Monosymmetric Cross-sections Research Articles

The transformation matrix in the 7DOFs beam formulation

A non-negligible percentage of steel products for civil and industrial constructions is realised with members characterised by a mono-symmetric cross-section. The non-coincidence between the centroid and the shear centre reflects in a quite complex behaviour that is remarkably influenced by the warping effects. Usually, frame design is based on the output data of commercial Finite Element Analysis Packages (FEAPs) but only few of them offer a refined beam element formulation able to reproduce the response of non-bisymmetric cross-section members (like the 7 degrees of freedom, DOFs, beam element). As a consequence, warping effects are usually neglected in routine design, leading in several cases to assess non-correct internal stresses, global displacements and critical buckling multipliers.This paper is focussed on mono-symmetric cross-section members and deals with the influence of the interaction between axial force, bending moments and bimoment in linear elastic range. In particular, the two strategies that FE developers usually adopt, in the 7 DOFs beam formulation, to pass from the local to the global reference system are herein shortly introduced and discussed. The effects of the associated transformation matrices have been investigated in few examples, also reproduced via closed expressions derived from the mixed torsion theory and FE shell models. Paper outcomes show that the results in terms of generalised displacements, internal forces and, consequently, stress distribution along the member cross-section are significantly different, strictly depending on the adopted transformation matrix. The importance of a correct evaluation of the bimoment distribution is stressed also by the fact that the new edition of the EN1993-1-1, expected in the next months, includes also the bimoment contribution for member checks.

Exact structural-property matrices and Timoshenko’s shape functions for nonprismatic frame elements defined by noncentroidal axes

When one is attempting to model framed structures and adopts beam-like finite elements formulated with reference to their centroidal axis—which avoids the interaction between axial and flexural effects—one must, in many practical situations, take into account geometrically complex curved-line finite elements. This fact, which can be quite disadvantageous, is typically called for when solving physically nonlinear problems or modeling nonprismatic beams having cross sections varying asymmetrically around their straight centerlines. This paper proposes then a flexibility-type method based on the principle of virtual forces (PVF) to obtain, according to the Timoshenko beam theory (TBT), exact expressions for the structural properties of nonprismatic frame elements described with reference to noncentroidal axes. The main advantage of this method is that it allows, through the use of straight-line segments, the modeling of complex variable-rigidity frame elements. To interpolate the variable sectional rigidities over the elements, one adopts polynomials of different orders. In addition, this new method allows one to obtain the exact Timoshenko’s shape functions (TSFs) for nonprismatic frame elements with noncentroidal axes. Such shape functions are needed to evaluate deformation-dependent structural properties, such as geometric stiffness. We validate the robustness of the proposed formulation by solving in-plane tapered beams possessing thin-walled monosymmetric cross sections. Using analytical and highly accurate 3D ANSYS responses, this work determines and compares responses described in terms of displacements, rotations, internal forces, and stresses.

Dynamic response of a thin-walled curved beam with a mono-symmetric cross-section under a moving mass

This paper presents analytical solutions for the dynamic response of thin-walled curved beams induced by moving vehicles. The solutions encompass vertical, torsional, radial, and axial motions. A comprehensive set of governing equations for the dynamic response of thin-walled curved beams is established by considering the mass inertia of moving vehicles, rotary inertia, and warping resistance. The solutions are obtained for the four-directional response of curved beams, involving vertical, torsional, radial, and axial motions, based on the Fourier finite integral transformation, the Laplace–Carson transformation, their inverse transformations, and the Galerkin approach. A comparison of the results calculated by the analytical solutions in this study are compared with those calculated by the moving-force solutions using the Galerkin approach reported in a related literature, demonstrates the reliability and superiority of the analytical solutions. Hence, an extensive parametric study is carried out to investigate the effect of the mass inertia of moving vehicles, velocities, higher-order modes of vibrations, and radii of curvature on the dynamic responses of beams. As per the results, the mass and velocity of the moving vehicles play important roles in the dynamic response of curved beams. Furthermore, the dynamic response for thin-walled curved beams increases significantly with heavier vehicles and higher velocities, confirming that it is necessary to consider the mass inertia effect to study the dynamic response of curved beams.

The effect of cross-section geometry on the lateral–torsional behavior of thin-walled beams: Analytical and numerical studies

In this paper, the elastic lateral–torsional behavior of simple beams is discussed by presenting a novel analytical solution and performing numerical studies. The motivation of the presented research is the observation that classic analytical prediction and finite element prediction are, typically, significantly different when the second-order nonlinear behavior of beams with initial imperfections is analyzed. To understand and explain the observed differences, a novel analytical model is worked out for the geometrically nonlinear analysis of beams with initial geometric imperfections. The advancement in the presented analytical solution is the explicit consideration of the changing geometry as the load increases. The most important steps of the derivations are summarized, and the resulting formulae are briefly discussed. The derivations are done for general cross-sections, however, the bending is assumed to act in one of the principal planes. Numerical studies are also presented, focusing on mono-symmetric cross-sections. As part of the numerical studies, first, the results of the new analytical formulae are compared to those from shell finite element analysis. The results suggest that the new formulae can capture the most essential elements of the behavior observed in the shell finite element calculations, justifying that the cross-section shape might have a significant effect on the nonlinear lateral–torsional behavior of beams. Then the effect of the lateral–torsional buckling is predicted by calculating buckling reduction factors, using the results of the geometrically nonlinear finite element calculations. The capacity prediction results, again, justify that the cross-section shape, as well as the sign of the assumed geometric imperfection, might have a non-negligible effect on the buckling reduction factors, and on the capacity of the member.

On the band gap formation in locally-resonant metamaterial thin-walled beams

This paper is concerned with flexural and torsional waves propagation in locally-resonant thin-walled beams with open cross section. A beam with monosymmetric cross section hosting a periodic array of translational and torsional resonators is considered. The resonators are designed to protect the beam from both vertical and lateral dynamic loads; while vertical loads acting in the symmetry plane induce flexural waves only, vertical loads acting out of the symmetry plane and lateral loads induce coupled flexural–torsional waves due to the monosymmetry of the beam cross section. Aim of the paper is to investigate the dispersive properties of the locally-resonant beam under consideration, focusing on coupled flexural–torsional waves. For this purpose, a homogenization approach is adopted to derive a simplified dispersion equation of the infinite locally-resonant beam. Starting from the observation that the latter equation is equivalent to the dispersion equation of a uniform infinite beam with homogenized inertial properties, it is possible to identify the location and the amplitude of band gaps; remarkably, it is found that band gaps for coupled flexural–torsional waves are located in the frequency region where the matrix of the homogenized inertial properties is negative definite. Further, it is demonstrated that band gaps arise when translational and torsional resonators are simultaneously connected to the beam and are tuned to the same frequency; on the other hand, band gaps are not guaranteed if only translational resonators or only rotational resonators are connected to the beam. Alternative exact methods based on the transfer matrix approach and the theory of generalized functions are employed to validate and substantiate the obtained results.

Neural network models for the critical bending moment of uniform and tapered beams

Most design standards require the calculation of the elastic critical bending moment for the design and verification of steel beams. Formulae exists for uniform beams with double or mono-symmetric cross-sections but, for tapered beams, simple design formulae are yet to be developed because of the complexity associated with the non-uniform geometry of these members.This work proposes a neural network model to calculate the critical moment. The model is developed using the Backpropagation and Levenberg–Marquardt algorithms and considering 60549 data samples for training and 8526 samples for validation. The samples are generated by a numerical model using the Finite Element Method (FEM). An innovative methodology to reduce the number of samples necessary to train the model is implemented based on the concept of random sample generation with constrained geometric proportions.The accuracy of the developed model is further illustrated on some particular cases and against the FEM results of other authors. For the uniform beams, the results of the proposed model are compared against those from existing formulae for uniform members showing its improved accuracy.Finally, it is expected that this investigation demonstrates the benefits of the use of neural networks based solutions as fast assessment tools during the search of optimal structural solutions in the early design stages.

Design of aluminium alloy channel sections under minor axis bending

In recent years, numerous research works have been reported on the flexural response of aluminium alloy tubular cross-sections. However, studies on monosymmetric cross-sections and particularly channel (C-) sections are limited, albeit their increased usage in structural applications. This paper aims to address this knowledge gap providing an improved understanding about the minor axis bending behaviour of C-sections through an experimental and numerical investigation. In total 14 specimens made from 6082-T6 heat-treated aluminium alloy were subjected to four-point bending. Tensile coupon tests were also performed to determine the mechanical properties of the examined aluminium alloy. The obtained experimental results are analysed and discussed. A series of geometrically and materially nonlinear analyses were also carried out to study the flexural performance of C-sections in two aluminium alloys and two bending orientations over a range of cross-sectional aspect ratios and slendernesses. The experimental and numerical results are utilised to assess the European design standards. The applicability of the Continuous Strength Method and the Direct Strength Method is also evaluated. An alternative design method based on the plastic effective width concept is proposed for slender C-sections subjected to minor axis bending. This method accounts for the inelastic reserve capacity which is in accordance with the experimental and numerical observations.

25. THE ROLE OF DELETERIOUS RARE VARIANTS IN ADHD RISK

The results of an extensive experimental campaign on centric and eccentric compressive behavior of thin-walled steel uprights of pallet-rack systems are presented. The analyzed cold-formed profiles are perforated through their whole length and have open, mono-symmetric cross-sections. Experimental results are extended through finite-element simulations, including geometrical and material nonlinearities, to obtain detailed three-dimensional compression-bending-bending strength-domains, which are compared to building-code prescriptions. Regulations are safety preserving, with an average difference of about 10%, and maximum difference for bending about the weak axis. Unlike in regulatory prescriptions, the experimental-numerical results show unsymmetrical behavior about the weak axis and nonlinear, convex bending-bending interaction.

Lateral-torsional buckling of partially protected steel I-beams exposed to fire

This paper proposes a new model for thin-walled open profiles able to predict bending-torsion buckling of thermally restrained steel I beams with bi- and mono-symmetric cross-sections. Refined nonlinear kinematics accounts for the shifts of the centroid and shear centre generated by the thermal gradient. The eigenvalue problem derived from geometrically nonlinear balance equations is solved via power series and Newton-Raphson technique. Several examples are presented; the results are compared with those provided by Abaqus simulation and the literature.

Analytical Solutions for the GNI Analysis for Lateral-torsional Buckling of Thin-walled Beams With Doubly-symmetric and Mono-symmetric Cross-sections

When a beam is subjected to bending, lateral-torsional buckling (LTB) is a potential failure mode. LTB has extensively been studied in the past decades. It can be observed, however, that the vast majority of the research was devoted to beams with doubly-symmetrical crosssections subjected to major-axis bending. This is reflected also in the current design standards, e.g. in the Eurocode for steel [1]: the formulae for LTB check are most surely worked out for major-axis bending of doubly-symmetric cross-sections, and when it comes to N-M interaction check, mono-symmetric and asymmetric cross-sections are formally excluded. In the practice mono-symmetric and asymmetric cross-sections are used, too, they are especially typical in cold-formed steel (CFS) construction. Though there is a specific Eurocode part [2] for CFS members, the LTB, and especially the N-M interaction check is fairly simplified. Thus, it seems to be important to better understand the LTB behaviour of beams with monosymmetric or asymmetric cross-sections.

Elastic lateral-torsional instability of monosymmetric shear deformable fixed arches under a localized uniform radial load

This paper investigates the elastic lateral-torsional instability of shear deformable fixed arches having a monosymmetric cross-section when subjected to a localized uniform radial load. Hitherto, this topic has not been addressed in the open literature. The complex in-plane internal actions in a fixed arch with a monosymmetric section are produced by a localized uniform radial load and are affected by the arc length of localized load segment. When the in-plane internal actions reach certain magnitudes, the fixed arch may fail suddenly in a lateral-torsional instability mode. The lateral-torsional instability loads of such arches considering shear deformations under a localized uniform radial load are difficult to obtain and are investigated in this paper. The strain-displacement relationships for the longitudinal normal axial and shear strains in the arch with a monosymmetric section are derived. In-plane analyses for the internal forces produced by the localized uniform radial load are firstly carried out. The analytical solution for the lateral-torsional instability load accounting for shear deformations and monosymmetry is then obtained using the principle of stationary potential energy together with the Rayleigh-Ritz method and compared with ANSYS numerical results. It is found that the monosymmetry parameter, the shear deformations, the slenderness ratio and the arc length of the localized load segment have considerable effect on the lateral-torsional instability load of fixed arches with a monosymmetric section.

An optimisation strategy for the (in- and out-of-plane) resistance of steel beams in demountable composite floor systems

Demountable and reusable composite floor systems enable the decoupling between the use of construction materials and economic activity, and hereby contribute to the sustainability of the built environment. Efficient material use through optimised cross-section design reduces construction material demand. Demountable steel-concrete composite floor systems are perceived as competitive when consisting of steel beams and large prefabricated concrete floor elements, with composite interaction achieved by demountable shear connectors. Compared to traditional monolithic floor systems, the demountable composite floor systems have an increased sensitivity to lateral-torsional buckling during execution, mostly because of unsymmetrical loading and the absence of rotational constraints in the execution phase. This increased sensitivity implies that the cross-section of the steel beam should not only be designed based on the required in-plane resistance, but should also maximise the out-of-plane resistance. The Energy method and Rayleigh-Ritz methods are combined to develop a prediction model for the critical bending moment of monosymmetrical web-tapered steel beams. The key cross-sectional dimensions and parameters that affect the in-plane and out-of-plane resistance are identified. An overarching strategy for the concurrent optimisation of the in-plane and out-of-plane resistance of monosymmetrical cross-sections is presented without compromising on material efficiency. The beneficial effects of the proposed optimisation strategy are quantified through a case study example.

Stationary complementary energy high-order theory for the static analysis of beams

A high-order theory is developed for the analysis of beams with general mono-symmetric cross-sections. The theory represents the nonlinear distribution of the longitudinal normal stress across the section depth by a polynomial series expansion up to any order as specified by the analyst. The corresponding shear and transverse normal stresses are obtained by satisfying the 2D infinitesimal stress-flow equilibrium conditions. The resulting statically admissible stress fields are then used in conjunction with the principle of stationary complementary strain energy to formulate the governing compatibility equations and boundary conditions. Closed form solutions are then developed for general loading and boundary conditions. Comparisons with the theory of elasticity and 3D finite element analysis predictions showcase the ability of the present theory to naturally capture shear deformation effects, transverse normal stress effects, nonlinear longitudinal normal stress distributions in deep beams, as well as the effect of support height. Unlike conventional beam solutions that are based on postulated kinematic assumptions, which tend to converge to the displacement response from below, the present theory avoids introducing any kinematic assumptions and is shown to converge to the solution from above. The theory is applicable to beams with doubly symmetric or mono-symmetric cross-sections, with isotropic or orthotropic materials, and subjected to general loading and boundary conditions. The theory is shown to offer advantages compared to other theories when modelling deep beams and/or beams with supports that are offset from the centroidal axis.

Study of the stability of weightless thin-walled straight columns under centrically applied terminal compressive force by using equivalent forces

There are three possible modes of buckling of thin-walled straight steel columns: flexural buckling, torsional buckling, and flexural-torsional buckling. These modes of buckling are considered in the specification for the design of steel structures, such as Eurocode 3, which includes the particular case of torsional-flexural buckling of centrically loaded members with monosymmetric cross-sections. The system of differential equations that governs the stability of centrically loaded weightless members was presented in the mid-twentieth century and has been widely addressed in both steel structures and instability books. In this work, a simpler way to obtain the differential equations of stability for both torsional and flexural-torsional buckling modes by using equivalent forces is presented. The presented idea is especially useful in the academic context of civil engineering. Students and faculty members will appreciate the deduction of the instability equations governing the equilibrium in a few simple steps.

Evaluation of European approaches applied to design of TWCF steel members

Thin-walled cold-formed (TWCF) steel members are widely used in aerospace, automobile and architecture industries. Among the various advantages, mainly associated with the relatively simple and very cheap techniques required for their production and shaping, it is important to mention the relevant high strength to weight ratio. Beside the benefits, it has to be pointed out that the TWCF member response is quite difficult to be predicted because of the significant influence of local, distortional and overall buckling phenomena, owing to the frequent presence of open mono-symmetric cross-sections.The present paper can be intended as a typical technical paper whose results are useful not only for research activities, but, especially, for the routine design. It summarises a study aimed at increasing the safety of two European approaches that should be potentially adoptable for designing TWCF steel members. The grade of reliability of these existing European procedures has been evaluated by predicting the performance of more than 240 TWCF members tested in laboratory, selected in the framework of 13 experimental studies, taken from the literature. Being the assessed load carrying capacity sometimes significantly greater than the characteristic-experimental one, suitable safety factors have been consequently calibrated and proposed for each cross-section type, to be directly adopted in routine design to match the requirements associated with the limit state philosophy. Finally, the strength of additional TWCF beam-column specimens has been predicted to assess the efficiency of the proposed coefficients for practical cases frequently encountered in routine design. A complete benchmark is proposed in Appendix A, where the application of the considered design procedures for one selected case is reported.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Abstract This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

Elastic critical force of centrically loaded member with asymmetric and monosymmetric cross-sections at various boundary conditions: A parametric study

The system of governing differential equations of stability of centrically loaded members with rigid open cross-sections was developed by Kappus in 1937 and by Vlasov in 1940. In 1941 Goľdenvejzer published a solution of this system by an approximate method. He proposed a formula for torsional-flexural critical force calculation which can take into account cases when the flexural boundary conditions of member are different to the torsional boundary conditions. This is allowed by introducing a factor α which depends on the combination of flexural and torsional boundary conditions. Goľdenvejzer investigated members with only 14 specific combinations of boundary conditions which are more frequently used in practice.The authors used Goľdenvejzer‘s approximate method for 9 different shapes of monosymmetric and 5 different shapes of asymmetric cross-sections to produce the parametric study. In the study all 100 theoretical possible combinations of boundary conditions for monosymmetric cross-sections and all 1000 theoretical combinations of boundary conditions for asymmetric cross-sections were investigated. The new α-factors were proposed to decrease the errors in the critical forces values. The finite element method with 1D elements was used for the proposition of new α-factors as well as for the verification of all results.

Flexural and Buckling Analysis of Laminated Composite Beams with Bi- and Mono-Symmetric Cross-Sections

Flexural and Buckling Analysis of Laminated Composite Beams with Bi- and Mono-Symmetric Cross-Sections

Stability of centrically loaded members with monosymmetric cross-section at various boundary conditions

The system of governing differential equations of stability of centrically loaded members with rigid open cross-sections was presented by Kappus in 1937 and by Vlasov in 1940. In 1941 Goľdenvejzer published a solution of this system by Bubnov-Galerkin's variational method using eigenfunctions of flexural vibration modes as fundamental functions. The result is a formula for torsional-flexural critical force calculation.The simplified formula for torsional-flexural critical force calculation is in EN 1993-1-3 as formula (6.35). This formula gives exact values of elastic critical force for torsional-flexural buckling of centrically loaded members with monosymmetric cross-section only if the flexural boundary conditions are the same as the torsional boundary conditions at both member ends. The Goľdenvejzer's formula differs from (6.35) by allowing different flexural boundary conditions to the torsional ones. Goľdenvejzer takes this into account by introducing factor α.The purpose of this paper is to verify whether the factor α can correctly include the influence of combinations of flexural and torsional boundary conditions which Goľdenvejzer did not investigate, and to verify if the simplified formula (6.35) may be used for any boundary conditions. The authors used Goľdenvejzer's method for 8 different shapes of monosymmetric cross-sections for all 100 theoretical possible combinations of boundary conditions, and for two different fundamental functions: a) eigenfunctions of flexural vibration modes (which were also used by Goľdenvejzer) and b) eigenfunctions of buckling modes (which is a more natural choice). All results are verified by the finite element method with 1D member elements. Research fieldStability of metal members under axial compression.

Finite element formulation for the analysis of multilayered beams based on the principle of stationary complementary strain energy

A family of finite elements for the analysis of orthotropic multilayered beams with mono-symmetric cross-sections is developed based on the principle of stationary complementary energy. The longitudinal normal stress field is postulated as polynomial and Heaviside step function series and substituted into the infinitesimal equilibrium conditions to develop expressions for the shear and transverse stress fields. The statically admissible stress fields thus derived are then adopted within the complementary energy variational principle framework to develop a family of finite elements. The distinguishing features of the solution are: (i) it captures the nonlinear distribution of the stress fields along the section depth and steep stress gradients typically occurring near bondline ends of multilayer beams, (ii) unlike conventional solutions based on the principle of stationary potential energy which predict jumps in the shear and peeling stresses at interfaces of adjacent layers, the present solution satisfies equilibrium in an exact infinitesimal sense at layer interfaces and thus ensures continuity of the stress fields across the interface, (iii) it naturally captures the effects of transverse shear and transverse normal stresses, and (iv) it consistently converges to the displacements from above, in contrast to conventional finite element solutions where convergence is typically from below. The versatility of the solution is then illustrated in applications involving wood beams and steel beams strengthened with GFRP plates and sandwich beams with soft cores.