Analytical method for thin-walled members of closed sections

In a previous paper,¹ based on a mixed variational principle, an analytical method for thin-walled members of open cross-section in general bending and torsion has been derived. In this paper the solution of thin-walled members of closed sections will be presented.     

Buckling analysis of thin-walled sections under general loading conditions

This paper presents an investigation of the buckling behaviour of thin-walled sections subjected to general loading conditions. The semi-analytical finite strip method is used. The existing results are only for sections subjected to a uniform loading, namely: uniform compression, uniform bending and uniform distributed loads, which are applied at the shear centre. For a general loading condition, we proposed the realizing linear analysis first to give longitudinal stresses. The stiffness matrix is provided in the standard manner. Each strip is divided into cells and longitudinal stresses are recorded in these cells. The integrations are performed on each cell domain and the sum of them provides the geometric matrix of the strip.     

高强冷弯薄壁型钢卷边槽形截面构件设计可靠度分析

基于已有的承载力试验研究结果,对屈服强度550MPa高强冷弯薄壁型钢中常用的卷边槽形截面轴压构件和偏压构件的计算模式不定性进行了分析,并统计分析了高强冷弯薄壁型钢强度不定性、几何特性不定性。在此基础上,采用改进一次二阶矩方法,按现有规范的抗力分项系数要求,计算了高强冷弯薄壁型钢卷边槽形截面轴压构件和偏压构件不同可能荷载组合下的可靠指标。结果表明：对于宽厚比符合规范要求的屈服强度550MPa高强冷弯薄壁型钢卷边槽形截面轴压构件和偏压构件,按现有规范的抗力分项系数计算得到的可靠指标均能满足目标可靠指标的要求,证明了所采用的承载力计算方法的适用性;但对于宽厚比超出规范要求的轴压和偏压构件,计算得到的可靠指标不能满足目标可靠指标的要求。     

Buckling analysis of thin-walled cold-formed steel structural members using complex finite strip method

In this paper, a generalised complex finite strip method is proposed for buckling analysis of thin-walled cold-formed steel structures. The main advantage of this method over the ordinary finite strip method is that it can handle the shear effects due to the use of complex functions. In addition, distortional buckling as well as all other buckling modes of cold-formed steel sections like local and global modes can be investigated by the suggested complex finite strip method. A combination of general loading including bending, compression, shear and transverse compression forces is considered in the analytical model. For validation purposes, the results are compared with those obtained by the Generalized Beam Theory analysis. In order to illustrate the capabilities of complex finite strip method in modelling the buckling behavior of cold-formed steel structures, a number of case studies with different applications are presented. The studies are on both stiffened and unstiffened cold-formed steel members.     

Buckling analysis of thin-walled sections under localised loading using the semi-analytical finite strip method

Thin-walled sections under localised loading may lead to web crippling of the sections. This paper develops the Semi-Analytical Finite Strip Method (SAFSM) for thin-walled sections subject to localised loading to investigate web crippling phenomena. The method is benchmarked against analytical solutions, Finite Element Method (FEM) solutions, as well as Spline Finite Strip Method (SFSM) solutions. The paper summarises the SAFSM theory then applies it to the buckling of plates, and channel sections under localised loading. Multiple series terms in the longitudinal direction are used to compute the pre-buckling stresses in the plates and sections, and to perform the buckling analyses using these stresses. Solution convergence with increasing numbers of series terms is provided in the paper. The more localised the loading and buckling mode, the more series terms are required for accurate solutions. The loading cases of Interior One Flange (IOF) and Interior Two Flange (ITF) are investigated in this paper using simply supported boundary conditions.     

加劲钢柱腹板受压稳定性分析

基于大型通用软件ANSYS,分析了6种加劲肋参数下的钢柱弹性屈曲特性,由分析结果得到了加劲肋的厚度越大,对腹板的约束越大,并有效的提升了钢柱腹板的一阶临界屈曲应力系数,120 mm宽的加劲肋由于自由伸出长度过大,会发生过早的屈曲,而导致腹板屈曲失效。     

Buckling mode identification of thin-walled members by using cFSM base functions

The objective of this paper is to demonstrate an approximate method whereby eigen buckling modes from a shell finite element method (FEM) analysis of a thin-walled member can be quantified in terms of the fundamental buckling classes, namely, global, distortional, local, or other. The buckling classes are defined using the mechanical definitions employed in the constrained finite strip method (cFSM). The cFSM base vectors are used to approximate an arbitrary FEM buckling mode. The resulting identification and its associated error is investigated, including dependency on FEM discretization, the number of cFSM functions considered, boundary conditions, and loading. The long-term goal of the work is to provide a generalized method for identification of local, distortional, and global buckling modes for arbitrary thin-walled members modeled in general purpose finite element codes.     

Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections

In this paper, the analysis of the behavior of thin-wa lled beams, derived from Proki ’s work, is carried out by using beam theory with a single warping function valid for arbitrary form of cross sections, without any distinction between open and closed profiles and without using sectorial coordinates. The finite element method is used and numerical examples show the accuracy of the solution by comparison with other numerical or analytical results. For the stability analysis, analytical and numerical calculations of critical loads are given for beams submitted to bending moment and centrally applied forces. Equilibrium equations are established from the principle of virtual work. Critical loads are calculated by considering that a structure already in equilibrium reaches instability if there is one or more than one equilibrium position for the same loading. Results with this formulation are compared to those obtained with classical warping functions.     

General analysis of asymmetric thin-walled members

This paper presents the closed-form solution for the deflection and stresses of an asymmetric thin-walled member. The method is based on the assumption that the cross-section can deform out of plane when warping and shear-lag effects are significant. The out-of-plane deformation is represented by a linear warping function plus a truncated series of complete eigenfunctions. The differential equations are derived by using the principle of Minimum Potential Energy and solved by a symbolic manipulator. An example is used to illustrate the application of the method and the use of the closed-form solutions obtained. Results of the example are compared to the results of a finite element analysis and other approximate models. The comparison indicates that the maximum error of the proposed method is within 0·1% of the value obtained by the finite element method in lateral displacement and 0·2% in axial displacement.     

Buckling mode identification of perforated thin-walled members by using GBT and shell FEA

The paper presents an original method based on the Generalised Beam Theory (GBT) whereby the general buckling modes, provided by the shell Finite Element Analysis (SFEA) of perforated thin-walled members, are expressed in terms of the fundamental (pure) buckling types (global, distortional and local). The contribution of each pure buckling mode to a coupled instability can be quantified, allowing a better understanding of the member buckling behaviour and post-buckling strength reserve. The main advantage of this method lies in using only the GBT cross-sectional pure deformation modes instead of member pure modal shapes. There are no restrictions regarding the element cross-sectional shape, loading and boundary conditions.     

Structural analysis of assemblages of thin-walled members

A matrix method for the analysis of structural systems composed of thin-walled members is presented. The matrix displacement analysis includes the effects of thin-walled non-uniform torsion theory, cross-section asymmetry, eccentric restraint as well as joint types peculiar to thin-walled members. The method is used for a prediction of the elastic behaviour of a set of representative test frames. The test frames were pitched-roof portals constructed from channel sections bent about their major-axis and supported by eccentric restraints simulating purlins and girts.     

GBT-based buckling analysis of thin-walled members with non-standard support conditions

This paper reports on the use of a recently developed Generalised Beam Theory (GBT) formulation, and corresponding finite element implementation, to analyse the local and global buckling behaviour of thin-walled members with arbitrary loading and support conditions — this formulation takes into account longitudinal normal stress gradients and the ensuing pre-buckling shear stresses. After presenting an overview of the main concepts and procedures involved in the performance of a GBT-based (beam finite element) member buckling analysis, one addresses in detail the incorporation of non-standard support conditions, such as (i) full or partial localised displacement or rotation restraints, (ii) rigid or elastic intermediate supports or (iii) end supports corresponding to angle connections. In order to illustrate the application and capabilities of the proposed GBT-based approach, one presents and discusses numerical results concerning cold-formed steel (i) lipped channel beams and (ii) lipped I-section beams and columns with various “non-standard” support conditions — while the beams are acted by uniformly distributed or mid-span point loads, applied at the shear centre axis, the columns are subjected to uniform compression. In particular, it is possible to assess the influence of the different support conditions on the beam and column buckling behaviour (critical buckling loads and mode shapes). For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the code Ansys.     

Coupled stability analysis of thin-walled composite beams with closed cross-section

For the coupled stability analysis of thin-walled composite beam with closed cross-section subjected to various forces such as eccentric constant axial force, end moments, and linearly varying axial force, the efficient numerical method to evaluate the element stiffness matrix is newly presented based on the homogeneous form of simultaneous ordinary differential equations. The general bifurcation type of buckling theory for thin-walled composite box beam is developed based on the energy functional corresponding to semitangential rotations and semitangential moments. The coupled stability equations including variable coefficients and the force–displacement relationships are derived from the energy principle and explicit expressions for displacement functions are presented based on power series expansions of displacement components. The element stiffness matrix is evaluated by applying member force–displacement relationships to these displacement functions. In addition, the finite element model based on the cubic Hermitian interpolation polynomial is presented. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the finite element solutions using the Hermitian beam elements and the available results from other researchers. Particularly, the influence of the eccentricity and the force ratio of axial forces, the fiber orientation, and the boundary conditions on the buckling behavior of composite box beam are parametrically investigated. Also the emphasis is given in showing the phenomenon of buckling mode change.     

Buckling modes identification from FEA of thin-walled members using only GBT cross-sectional deformation modes

This paper presents the latest developments of an original method based on Generalized Beam Theory (GBT) capable to identify the fundamental deformation modes of global, distorsional or local nature, in general buckling modes provided by the shell finite element analysis (FEA) of isotropic thin-walled members. This method has the advantage of using only the GBT cross-sectional deformation modes instead of the member base mode shapes. The participation of each fundamental buckling mode can be calculated, allowing an accurate and quantitative evaluation of the coupled instability. There are no restrictions regarding the element cross-sectional shape, loading and quite recently discovered, boundary conditions.     

Lateral buckling analysis of thin-walled laminated composite beams with monosymmetric sections

Lateral buckling of thin-walled composite beams with monosymmetric sections is studied. A general geometrically nonlinear model for thin-walled laminated composites with arbitrary open cross-section and general laminate stacking sequences is given by using systematic variational formulation based on the classical lamination theory. All the stress resultants concerning bar and shell forces are defined, and nonlinear strain tensor is derived. General nonlinear governing equations are given, and the lateral buckling equations are derived by linearizing the nonlinear governing equations. Based on the analytical model, a displacement-based one-dimensional finite element model is developed to formulate the problem. Numerical examples are obtained for thin-walled composite beams with monosymmetric cross-sections and angle-ply laminates. The effects of fiber orientation, location of applied load, modulus ratio, and height-to-span ratio on the lateral buckling load are investigated. The torsion parameter and a newly-defined composite monosymmetry parameter are also investigated for various cases.     

Torsional analysis of thin-walled composite beams with single- and double-celled sections

The exact solutions for twist angle and fiber stresses of thin-walled composite box beams with single- and double-celled sections subjected to torsional moment are presented by introducing fourteen displacement parameters. For this, a general thin-walled composite box-beam theory including the effects of elastic couplings and restrained warping is developed based on the Vlasov’s assumptions. The equilibrium equations and the force–deformation relations are derived from the energy principle. A system of linear algebraic equations with non-symmetric matrices is constructed by introducing fourteen displacement parameters and by transforming the higher order simultaneous differential equations into first-order ones. This numerical technique determines eigenmodes corresponding to multiple zero and non-zero eigenvalues and derives exact displacement functions for displacement parameters based on the undetermined parameter method. Finally, the exact stiffness matrix is determined using the member force–deformation relations. The theory developed by this study is validated by comparing several torsional responses from the present approach with those from the finite element beam model that uses third-order Hermitian polynomials and detailed two-dimensional analysis results using the shell elements of ABAQUS for coupled composite beams with single- and double-celled sections.     

Buckling analysis of thin-walled sections by semi-analytical Mindlin–Reissner finite strips: A treatment of drilling rotation problem

Drilling rotation problem may arise if the Mindlin–Reissner plate theory is adopted to develop the semi-analytical finite strip method for the analysis of thin-walled sections. Through the introduction of a fictitious in-plane shear strain in the standard deformation–displacement relation of the Mindlin–Reissner plate theory, a drilling rotation is added. This strategy is applied for 2-nodal, 3-nodal and 4-nodal line strips in dealing with buckling problems of thin-walled sections. Numerical results show the potential of the proposed technique.     

Flexural–torsional post-buckling analysis of thin-walled elements with open sections

The post-buckling analysis of thin-walled elements under compression is investigated. A nonlinear model is developed by using nonlinear relationships between curvatures and bending moments. Warping and shortening effects are considered in the torsion equilibrium equation. Based on Galerkin's method, a nonlinear algebraic system is obtained for simply supported boundary conditions. The three resulting equations in bending and torsion are highly coupled and the Newton–Raphson algorithm with displacement control is adopted for the solution. The post-buckling equilibrium curves are obtained for various sections shapes, such as bisymmetric and monosymmetric sections. The importance of the shortening effect is outlined.     

Shear lag analysis for thin-walled members by displacement method

The classical theory of thin-walled members is unable to reflect the shear lag phenomenon since it is based on the assumption of no shear deformation.This paper presents a displacement method based on the potential variational principle and discusses the shear lag phenomenon. Compared with the mixed method, the displacement method is simpler in the analysis process and hence is easier to apply and to be combined with the finite element method. Numerical examples show that quite good results can be obtained. The solution also includes the results from the classical theory.     

GBT-based first-order analysis of elastic-plastic thin-walled steel members exhibiting strain-hardening

This paper addresses the development and illustrates the application of a generalised beam theory (GBT) formulation intended to perform first-order elastic-plastic analyses of thin-walled members made of isotropic non-linear materials exhibiting strain-hardening. After presenting an overview of the main concepts and procedures involved in the above GBT formulation, its application is illustrated through the analysis of (i) simply supported Z-section beams and (ii) fixed-ended lipped channel beams. In both cases, a bilinear elastic-plastic material model is adopted, which exhibits three strain-hardening levels, namely E_sh = 0 (perfectly plastic model), E_sh = E/100 and E_sh = E/50. The results presented and discussed consist of equilibrium paths, modal participation diagrams, displacement profiles, beam deformed configurations and stress diagrams and contours. For validation purposes, most of the GBT results are compared with values obtained from shell finite element analyses ? with a few relatively minor exceptions, a very good correlation is always found. Finally, the paper closes with some remarks concerning the influence of the strain-hardening slope on the structural behaviour of thin-walled steel beams.