Buckling analysis of thin-walled open members—A complementary energy variational principle

This first of two companion papers develops a new variational principle for the buckling analysis of thin-walled members based on the principle of stationary complementary energy. Some of the aspects of the Vlasov thin-walled beam theory (the rigid cross section assumption, and the stress expressions) are postulated to describe the behavior of members while other aspects of the theory (i.e., the zero shear strain assumption at mid-surface) are discarded. Koiter's formulation based on polar decomposition theory in finite elasticity is adopted to formulate expressions for statically admissible stress resultant fields. The stationarity conditions of the complementary energy expression are then evoked to yield the conditions of neutral stability and associated boundary conditions in which the rotation fields appear explicitly. The formulation seamlessly incorporates shear deformation effects and load position effects. Also, the Wagner effect and the mono-symmetry property which arise in displacement based formulations arise in the present formulation in a natural way.     

Buckling analysis of thin-walled open members—A finite element formulation

In a companion paper, a variational principle based on the principle of stationary complementary energy was developed for the buckling analysis of thin-walled members with open cross-sections. In this paper, the variational principle is adopted to formulate a finite element buckling solution. The formulation successfully incorporates shear deformation effects, a feature that is neglected in most available buckling solutions. By adopting a non-orthogonal coordinate system, the solution successfully captures the transverse load position effect relative to the shear center. A series of examples demonstrate the validity of the finite elements formulated and their applicability to a wide variety of buckling problems. Examples include column flexural and torsional buckling, lateral torsional buckling of beams with a variety of end conditions and subjected to a variety of moment gradients. The formulation is shown to be applicable to beams with mono-symmetric sections. In all cases, the validity of the new solution is assessed and established through comparisons to well-established closed-form and/or numerical solutions.     

Variationally-based theories for buckling of partial composite beam-columns including shear and axial effects

This paper is focused on elastic stability problems of partial composite columns: the conditions for the axial load not to introduce any pre-bending effects in composite columns; the equivalence, similarities and differences between different sandwich and partial composite beam theories with and without the effect of shear, with and without the effect of axial extensibility, and also the effect of eccentric axial load application. The basic modelling of the composite beam-column uses the Euler-Bernoulli beam theory and a linear constitutive law for the slip. In the analysis of this reference model, a variational formulation is used in order to derive relevant boundary conditions. The specific loading associated with no pre-bending effects before buckling is geometrically characterized, leading to analytical buckling loads of the partial composite column. The equivalence between the Hoff theory for sandwich beam-columns, the composite action theory for beam-columns with interlayer slip and the corresponding Bickford-Reddy theory, is shown from the stability point of view. Special loading configurations including eccentric axial load applications and axial loading only on one of the sub-elements of the composite beam-column are investigated and the similarity of the behaviour to that of imperfect ordinary beam-columns is demonstrated. The effect of axial extensibility on kinematical relationships (according to the Reissner theory), is analytically quantified and compared to the classical solution of the problem. Finally, the effect of incorporating shear in the analysis of composite members using the Timoshenko theory is evaluated. By using a variational formulation, the buckling behaviour of partial composite columns is analysed with respect to both the Engesser and the Haringx theory. A simplified uniform shear theory (assuming equal shear deformations in each sub-element) for the partial composite beam-column is first presented, and then a refined differential shear theory (assuming individual shear deformations in each sub-element) is evaluated. The paper concludes with a discussion on this shear effect, the differences between the shear theories presented and when the shear effect can be neglected.     

Elastic buckling load of multi-story frames consisting of Timoshenko members

The objective of this paper is to propose a method for the evaluation of the elastic critical buckling load of columns in frames consisting of members susceptible to non-negligible shear deformations, such as built-up members in steel frames, based on Engesser's approach. To that effect, a stability matrix is proposed and three general stability equations are derived for the cases of unbraced, partially braced and braced frames. Indicative graphic interpretation of the solutions for the stability equations of the braced and unbraced cases is shown. Slope-deflection equations for shear-weak members with semi-rigid connections are also derived and used for the presentation of a complete set of rotational stiffness coefficients, which are then used for the replacement of members converging at the bottom and top ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members, as well as the eventual presence of axial force, are considered. Five examples are presented, dealing with braced, unbraced and partially braced frames, with rigid and semi-rigid beam to column connections, loaded with concentrated or uniformly distributed loads, in a symmetrical or non-symmetrical pattern. In all cases the proposed approach is in excellent agreement with finite element results.     

薄壁受压构件的非弹性有限条法

最近几年,证实了有限条法(FSM)分析薄壁结构构件屈曲模态的有效性。提出了一个能估算均匀压力下无初始曲率薄壁非弹性构件的屈曲应力的FSM公式。在该公式中,利用塑性流方程考虑了塑性,得到了一个考虑非弹性剪切刚度的表达式,避免了之前的塑性流理论关于屈曲模型问题的缺陷。     

Effective section method: A general direct method for the design of steel cold-formed members under local–global buckling interaction

Thin-walled steel cold-formed members usually display local–global buckling interaction which strongly affects the structural strength of columns and beams. The local bucking of slender folded sections develops in analogy with single plate buckling, including the interaction between the plate elements of the cross-section and can be identified with appropriate first-order stability analysis and the consequent results of the critical buckling loads and the associated modes. The effective section method, ESM, as an extension of the original effective area method, EAM, was conceived for the design of cold-formed members on the basis of the actual local buckling results of the section, together with calibrated formulations for column and beam resistance. The strength equations were taken from the direct strength method, DSM, as it is presented in the North American AISI standard. In addition, as a consequence of its proposed formulation the effective section method is able to be applied side-by-side with the traditional effective width method, EWM, allowing its inclusion in the main part of the codes and specifications for the design of cold-formed members and improving its dissemination. The ESM was proposed in combination with equations and tables that enables designers to directly access the critical local buckling compressive force and bending moment of usual sections, resulting in a contribution that improves the attractiveness of the method.     

Shear buckling characteristics of cold-formed steel channel beams

Cold-formed steel members are increasingly used as primary structural elements in the building industries around the world due to the availability of thin and high strength steels and advanced cold-forming technologies. Cold-formed lipped channel beams (LCB) are commonly used as flexural members such as floor joists and bearers. However, their shear capacities are determined based on conservative design rules. For the shear design of LCB web panels, their elastic shear buckling strength must be determined accurately including the potential post-buckling strength. Currently the elastic shear buckling coefficients of LCB web panels are determined by assuming conservatively that the web panels are simply supported at the junction between their flange and web elements. Hence finite element analyses were conducted to investigate the elastic shear buckling behavior of LCBs. An improved equation for the higher elastic shear buckling coefficient of LCBs was proposed based on finite element analysis results and included in the ultimate shear capacity equations of the North American cold-formed steel codes. Finite element analyses show that relatively short span LCBs without flange restraints are subjected to a new combined shear and flange distortion action due to the unbalanced shear flow. They also show that significant post-buckling strength is available for LCBs subjected to shear. New equations were also proposed in which post-buckling strength of LCBs was included.     

Ultimate capacity of battened columns composed of four equal slender angles

Cold-formed steel structural members play a great role in modern steel structures due to their high strength and light weight. The behavior and strength of battened column members composed of slender angle sections are mainly governed by local buckling of angle legs or torsional buckling of the angle between batten plates. Moreover, local buckling depends on the interaction between the width–thickness ratio of angle leg, overall slenderness ratio of angle between batten plates and overall slenderness of column. Theoretical study has been carried out by a nonlinear material and geometrical finite element model. Numerous cases of slender battened column sections having different width–thickness angle leg ratios, overall slenderness ratios between batten plates and overall slenderness ratios are chosen in this study. Complete ultimate strength curves are drawn and different failure modes are studied by taking different member lengths, which produce local or torsional buckling of single angles between batten plates or overall buckling of the member. Empirical equations for the effect of shear deformation factor and the ultimate axial load capacities of members formed of battened slender angle sections are proposed. Strengths of axially loaded battened members predicted using finite element as well as the proposed empirical equations is compared with the design strengths obtained using North American and European codes. It is concluded that the design strengths predicted by North American and European codes are generally conservative, and these design rules have been shown to be reliable using reliability analysis.     

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.     

Flexural–torsional buckling of thin-walled beam members based on shell buckling theory

This paper summarized currently available techniques of setting up flexural–torsional buckling theory of thin-walled members. It is found that all the existing methods introduced a nonlinear load potential in their total potentials, while based on the classical variational principle for stability of a solid structure, no such load potential should be included. This situation has led to an inconsistency between some widely referenced monographs in buckling theories of beams with mono-symmetrical cross-sections.

This paper provides a new theory for flexural–torsional buckling of thin-walled members based on the classical variational principle and the theory for thin-walled shells. No nonlinear load potential is included, but a new term: nonlinear strain energy from transverse stresses, which has been neglected in previous theories of thin-walled members, is introduced in. It is found that the nonlinear load potential is not equivalent to the contribution of transverse stresses for beams with mono-symmetrical cross-sections, which causes the inconsistency mentioned above.

The comparison shows that the proposed theory and the traditional theory are the same for most cases encountered in practice.     

Thermal buckling of rotationally restrained steel columns

The in-plane elastic buckling of a steel column under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or an elastic foundation. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain-displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the buckling of columns in three different thermal loading cases is studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The thermal gradient plays a positive role in improving the stability of columns. Furthermore, the effect of thermal gradients decreases while increasing the rotational restraint stiffness and decreasing the slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the rotational stiffness of thermal restraints will increase the critical temperature.     

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.     

Cellular buckling in I-section struts

An analytical model that describes the interactive buckling of a thin-walled I-section strut under pure compression based on variational principles is presented. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weakly stable global buckling mode and the strongly stable local buckling mode. The resulting behaviour is highly unstable and when the model is extended to include geometric imperfections it compares excellently with some recently published experiments.     

Shear strength and design of trapezoidally corrugated steel webs

Due to the accordion effect, corrugated steel webs are only able to resist shear force. The shear force in the web can cause three different buckling modes: local, global and interactive shear buckling. Although several researchers have been investigating it, the shear buckling behavior of the corrugated webs has not yet been clearly explained, this leads to conservative design. This paper presents the shear strength and design of trapezoidally corrugated steel webs. Firstly, global shear buckling equations are rearranged in order to derive the global shear buckling coefficient. The interactive shear buckling coefficient and the shear buckling parameter for corrugated steel webs are then proposed based on the 1st order interactive buckling equation. The inelastic buckling strength is determined from the buckling curves based on the proposed shear buckling parameter. A series of tests are conducted to verify the proposed design equations. From the test results of this study and those provided by previous researchers, it was found that the proposed shear strengths provide good predictions for the shear strength of the corrugated steel webs.     

Buckling formulation for shear deformable thin-walled members—II. Finite element formulation

The variational principle developed in a companion paper is adopted to formulate a new finite element for the buckling analysis of members of doubly symmetric cross-sections. A series of examples demonstrate the convergence characteristics of the element and its applicability to a wide variety of problems. The examples include column flexural buckling, lateral–torsional buckling of beams subjected to moment gradients, buckling of members under combinations of axial force and bending moment, and eccentrically supported structural members. In all cases, the validity of the solutions is assessed through comparisons to well-established closed-form and/or other established numerical solutions.     

薄壁构件弯扭失稳的一般理论

在薄壁构件的弯扭失稳问题上近20年以来存在着传统的和较新的两种不同的理论,这两种理论的并存使得一些国家的规范和一些著作出现不一致,例如我国的(GB50017—2003)和N.S.Trahair的著作犤17犦仍然采用传统理论的结果,而(GB50018—2003)、ISO的钢结构材料标准和美国的金属结构稳定设计解说却采用较新理论的结果。作者发现被忽视的横向正应力也是影响薄壁构件稳定的重要因素。本文在考虑非线性横向应变能的基础上,导出了薄壁构件的总势能方程。从板的理论出发,全面考虑微元上各种应力以及外荷载的影响,用假想荷载法导出了薄壁构件的弯扭失稳平衡微分方程。这组平衡微分方程和本文提出能量法的是一致的,这进一步证实了本文提出理论的正确性。     

Non-linear model for stability of thin-walled composite beams with shear deformation

A geometrically non-linear theory for thin-walled composite beams is developed for both open and closed cross-sections and taking into account shear flexibility (bending and warping shear). This non-linear formulation is used for analyzing the static stability of beams made of composite materials subjected to concentrated end moments, concentrated forces, or uniformly distributed loads. Composite is assumed to be made of symmetric balanced laminates or especially orthotropic laminates. In order to solve the non-linear differential system, Ritz's method is first applied. Then, the resulting algebraic equilibrium equations are solved by means of an incremental Newton–Rapshon method. This paper investigates numerically the flexural–torsional and lateral buckling and post-buckling behavior of simply supported beams, pointing out the influence of shear–deformation for different laminate stacking sequence and the pre-buckling deflections effect on buckling loads. The numerical results show that the classical predictions of lateral buckling are conservative when the pre-buckling displacements are not negligible, and a non-linear buckling analysis may be required for reliable solutions.     

Stability of variable thickness shear deformable plates—first order and high order analyses

This work presents analysis of the buckling loads for thick elastic rectangular plates with variable thickness and various combinations of boundary conditions. Both the first order and high order shear deformation plate theories have been applied to the plate's analysis. The effects of higher order non-linear strain terms (curvature terms) are considered as well. The governing equations and the boundary conditions are derived using the principle of minimum of potential energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the stability analysis of compressed members with variable cross-section, which provides for the derivation of the exact stiffness matrix of tapered strips including the effect of in-plane forces. The results from the two shear deformation theories are compared with those obtained by the classical thin plate's theory and with published results.     

Comprehensive stability design of planar steel members and framing systems via inelastic buckling analysis

This paper presents a comprehensive approach for the design of planar structural steel members and framing systems using a direct computational buckling analysis configured with appropriate column, beam and beam-column inelastic stiffness reduction factors. The stiffness reduction factors are derived from the ANSI/AISC 360-16 Specification column, beam and beam-column strength provisions. The resulting procedure provides a rigorous check of all member in-plane and out-of-plane design resistances accounting for continuity effects across braced points as well as lateral and/or rotational restraint from other framing. The method allows for the consideration of any type and configuration of stability bracing. With this approach, no member effective length (K) or moment gradient and/or load height (C _b ) factors are required. The buckling analysis rigorously captures the stability behavior commonly approximated by these factors. A pre-buckling analysis is conducted using the AISC Direct Analysis Method (the DM) to account for second-order effects on the in-plane internal forces. The buckling analysis is combined with cross-section strength checks based on the AISC Specification resistance equations to fully capture all the member strength limit states. This approach provides a particularly powerful mechanism for the design of frames utilizing general stepped and/or tapered I-section members.