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.     

Distortional buckling of thin-walled inclined lipped channel beams bending about the minor axis

Based on the Generalised Beam Theory (GBT), two computing models are presented to analyse the distortional critical stress of cold-formed thin-walled inclined lipped channel beams bending about the minor axis. The computing model of rigid-body motion ignores the transverse bending deformation of the flange. However, the bending deformation of the flange is accounted for in the transverse bending computing model. Based on the transverse bending computing model, this paper puts up a simple method to take into account the in-plane bending of the flange. The results given by the rigid-body motion computing model does not correlate as well as those given by the transverse bending computing model with the results available in the literature. The accuracy of the transverse bending computing model is verified through comparison of its results with the known results. The comparisons demonstrate the importance of the bending deformation of the flange on the distortional buckling of cold-formed thin-walled channel beams bending about the minor axis.     

Distortional buckling of thin-walled inclined lipped channel beams bending about the minor axis

Based on the Generalised Beam Theory (GBT), two computing models are presented to analyse the distortional critical stress of cold-formed thin-walled inclined lipped channel beams bending about the minor axis. The computing model of rigid-body motion ignores the transverse bending deformation of the flange. However, the bending deformation of the flange is accounted for in the transverse bending computing model. Based on the transverse bending computing model, this paper puts up a simple method to take into account the in-plane bending of the flange. The results given by the rigid-body motion computing model does not correlate as well as those given by the transverse bending computing model with the results available in the literature. The accuracy of the transverse bending computing model is verified through comparison of its results with the known results. The comparisons demonstrate the importance of the bending deformation of the flange on the distortional buckling of cold-formed thin-walled channel beams bending about the minor axis.     

Multi-cell carbon fibre composite box beams subjected to torsion with variable twist

The structural performance of multi-cell carbon fibre composite box beams when subjected to constrained torsional loading is examined in this paper. A simplified analytical procedure for determining the constrained torsional response of a specific class of multi-cell carbon fibre composite box beams is outlined in some detail. The constrained condition analysed is that of the cantilevered multi-cell beam with torque applied at the free end of the beam. Overall elastic couplings in the beams between bending, torsion and axial effects are eliminated from the analysis process through the use of constituent laminates for the thin walls of the cross-sections which are symmetrically layed-up about their own mid-planes and in such a manner that they possess in-plane orthotropy. The analysis procedure employed makes use, essentially, of the existing theories of torsion developed for isotropic construction and these are then suitably modified to account for the non-isotropic nature of the typical carbon fibre composite material. The resulting approach is shown to be able to predict the structural response of the multi-cell composite beams with a considerable degree of accuracy and comparisons between the theory and the results from finite element numerical modelling are shown to give close agreement. The torsional and warping rigidities of the composite multi-cell beams are calculated in a procedure which makes use of the appropriate membrane engineering elastic constants of the individual thin composite walls and the constituent thin walls can have different lay-up configurations provided the stiffness distribution around the sections is of a symmetrically disposed nature in order to preclude the influence of overall elastic couplings.     

Dynamic response of axially loaded monosymmetrical thin-walled Bernoulli–Euler beams

The dynamic bending–torsion coupled vibrations of elastic axially loaded slender thin-walled beams with monosymmetrical cross-sections are investigated by using normal mode method. The Bernoulli–Euler beam theory is employed and the effects of warping stiffness and axial force are included in the present formulations. The theoretical expressions for the displacement response of axially loaded slender thin-walled beams subjected to concentrated or distributed loads are presented. The method is illustrated by its application to two test examples to describe the effects of warping stiffness and axial force on the dynamic behavior of thin-walled beams. The numerical results for the dynamic bending displacements and torsional displacements are given. The proposed theory is fairly general and can be used for thin-walled beam assemblage of arbitrary boundary conditions subjected to various kinds of loads.     

Predicting the ultimate bending capacity of concrete beams from the “relaxation ratio” analysis of AE signals

This paper presents an alternative approach to the problem, based on “testing” the real structure rather than trying to model it. Experiments on reinforced concrete (RC) beams, representative of bridge beams, are described. The beams were loaded in cycles up to failure whilst recording the acoustic emissions (AE) generated. The analysis of the AE signals was then undertaken based on a proposed new parameter, named the “relaxation ratio”. This quantifies the AE energy recorded during the unloading and loading phases of a cycle test and it showed a clear correlation with the bending failure load of the RC beams. A change in trend was noted when the load reached approximately the 45% of the ultimate bending load. The results appeared to be influenced by factors such as the concrete strength and loading rate and further work is needed to extend the results to full scale testing of bridge beams.     

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

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

Bending and buckling of inflatable beams: Some new theoretical results

The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.     

Influence of initial deflection of plate girder webs on fatigue crack initiation

When a thin-walled plate girder is subjected to repeated bending, it is possible that three types of fatigue cracks may be initiated. Fatigue cracks initiated in the tension flange and at the terminal point of the transverse stiffeners in the tension zone of the web are the most common fatigue cracks in thick beams. In addition, fatigue cracks which occur at the toe on the web side of fillet welds connecting a compression flange to a web, can only be observed in thin-walled plate girders. The crakcs are caused by secondary bending stresses produced by the out-of-plane movement of unavoidable initial deflections of the web under repeated in-plane bending.This influence of initial web deflections on the out-of-plane deformations of the web or the secondary bending stresses at the toe is examined by finite element analysis. Comparisons between the analytical results and the past fatigue test results conclude that not only the magnitudes of the initial web deflections, but also their shapes will greatly influence an increase in the secondary bending stresses and then the initiation of the fatigue cracks along the upper boundary of the web.     

Out-of-plane behaviour of partially composite or sandwich beams by exact and Finite Element Methods

The out-of-plane vibrations of composite beams with interlayer slip or three-layer sandwich beams are theoretically and numerically investigated in this paper for general boundary conditions. The governing dynamics equations are derived by applying the Hamilton's principle. A Finite Element Resolution is presented for general boundary conditions, and compared to the exact solution based on the resolution of a tenth-order differential equation. The Finite Element Method may exhibit slip locking phenomenon for very stiff connection, a phenomenon widely investigated in the past for the in-plane behaviour of partially composite beams or sandwich beams. This slip locking, analogous to the shear locking for Timoshenko beams, can be faced with some relevant interpolation shape functions of the same order for each kinematics variables, namely the deflections and the torsion angle. The numerical results are presented for layered wood beams and laminated glass beams, with particular emphasis on the rate of convergence of the natural frequencies with respect to the number of Finite Elements. It is theoretically and numerically shown that the elastic spectra of the symmetrical composite beam are composed of two independent spectrums. One spectrum is independent of the connection parameter and can be studied using the solution of the non-composite action, whereas the second spectrum can be obtained from the resolution of a third-order polynomial equation using the Cardano's method. We show the phenomenon of cut-on frequency for this out-of-plane problem, a phenomenon already noticed for the in-plane Timoshenko beam vibrations. The exact method associated to a 10 degrees-of-freedom shape function can be formally associated with the dynamics stiffness method. The numerical and the exact approaches lead to the same dimensionless spectra, up to four digits.     

Theoretical shape optimization of cold-formed thin-walled channel beams with drop flanges in pure bending

The paper is devoted to cold-formed thin-walled channel beams with open or closed profile of drop flanges. Geometric properties of two C-sections are described with consideration of warping functions and warping inertia moments. Optimization criterion and the dimensionless objective function as a quality measure are defined. Constraints of feasible solutions are strength, global and local buckling conditions and also geometric condition. Analytical solutions of the problems of global and local buckling for thin-walled beams are presented. Results of numerical investigation of optimization problem are compared and presented in tables and figures.     

Effective shaping of cold-formed thin-walled channel beams with double-box flanges in pure bending

The paper is devoted to cold-formed thin-walled channel beams with double-box flanges. Geometric properties of the C-section are described in terms of dimensionless parameters. The warping function and the warping inertia moment are analytically determined. The optimization criterion and the dimensionless objective functions are defined as a quality measure. The space of feasible solutions is constrained by the strength, global, local buckling, and geometric conditions. Analytical solutions of the problems of global and local buckling for thin-walled beams are presented. Results of the numerical calculations of the optimal shaping problem are presented in tables and figures.     

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.     

General analysis of asymmetric thin-walled members

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

Analytical and parametric investigations on lateral torsional buckling of European IPE and IPN beams

Lateral torsional buckling is one of the main failure modes controlling the strength of the slender thin-walled members. A transversely or transversely and axially combined loaded member that is bent with respect to its axis of greatest flexural rigidity may buckle laterally and twist as applied load reaches its critical value unless the beam is provided with a sufficient lateral support. This study intends to present a unique convenient equation that it can be used for calculating critical lateral-torsional buckling load of simply supported European IPE and IPN beams. First, an analytical model is introduced to describe lateraltorsional buckling behavior of beams with monosymmetric cross-section. The analytical model includes first order bending distribution, load height level and monosymmetry property of the section. Then, parametric study is carried out using the analytical solutions in order to establish a simplified equation with dimensionless coefficients. The effect of slenderness and loading positions on lateral-torsional buckling behavior of IPE and IPN beams are studied. The proposed solutions are compared to finite element simulations where thin-walled shell elements and beam elements including warping are used. Good agreement between the analytical, parametric and numerical solutions is demonstrated. It is found out that the lateral-torsional buckling load of European IPE and IPE beams can be determined by presented equation and can be safely used in design procedures.

     

任意开口薄壁截面圆弧曲梁弯扭精确分析

现有薄壁曲梁弯扭理论缺乏严密的理论推导。本文基于薄壁构件分析的两个基本假定，导出了任意开口薄壁截面圆弧曲梁的翘曲位移、正应力、剪应力及其各自合力以及平衡微分方程的精确表示式。为便于应用，本文还给出了一套具有良好精度的内力简化公式及相应的平衡微分方程。     

Mechanics of thin-walled curved beams made of composite materials, allowing for shear deformability

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled curved beams with open and closed cross-sections. In the present model two important concepts concerning to composite thin-walled curved beams are addressed. The first one is the incorporation in the model of what is called full shear deformability, i.e. shear flexibility due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses that can be adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Work. In order to obtain the motion equations of the model a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation, is employed. This model allows the study of many problems of statics, free and forced vibrations with arbitrary initial stresses and linear stability of composite thin-walled curved beams with general cross-sections. A discussion about the constitutive equations is performed in order to explain characteristic features of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed to obtain the numerical approximations to the general equations of thin-walled shear-deformable composite curved beams. For this kind of structural member, iso-parametric finite elements are introduced. Numerical examples are carried out in several topics of statics, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametric studies are performed in order to show the influence of shear deformability on the mechanics of the thin-walled composite curved-beams with open and closed cross-sections as well as to illustrate the utility of the model.     

Mechanics of shear deformable thin-walled beams made of composite materials

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form the shear deformability by means of two features. The first one may be addressed as a mechanical aspect where the effect of shear deformability due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Works. The model is developed by employing a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation. This model allows studying many problems of static's, free vibrations with or without arbitrary initial stresses and linear stability of composite thin-walled beams with general cross-sections. A discussion about the constitutive equations is performed, in order to explain distinctive aspects of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed with the aim to obtain solutions to the general equations of thin-walled shear deformable composite beams. A non-locking fourteen-degree-of-freedom finite element is introduced. Numerical examples are carried out in several topics of static's, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametrical studies are performed in order to show the influence of shear flexibility in the mechanics of the thin-walled composite beams as well as to illustrate the usefulness of the model.     

耦合框架平面外变形钢板装配式屈曲约束支撑#br# 钢筋混凝土框架抗震性能试验研究

为了研究双向地震作用下屈曲约束支撑（Buckling-restrained brace,BRB）钢筋混凝土框架平面内抗震性能以及BRB与钢筋混凝土框架在平面外方向的相互作用,考虑框架平面外变形、节点板平面外刚度和屈曲约束支撑外伸段平面外刚度的影响,对4个钢板装配式屈曲约束支撑钢筋混凝土框架及1个钢筋混凝土空框架进行了平面内拟静力试验研究。研究结果表明：①耦合框架平面外变形的BRB钢筋混凝土框架和无框架平面外变形的BRB钢筋混凝土框架平面内层间剪力与层间位移滞回曲线饱满稳定,具有良好的抗震性能;②框架平面外变形在一定程度上减小了钢板装配式BRB钢筋混凝土框架平面内初始水平刚度和承载力,增大了钢板装配式BRB和节点板平面外变形量;③节点板面外刚度的提高增加了框架节点平面外作用力;④钢板装配式BRB面外刚度的增加能减小自身平面外变形,减小其对钢筋混凝土框架平面外作用力。     

Behaviour and strength of hollow flange channel sections under torsion and bending

Hollow flange channel section is a cold-formed high-strength and thin-walled steel section with a unique shape including two rectangular hollow flanges and a slender web. Due to its mono-symmetric characteristics, it will also be subjected to torsion when subjected to transverse loads in practical applications. Past research on steel beams subject to torsion has concentrated on open sections while very few steel design standards give suitable design rules for torsion design. Since the hollow flange channel section is different from conventional open sections, its torsional behaviour remains unknown to researchers. Therefore the elastic behaviour of hollow flange channel sections subject to uniform and non-uniform torsion, and combined torsion and bending was investigated using the solutions of appropriate differential equilibrium equations. The section torsion shear flow, warping normal stress distribution, and section constants including torsion constant and warping constant were obtained. The results were compared with those from finite element analyses that verified the accuracy of analytical solutions. Parametric studies were undertaken for simply supported beams subject to a uniformly distributed torque and a uniformly distributed transverse load applied away from the shear centre. This paper presents the details of this research into the elastic behaviour and strength of hollow flange channel sections subject to torsion and bending and the results.