A new approach for thin-walled member analysis in the framework of GBT

A new approach is illustrated for the cross-sectional analysis to be performed in the context of the Generalised Beam Theory (GBT). The novelty relies in formulating the problem in the spirit of Kantorovich’s semi-variational method, namely using the dynamic modes of an unconstrained planar frame as in-plane deformation modes. Warping is then evaluated from the post-processing of these in-plane modes, thus reversing the strategy of the classical GBT procedure. The new procedure does not require several steps of the classical algorithm for the determination of the conventional modes, in which bending, shear and local modes are evaluated separately, and is applicable indifferently to open, partially-closed and closed sections. The efficiency and ease of use of the method are outlined by means of two examples, aimed to describe the linear–elastic behaviour of thin-walled members.     

一种基于广义梁理论的薄壁构件分析的新方法

广义梁理论中的横截面分析证实了一种新的薄壁构件分析方法。这个创新依赖于康托罗维奇的变分法,即采用无约束平面框架动力模式取代平面内变形的模式。通过平面内模型的后处理程序计算构件的翘曲变形,这与经典的广义梁理论的做法相反。新的计算程序比经典算法的步骤要少,可以分别计算弯曲、剪力和局部模态,并且适用于开口、部分封闭和封闭截面。通过两个实例说明了该方法对薄壁结构的线弹性性能分析的有效性和简易性。     

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.     

On the shear deformation modes in the framework of Generalized Beam Theory

This paper extends previous work concerning the determination of cross-section deformation modes in thin-walled members with arbitrary polygonal cross-section, in the framework of Generalized Beam Theory (Gonçalves et al., 2010 [1]). In particular, the paper addresses the so-called “natural shear deformation modes” (i.e. the deformation modes that involve non-null membrane shear strains and are independent of the cross-section discretization employed), which are relevant for capturing the behaviour of thin-walled members with complex multi-cell cross-sections undergoing torsion and/or distortion. The contributions of the paper are (i) the derivation of fundamental properties of the shear modes, (ii) the proposal of an efficient mode extraction procedure and (iii) the development of analytical results for several particular cases. In order to illustrate the application of the proposed mode extraction procedure and demonstrate the validity of the derived properties, several cross-sections are analyzed, including complex multi-cell tubes.     

GBT formulation to analyse the behaviour of thin-walled members with variable cross-section

The generalised beam theory (GBT) provides a general solution for the linear/non-linear analysis of prismatic thin-walled structures, using bar elements capable of describing the cross-section rigid-body motions and distortions. Nowadays GBT is fully developed for thin-walled members having a large variety of constant cross-sections. This paper provides the extension of GBT for the special case of thin-walled members with variable open cross-section and the limits of its applicability.     

GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections

This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and (ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip or shell finite element analyses.     

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.     

Global buckling analysis of plane and space thin-walled frames in the context of GBT

This paper reports research work concerning the use of Generalised Beam Theory (GBT) to analyse the global buckling behaviour of plane and space thin-walled frames. Following a brief overview of the main concepts and procedures involved in the performance of a GBT buckling analysis, one presents in detail the formulation and numerical implementation of a GBT-based beam finite element that includes only the first four (rigid-body) deformation modes — namely, one describes (i) the kinematical models developed to simulate the warping transmission at frame joints connecting two or more non-aligned U- and I-section members, (ii) the procedures adopted to handle the effects stemming from the non-coincidence of the member centroidal and shear centre axes (cross-sections without double symmetry), and (iii) the definition of joint elements, which involves providing a relation between the connected member GBT degrees of freedom and the joint generalised displacements. Finally, one presents and discusses numerical results that make it possible to illustrate the application and show the capabilities of the above GBT-based finite-element formulation and implementation. For validation purposes, the GBT-based results (critical buckling loads and mode shapes) are also compared with values yielded by shell (mostly) and beam finite element analyses carried out in the code ANSYS.     

对广义梁理论中的平面和空间薄壁构件的整体屈曲分析

采用广义梁理论（GBT）对平面和空间薄壁构件进行整体屈曲分析。简要概述主要概念和广义梁理论屈曲分析中所采用的程序。基于广义梁理论的梁有限元分析考虑了4种刚体变形模式,即：i）运动学模型,用于模拟连接2个或更多U型和I型构件节点的翘曲传输作用;ii）采用程序处理构件重心和剪力中心轴（横截面非双向对称）不重合的效应;iii）节点构件的定义,这些节点可提供连接构件的自由度和节点广义位移之间的联系。最后,介绍并讨论数值结果,其可能有助于基于广义梁理论的有限元公式的制定和实施。并将基于GBT分析的结果（临界屈曲载荷和模态）与ANSYS程序中壳单元和梁单元的建模分析结果进行了对比验证。     

Modal decomposition of thin-walled member collapse mechanisms

Following recent investigations on the decomposition of elastic buckling modes into combinations of structurally meaningful deformation modes, this work presents a novel extension of the above procedure to elastic–plastic collapse mechanisms and highlights the relevant role that this concept may play in the mechanical knowledge/interpretation of thin-walled member failures. In order to achieve the sought decomposition, a code based on a Generalised Beam Theory (GBT) formulation developed to perform first-order elastic–plastic analyses of thin-walled members is employed. Five illustrative examples are presented and discussed, and the results displayed, namely load-deflection curves, deformed configurations and stress contours, are validated through the comparison with values provided by shell finite element analyses. The most relevant modal results addressed consist of (i) load-deflection curves determined on the basis of pre-selected deformation mode sets, (ii) modal participation diagrams and (iii) modal amplitude functions. These results make it easy to characterise and interpret the mechanics associated with the thin-walled member elastic–plastic failures (as well as with the various loading stages), which may be of great importance in the improvement/development of existing/new design methods (e.g, yield-line theory, direct strength method).     

A beam model for large displacement analysis of flexibly connected thin-walled beam-type structures

This paper presents a beam formulation for large displacement analysis of beam-type structures with flexible connections. Within the framework of updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping and the second-order displacement terms due to large rotations, the equilibrium equations of a straight beam element are firstly developed. Due to the nonlinear displacement field, the geometric potential of semitangential moment is obtained for both the internal torsion and bending moments, respectively. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends. To account for the flexible connection behaviour, a special transformation procedure is developed. The numerical algorithm is implemented in a computer programme and its reliability is validated trough several test examples.     

通过有限元法计算受压构件的纯畸变弹性屈曲荷载

基于有限元法,提出一个分析方法,可以计算纯畸变弹性屈曲荷载。计算中采用的有限元模型为非耦合屈曲变形模式。具体方法分为2步:第1步,采用一般梁理论(GBT)分析构件横截面,在此可以给有限元模型施加约束条件;第2步,对受约束有限元模型进行线性屈曲分析,确定纯畸变荷载。将此法应用于开口薄壁构件和冷弯构件,得到的畸变荷载非常准确,与一般梁理论和约束有限条法(cFSM)计算的荷载值一致。     

A unified energy formulation for the stability analysis of open and closed thin-walled members in the framework of the generalized beam theory

Generalized beam theory—GBT—is among the most adequate tools for the analysis of thin-walled prismatic elements. It enables the analysis of the distortion of the element cross-section and local buckling of individual walls in a unified manner that incorporates the results from classical bending theory. The basis of this theory was developed in the 1960s by Schardt for first and second order elastic behaviour of thin-walled members.Open and closed thin-walled members present the distinctive difference of the unknown shear flow that characterizes the latter. More specifically, shear strains must follow an elasticity law, as opposed to the simplifying assumptions for open cross-sections.It is the purpose of the present paper to present a unified energy formulation for the non-linear analysis of both open and closed sections in the framework of GBT, able to deal with all modal interaction phenomena between local plate behaviour, distortional behaviour and the more classical global (flexural, torsional and flexural–torsional) response. Finally, an application to the stability analysis of a compressed thin-walled column is presented and discussed.     

A Koiter's perturbation strategy for the imperfection sensitivity analysis of thin-walled structures with residual stresses

This paper suggests a strategy for the imperfection sensitivity analysis of elastic thin-walled structures with notable residual stresses. The analysis is carried out by means of a Koiter's perturbation approach. The concept of imperfection, traditionally associated with geometric and load factors, is extended in this paper to the residual stresses. The strategy is implemented in a FEM code. A comparison of the obtained results allows a discussion on the accuracy and the influence of the different coefficients connected to the asymptotic analysis of the residual stresses.     

A model for warping transmission through joints of steel frames

A simple approach is developed in this paper which considers the effect of partial warping continuity through the joints of thin-walled steel frames when using beam finite element analysis. Using a condensed stiffness matrix for the joint generated by the substructuring technique, warping springs are introduced to represent the condition of partial warping restraint at intersections between members. The performance of the proposed model is demonstrated through a number of numerical examples. Excellent agreement is achieved between the results of beam finite element analysis using the suggested joint model and accurate shell finite element analysis.     

A probabilistic formulation of the damage evolution law

The inhomogeneous character of composites has many consequences regarding their behaviour. Using a two-level approach (micro- and macro-), a scalar is constructed where the damage of a representative volume of material is no longer considered as deterministically known, but results from a probabilistic formulation. A probabilistic elastic—brittle behaviour is assumed at the micro-level and the evolution law of damage results as a sum, through a loose parallel bundle of size N_t (Daniel's model) over all the micro-states. The approach developed appears as an intermediate model between those arising in the fields of Damage Mechanics and Probabilistics Brittle Fracture.

The model has been implemented in a finite element code and some significant results are presented. A concrete specimen is loaded in pure compression and pure tension, with the external boundary conditions remaining deterministic. The interaction between the external loading and the inhomogeneities creates some anisotropy and it is found that the specimen behaves more like a structure than like a theoretical volume of material.     

The development of the direct strength method for welded steel members with buckling interactions

The direct strength method (DSM) has been adopted by the NAS (2004) and AS/NZS 4600 (2005) for the design of cold-formed steel members. The method can be successfully applied to the design of welded and hot-rolled sections. This paper reviews the development of the DSM for welded steel structural members. The design strength formulae for welded section columns and beams for the DSM are provided based on the tests performed on welded H-section, C-section, circular and rectangular hollow section columns fabricated from steel plates whose nominal yield stress is 235 MPa or 315 MPa. The comparison between the design strength of welded sections predicted by the DSM and that estimated by existing specifications is provided. This paper verifies that the DSM which adopts the nominal axial strength and flexural strength in the AISC (2010) or EC3 (2004) can properly predict the ultimate strength of welded section columns and beams.     

Homogeneous generalized yield criterion based elastic modulus reduction method for limit analysis of thin-walled structures with angle steel

A homogeneous generalized yield criterion (HGYC) expressed by piecewise polynomial is given for angle section. The element bearing ratio (EBR), reference EBR, and uniformity of EBR are defined in term of the HGYC. Then, a HGYC based elastic modulus reduction method for limit analysis of thin-walled structures with angle steel is presented based on the modulus adjustment strategy established on the EBR and the conservation criterion of energy. The method proposed can overcome the disadvantage of the conventional elastic modulus adjustment procedure where the nonhomogeneous generalized yield criterion is employed and the proportional loading condition is not satisfied.     

Structural characteristics of Hagia Sophia: II—A finite element formulation for dynamic analysis

The Hagia Sophia, a 6th century masonry edifice in Istanbul, is analysed with a linear finite element model including thickness shear deformations, the terms z/R, and rotatory inertia effects with consistent mass matrix to provide insight to the structure's behaviour under earthquake loading. The structure, including the primary main dome and supporting system, is modelled by using a two-dimensional model composed of eight-node shell element. The dynamic characteristics of the system are obtained by using the Wilson-θ method, step-by-step integration method, and also by applying the available ground motion data to understand its general structural behaviour in the future earthquakes.     

A shear deformable theory for the analysis of steel beams reinforced with GFRP plates

A shear deformable thin-walled beam theory is developed for the analysis of steel beams reinforced with a GFRP plate to one of the flanges. Starting with the principle of stationary potential energy, the governing equilibrium equations and boundary conditions are formulated for the problem. The theory results in two sets of fully coupled systems of equilibrium equations. The first system describes the longitudinal-flexural response of the system and involves four generalized displacement fields and the second system governs the lateral-torsional response and involves six generalized displacement fields. The resulting coupled systems are then solved numerically for practical problems. Detailed comparisons with three dimensional and shell solutions under ABAQUS show that the present theory provides reliable predictions for displacements and stresses. A comparison with results from a non-shear deformable theory illustrates the necessity of incorporating shear deformation effects in cases involving predominantly twisting responses.