of cold formed angles subjected to axial compressive force are determined numerically. A nonlinear finite element model has been developed using ABAQUS software. Four nodes isoparametric shell element that allows both geometric and material non‐linarites is used. The effect of geometric imperfections is included in the model. The parameters studied include, the angle leg width‐thickness ratios, overall member slenderness ratios. Further, Lipped as well as plain angles are considered in the study. Finally, results have been compared with the predicted strengths determined by the American specifications and the newly developed direct strength method, DSM.