Members Of Open Cross Section Research Articles

Co-Rotational Formulations for Geometrically Nonlinear Analysis of Beam-Columns Including Warping and Wagner Effects

The warping effects may predominate in geometrically nonlinear analysis of open cross-section members. The formulation of conventional beam-column elements incorporating the warping effects is cumbersome due to the method considering the inconsistency between the shear center and centroid. To develop a concise warping element formulation, this paper presents a transformation matrix to integrate the inconsistent effects into the element stiffness matrix. The co-rotational (CR) method used to establish the element equilibrium conditions in the geometrically nonlinear analysis is adopted to simplify the element formulation and improve the efficiency of nonlinear analysis. A new beam-column element explicitly considering the warping deformation and the Wagner effects is derived based on the CR method and the Euler–Bernoulli beam theory. A detailed kinematic description is provided for considering large deflections and rigid body motions. Based on the mechanical characteristic, the coordinate and the rigid body motion transformation matrices are given. The secant relationship is developed to evaluate the element internal forces accurately and effectively in each iteration. Several verification examples are provided to validate the proposed method’s reliability and robustness. The verifications demonstrate that the proposed element leads to considerable computational advantages. The results of this paper are useful for future upgrading of frame analysis software with warping degrees-of-freedom (DOFs).

Numerical and experimental post-buckling analysis of slender thin-walled GLARE members subjected to compressive loading

This study deals with a post-buckling analysis of thin-walled GLARE members subjected to axial compressive loading. Considered slender GLARE samples are a hybrid composite that consists of alternating thin layers of aluminium alloy sheets and unidirectional glass fibrereinforced prepregs. Comparative study was performed for different thin-walled, open cross–section members that included channel, Z-shape and top-hat-shaped slender sections. This study is mainly focused on assessing the performance of top-hat-shaped sections which provided the highest strength when subjected to the compressive test. Various layer arrangements of GLARE samples were investigated based on the fibres alignment in the composite layer. GLARE laminates were manufactured by autoclaving technique which provided the high-quality of multi-layered composite. Composite specimens were axially compressed in laboratory tests by means of static testing unit of Instron upgraded with Zwick/Roel control software. Electromechanical strength testing machine had a maximum capacity of 200kN for which applied screw type testing machine provided a displacement control loading. During experimental tests, deformations were measured by Aramis 3D non-contact optical equipment based on digital image correlation (DIC) which allowed investigating columns behaviour in full load range until fracture. The behaviour of thin-walled GLARE members was analysed with the main attention to postbuckling response. Simultaneously, numerical simulations by FEM were performed to predict the post-buckling equilibrium paths of the multi-layered GLARE structures modelled by SHELL element type. The non-linear problem was solved initially by the incremental Newton– Raphson procedure whereby the stiffness matrix was updated with each iteration and equilibrium paths were calculated based on nominal stress state in non-degraded structure. Secondly, the damage evolution law was introduced by the material property degradation method (MPDG). Hence, based on the damage variables the stiffness was gradually reduced in the area where failure was initiated which had a significant impact on the samples’ post-buckling response. Therefore, the influence of damage variables on the laminate's post-buckling equilibrium paths was also considered. For various GLARE lay-ups, numerical results were found to be in a good agreement with experimental tests. Acknowledgments The author has obtained funding as part of financing a doctoral scholarship from the National Science Center, No. UMO-2019/32/T/ST8/00605

Damage analysis of thin-walled GLARE members under axial compression – Numerical and experiment investigations

This paper presents an investigation of the progressive damage and failure behaviour of thin-walled Glare – Fibre Metal Laminate (FML) short columns subjected to in-plane compressive loading. The study concerns 7-layered top-hat-shaped open cross-section members that consist of aluminium and angle-ply composite plies. The compression tests were conducted by means of an ultimate static testing strength machine with displacement control loading, whereas wall deflections were monitored by a Digital Image Correlation (DIC) system Aramis®. Laboratory procedures also included ultrasonic non-destructive testing to assess the quality of specimens produced by the autoclave technique. Failure tests were compared with FE simulations with the implementation of the progressive failure algorithm. Based on the material property degradation method (MPDG) and Hashin failure criterion, damage variables were used to control material stiffness reduction after damage initiation. Progressive failure assessment of top-hat members by FEM included an attempt to estimate the post-buckling equilibrium paths and failure loads that were found to be in a good agreement with experiment evidence. High consistency of numerical and experiment damage modes was also achieved.

Generalized constrained finite strip method for thin-walled members with arbitrary cross-section: Primary modes

In this paper the generalization of the constrained finite strip method (cFSM) is discussed. cFSM is a special version of the semi-analytical finite strip method (FSM), where carefully defined constraints are applied which enforce the thin-walled member to deform in accordance with specific mechanics, e.g., to allow buckling only in flexural, lateral–torsional, or a distortional mode. In the original cFSM only open cross-section members are handled, here the method is extended to cover any flat-walled member, including those with closed cross-sections or cross-sections with open and closed parts. Moreover, in the original cFSM only 4 deformation classes are defined, here the deformation field is decomposed into additional, mechanically meaningful, sub-fields. Formal mechanical criteria are given for the deformation classes, and implementation of the criteria regardless of cross-section topology is illustrated. In this paper, the primary deformation classes are presented in detail. Primary deformations are associated with minimal cross-section discretization, i.e. nodal lines located at folds and ends only. This paper is accompanied by a companion, where secondary modes and additional practical aspects in the selection of base vectors for the deformation classes are discussed. With the proposed modifications the powerful cFSM capabilities of buckling mode decomposition and identification are extended to essentially arbitrary thin-walled cross-sections.

TORSION OF FUNCTIONALLY GRADED OPEN-SECTION MEMBERS

There exist some approximate analytical methods for torsion analysis of homogeneous open cross-section members; however, no analytical formulation has been presented for solving a torsion problem of inhomogeneous open cross-section members yet. In this paper, an approximate analytical method for the torsion analysis of thin- to moderately thick-walled functionally graded open-section members with uniform thickness is presented. The shear modulus of rigidity is assumed to have a variation across the thickness. The cross-section is decomposed into some straight, curved and end segments. The torsion problem is then solved in each segment considering some appropriate approximations. By presenting three examples, accuracy of the presented method with respect to thickness, corner radius, and material parameters are investigated. The results show that the proposed method is useful for torsion analysis of thin- to moderately thick-walled functionally graded open-section members.

A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method

This paper derives a new method for fully decomposing the elastic stability solution, of a thin-walled single-branched open cross-section member, into mechanically consistent buckling classes associated with global, local, distortional, and shear and transverse extension buckling modes. The method requires a set of formal mechanical definitions for each of the buckling classes. For global and distortional buckling the definitions employed successfully by generalized beam theory are utilized herein, while for local and other (shear and transverse extension) buckling, new definitions are provided. The mechanical definitions for a given buckling class represent a series of constraint conditions on the general deformations that the thin-walled cross-section may undergo. These constraint conditions are derived as explicit constraint matrices within the context of the finite strip method, and provide the desired decomposition of the buckling deformations of the member. The decomposition is full in the sense that the union of the deformation spaces of the decomposed buckling classes is the same as the general deformation space in the original finite strip method. The resulting method is termed the constrained finite strip method (cFSM). The two primary applications for cFSM are modal decomposition and modal identification. Modal decomposition reduces the general finite strip solution to a desired set of buckling classes and performs a useful model reduction that allows the results to focus on a particular buckling class, e.g., distortional buckling. Modal identification provides a means to quantify the extent to which a given buckling class is contributing to a general buckling deformation. Application of cFSM, including graphical representation of the buckling classes, and the advantages of modal decomposition and modal identification, are provided in a series of numerical examples.

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Application and examples

The objective of this paper is to provide implementation details of, and practical examples for, modal decomposition of the cross-section stability modes of thin-walled members by constraining a traditional finite strip method (FSM) solution. The theoretical development of the proposed method is provided in a companion to this paper [Ádány S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-walled Structures, submitted for publication, companion to this paper.] The constraint matrix, which is directly applied to the elastic and geometric stiffness matrices of a traditional FSM solution in order to constrain the deformations, is provided along with all formulae necessary in its construction. In addition, a completely worked out numerical example is provided to aid in implementing the constrained FSM solution. The authors implemented the constrained FSM in the open source program CUFSM. This modified version of CUFSM is then used to provide a series of numerical examples that illustrate (i) the advantages of performing modal decomposition, (ii) the importance of understanding and defining the deformation fields related to a desired mode, and (iii) the behavior of constrained FSM stability solutions compared with classical analytical solutions, GBT, and unconstrained FSM. Decomposition of the cross-section buckling classes related to global and distortional modes is demonstrated. Further, the impact of how to select the deformation fields and perform modal decomposition for cross-section stability modes within a class, e.g., for the traditional three global modes (weak-axis flexure, strong-axis flexure and flexural-torsional buckling), is explored and the impact of the deformation field definitions demonstrated. Comparisons of the constrained FSM solutions with other available solutions demonstrate the importance of properly determining when beam theory and plate theory should apply to the cross-section stability of thin-walled members.

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Derivation

This paper provides the first detailed presentation of the derivation for a newly proposed method which can be used for the decomposition of the stability buckling modes of a single-branched, open cross-section, thin-walled member into pure buckling modes. Thin-walled members are generally thought to have three pure buckling modes (or types): global, distortional, and local. However, in an analysis the member may have hundreds or even thousands of buckling modes, as general purpose models employing shell or plate elements in a finite element or finite strip model require large numbers of degrees of freedom, and result in large numbers of buckling modes. Decomposition of these numerous buckling modes into the three buckling types is typically done by visual inspection of the mode shapes, an arbitrary and inefficient process at best. Classification into the buckling types is important, not only for better understanding the behavior of thin-walled members, but also for design, as the different buckling types have different post-buckling and collapse responses. The recently developed generalized beam theory provides an alternative method from general purpose finite element and finite strip analyses that includes a means to focus on buckling modes which are consistent with the commonly understood buckling types. In this paper, the fundamental mechanical assumptions of the generalized beam theory are identified and then used to constrain a general purpose finite strip analysis to specific buckling types, in this case global and distortional buckling. The constrained finite strip model provides a means to perform both modal identification relevant to the buckling types, and model reduction as the number of degrees of freedom required in the problem can be reduced extensively. Application and examples of the derivation presented here are provided in a companion paper.

Non-linear analysis of thin-walled members of open cross-section

The paper presents a means of determining the non-linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin-walled open section finite element that lead to non-linear stiffness equations. These non-linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non-linear matrices is stated herein. It is shown that the method of solution of the non-linear stiffness matrices is problem dependent. The finite element procedure is used to study non-linear torsion that illustrates torsional hardening, and the Newton-Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post-buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non-linear finite element method of analysis. Copyright (C) 1999 John Wiley & Sons, Ltd.

Flexural-Torsional Buckling of Variable and Open Cross-Section Members

This work gives the exact solutions for the coupled flexural-torsional buckling loads of variable and open cross-section columns, loaded by variable axial force. Both the cross-section dimensions and the axial load can vary along the column as polynomial expressions. The proposed solution is based on a new finite-element method for getting the exact stiffness matrix for the member, including the effects of the axial loading. The buckling load is found as the load that causes the stiffness matrix determinant to become zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method. New, exact results are given for several other cases.

Higher stage polar materials and the application to thin-walled members.

Polar materials or oriented media are typical examples of generalized continua. These materials, to some extent, are composed of particles that behave like rigid bodies or deformable particles, unlike the geometrical points of the classical continuum theory. Extending these theories in this paper, higher stage innner structures are further introduced successively in the microstructure. This continuum model can be called "higher stage polar materials". The kinematics of the 2nd stage polar materials are studied as an example of the n-th stage polar materials, and 2nd stage strain measures are obtained in addition to the conventional 1st stage strain measures. Applying this model, cross-sectional distortions of thin-walled open cross-section members are analyzed. Constitutive equations and balance equations are obtained, which have the same form as the conventional formula. We can regard the concept of "bimoment" in the theory of thin-walled structures as the couple-stress of the 2nd stage polar materials.

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.

The Application of Flexural Methods to Torsional Analysis of Thin-walled Open Sections

The analysis of torsionally loaded thin-walled members of open cross-section is complicated by the presence of warping stresses. Warping normal and shear stresses arise from restraint of longitudinal warping of the cross-section due to end restraints or variations in the internal torque along the length of the member. In a typical thin-walled open member, such as a rolled or cold-formed steel section, warping normal stresses tend to be the critical torsional stress component. The usual method for analysis of warping stresses involves solution of the torsion differential equation to determine rotation of the member about the longitudinal axis. Torsional and warping stresses are functions of various order derivatives of the rotation with respect to the length. This method is described in detail in many sources, including Refs. 1 and 4-9. A recent AISC publication1 presents nondimensionalized solutions of the torsion differential equation for various end conditions and loading cases of a single span straight member, which can be used for determination of torsional stresses. For steel I sections, torsional loading can be resolved into opposite lateral forces acting on flanges and an upper bound to warping stresses can be obtained by determining the resulting bending normal and shear stresses in the flanges. This flexural analogy is applied to determination of warping normal and shear stresses in Refs. 6, 8 and 9. A more general bending-warping analogy, presented in Ref. 7, can be applied to C, Z and other sections commonly used in cold-formed construction.

A non-linear thin-walled beam theory

A non-linear thin-walled beam theory for elastic beams of open section is presented. The theory takes into account geometric non-linearities and longitudinal deformations caused by large cross-sectional rotation of the beam. The non-linear differential equations are derived by the minimum potential energy principle. The boundary conditions associated with the differential equations are obtained. It is shown that the set of equations reduces to the linear theory of Vlasov if non-linear terms are neglected. Also, the set of equations admits a simple solution in the special cases of large uniform torsion of thin-walled members of open cross-section. The torque-rotation relationship and axial strain-rotation relationship thus obtained are identical to the results obtained by Cullimore and Gregory.

Tier buildings with shear cores, bracing, and setbacks

The theory of bending and torsion in thin-walled members of open cross section is applied to shear cores included within the analytical model of the tier building. Representative types of bracing arrangements are also considered and their stiffnesses augment those of the beams, columns and shear walls. Changes in floor plan, referred to as “setbacks”, further complicate the analysis; but a relatively simple routine allows automatic handling of this characteristic. Computer programs named STATIER and DYNATIER include all of these features within the linear static and dynamic analyses of three-dimensional multi-story buildings. Results for a twenty-story structure demonstrate the significance of the interaction between a shear core and the skeletal framing. The effect of warping restraints upon the stresses in the shear core is large at the base but diminishes rapidly with height. The normal stress at the base associated with the bimoment is of the same order of magnitude as that due to the combination of biaxial bending and axial force.

Theory of bending, torsion and buckling of thin-walled members of open cross section

Theory of bending, torsion and buckling of thin-walled members of open cross section

Theory of bending, torsion and buckling of thin-walled members of open cross section

Theory of bending, torsion and buckling of thin-walled members of open cross section

Theory of bending, torsion and buckling of thin-walled members of open cross section

For decades, engineers have assessed and analysed steel frames using simple joints between beams and columns. These joints are often based on oversimplified assumptions using hinges or a direct transfer of beam displacements without any relative displacements. More seldom is the use of spring models that allow relative beam and column displacements at the joints. This despite the standardised component method approach, which can be used to determine the rotational spring stiffness of the relative rotation in a joint. This paper gives a background overview of essential developments in joint modelling and generalised thin-walled beam modelling, including torsional, distortional and related warping effects. For particular situations, some recent proposals for joint models can be applied to joints between thin-walled beams. On this basis, this paper presents a novel idea and a generic methodology that allows the interface between an extended number of generalised beam displacement modes and joints that are modelled using shell elements. The main novelty is the idea to transform from standard degrees of freedom of the interface into a reduced number of beam displacement mode related degrees of freedom. Thus, the number of degrees of freedom of the joint can be reduced to the corresponding total sum of beam modes that have been chosen for the modelling of each of the connected beam elements. The total number of degrees of freedom used for modelling the complete framework will depend on the selected number of modes in each beam element and on the number of extra internal modes chosen in the joint models. For enhanced structural analysis with advanced beam elements and joints that allow relevant distortions and built-in refined connection components, it is believed that this methodology will enable the full detailed analysis of large steel frameworks with a reasonable number of degrees of freedom.