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.     

Torsion analysis of thin-walled beams including shear deformation effects

The first part of the paper develops a theory for the torsional analysis for open thin-walled beams of general cross-sections which accounts for shear deformation effects. Statically admissible stress fields are postulated in agreement with those resulting from the Vlasov thin-walled beam theory. The principle of stationary complementary energy is then adopted to formulate the governing field compatibility condition under the stress fields postulated. The naturally arising boundary terms are found to relate the warping deformations to the internal force fields. A torsion beam example is solved using the new theory in order to illustrate its applicability to practical problems. The second part of the paper implements the solution numerically in a force-based finite element context. Two finite elements are developed by assuming linear and hyperbolic bimoment fields. The FEA solutions are shown to provide lower bound representations of the stiffness when compared to those based on conventional beam theories founded on postulated kinematic assumptions.     

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.     

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.     

Theory of initially twisted, composite, thin-walled beams

An asymptotically correct theory for initially twisted, thin-walled, composite beams has been constructed by the variational asymptotic method. The strain energy of the original, three-dimensional structure is first rigorously reduced to be a two-dimensional energy expressed in terms of shell strains. Then the two-dimensional strain energy is further reduced to be expressed in terms of the classical beam strain measures. The resulting theory is a classical beam model approximating the three-dimensional energy through the first-order of the initial twist. Consistent use of small parameters that are intrinsic to the problem allows a natural derivation for all thin-walled beams within a common framework, regardless of whether the section is open, closed, or strip-like. Several examples are studied using the present theory and the results are compared with a general cross-sectional analysis, VABS, and other published results.     

Distortional warping functions and shear distributions in thin-walled beams

In the analysis of thin-walled beams it is often necessary to consider the effects of distortion of the cross-section. The distortion in the plane of the cross-section generates axial warping displacements. On the basis of a known in-plane distortional displacement mode it is possible to derive a unique warping function and the related shear stress distributions. Local axial equilibrium is used to derive the main differential equation for determination of the distortional warping function and shear distributions. In closed single- or multi-celled cross-sections it is necessary to introduce circulation shear force flows around the cells to achieve compatibility of the axial displacement. Methods for analysis of open and closed cross-sections are generalized to include distortional displacement modes. It is shown that axial extension, flexure and torsional warping are included as special cases of distortion. A generalization of the conventional orthogonalization procedure and a normalization technique for distortional modes are also presented. A triple cell cross-section is used to illustrate the generalized calculation procedure and computed results are presented.     

Ultimate load analysis of thin-walled box beams considering shear lag effect

In this paper, the ultimate load of thin-walled box beams undergoing limited plastic strain is investigated with consideration of shear lag effect on the basis of the stress–strain relationship of elastic, linearly hardening materials. In the procedure, calculation formulae for strength increase coefficient, flange effective width ratio, critical values of plastic strain and shear lag coefficient are obtained for thin-walled box beams with elastic, linearly hardening materials. In addition, the relationships among the abovementioned parameters and conditions of boundary, load and aspect ratio L/2b (span length/beam width) of the box beams are established in this paper. For illustration, the numerical results of box beams under certain boundary and load conditions are presented and some conclusions are drawn which may offer references for the application of this procedure in structural design.     

Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams

The classical Vlasov theory for torsional analysis of thin-walled beams with open and closed cross-sections can be generalized by including distortional displacement fields. We show that the determination of adequate distortional displacement fields for generalized beam theory (GBT) can be found as part of a semi-discretization process. In this process the cross-section is discretized into finite cross-section elements and the axial variation of the displacement functions are solutions to the established coupled fourth order differential equations of GBT. We use a novel finite-element-based displacement approach in combination with a weak formulation of the shear constraints and constrained wall widths. The weak formulation of the shear constraints enables analysis of both open and closed cell cross-sections by allowing constant shear flow. We use variational analysis to establish and clearly identify the homogeneous differential equations, the eigenmodes, and the related homogeneous solutions. The distortional equations are solved by reduction of order and solution of the related eigenvalue problem of double size as in non-proportionally damped structural dynamic analysis. The full homogeneous solution is given as well as transformations between different degree of freedom spaces. This new approach is a considerable theoretical improvement, since the obtained GBT equations found by discretization of the cross-section are now solved analytically and the formulation is valid without special attention also for closed single or multi-cell cross-sections. Further more the found eigenvalues have clear mechanical meaning, since they represent the attenuation of the distortional eigenmodes and may be used in the automatic meshing of approximate distortional beam elements. The magnitude of the eigenvalues thus also gives the natural ordering of the modes.     

Nonlinear viscoelastic analysis of thin-walled beams in composite material

Laminated composites of polymeric matrix show anisotropic viscoelastic behaviour, enhanced by temperature and humidity effects. The consideration of anisotropy and viscoelasticity are important for the determination of deformations and, as a consequence, of deformation-related phenomena, as elastic and creep buckling. This paper studies the behaviour of thin-walled beams of composite material under flexure and buckling, taking account of creep effects. The analysis uses a nonlinear viscoelastic finite element code with shell elements, whose basic formulation is given. The use of shell elements allows a better representation of constitutive properties and boundary conditions. Comparison with available analytical results is made for several cases like flexure of an I beam, buckling of beam columns and lateral buckling of this beams. The results show good correlation.     

On free vibration analysis of thin-walled beams axially loaded

In this paper, a numerical–experimental study about natural frequencies of thin-walled beams axially loaded is presented. Moreover, the influence of axial load in the frequencies is studied. The equations of motion are based on Vlasov’s theory of thin-walled beams, which were modified previously to include the effects of shear flexibility, rotatory inertia in the stress resultants. Moreover, a constant axial load is incorporated to the formulation, both in the time and frequency domain. The differential equations are shown to be particularly suitable for analysis in the frequency domain using a state variables approach. A numerical investigation is carried out to reveal the influence of the axial load in several boundary conditions. Finally, free vibration experimental tests are presented, which allow verify the theory presented in this paper and provide good quality data that can be used for checking the accuracy and reliability of different theories.     

工字钢-混凝土组合梁弹性约束畸变屈曲研究

约束畸变屈曲是不同于侧向屈曲和畸变屈曲的一类特殊的屈曲形式,通常发生在组合梁负弯矩区。基于弹性地基压杆方法对组合梁弹性约束畸变屈曲进行了研究。将Svensson压杆模型进行改进,考虑了腹板参与部分,并推导了两种基于改进压杆模型的变轴力稳定计算表达式。借助于有限元方法,分析了现有变轴力弹性地基压杆方法用于组合梁约束畸变屈曲的求解精度,研究结果表明：弹性地基压杆方法对组合梁作用纯弯矩及三角形负弯矩情况符合良好,但对非线性弯矩分布情况精度较差。对约束畸变屈曲引入等效弯矩假设,并对其适用性进行了分析,提出了约束畸变屈曲等效弯矩假设临界长度简化公式,进而通过三步简化实现了连续组合梁弹性约束畸变屈曲计算。图16表7参17     

Numerical method for predicting the elastic lateral distortional buckling moment of a mono-symmetric beam with web openings

When used as floor joists, the new mono-symmetric LiteSteel beam (LSB) sections require web openings to provide access for inspections and various services. The LSBs consist of two rectangular hollow flanges connected by a slender web, and are subjected to lateral distortional buckling effects in the intermediate span range. Their member capacity design formulae developed to date are based on their elastic lateral buckling moments, and only limited research has been undertaken to predict the elastic lateral buckling moments of LSBs with web openings. This paper addresses this research gap by reporting the development of web opening modelling techniques based on an equivalent reduced web thickness concept and a numerical method for predicting the elastic buckling moments of LSBs with circular web openings. The proposed numerical method was based on a formulation of the total potential energy of LSBs with circular web openings. The accuracy of the proposed method's use with the aforementioned modelling techniques was verified through comparison of its results with those of finite strip and finite element analyses of various LSBs.     

Flexural–torsional behavior of thin-walled composite beams with closed cross-section

This paper investigates the static and dynamic characteristics of composite thin-walled beams that are constructed from a single-cell box. The structural model considered herein incorporates a number of nonclassical effects, such as material anisotropy, transverse shear, warping inhibition, nonuniform torsional model and rotary inertia. The governing equations were derived using extended Hamilton's principle and solved using extended Galerkin's method. The effects of fiber orientation on static deflection and natural frequencies are considered and a number of important conclusions are outlined.     

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.     

A large deformation–small strain formulation for the mechanics of geometrically exact thin-walled composite beams

This work presents a new formulation of the geometrically exact thin walled composite beam theory. The formulation assumes that the beam can undergo arbitrary kinematical changes while the strains remain small, thus compatibilizing the hypotheses of the strain measure and the constitutive law of the composite material. A key point of the formulation is the development of a pure small strain measure written solely in terms of scalar products of position and director vectors; the latter is accomplished through the obtention of a generalized small strain vector by decomposition of the deformation gradient. The resulting small strain measure is objective under rigid body motion. The finite element implementation of the proposed formulation is simpler than the finite strain theory implementation previously developed by the authors. Numerical experiments show that the present formulation is very accurate and computationally more efficient than the finite strain formulation, thus it is more convenient for most practical applications.     

Local and distortional buckling of thin-walled short columns

This paper assesses the applicability of Eurocode 3 (EC3) to the prediction of the compression capacity of short fixed-ended columns with different cross-sections. This compression capacity is determined by combining the effective width of plane elements due to local buckling and the effective stiffener thickness due to distortional buckling. Numerical calculations have been carried out in order to compare alternative methods for determining the minimum elastic distortional buckling stress in compression. The method given in EC3 does not correlate as well as Lau and Hancock's method with the results given by Generalized Beam Theory (GBT). The end boundary conditions have a significant influence on the distortional buckling strength, and thus also on the compression capacity of short columns. Selected experimental results from compression tests on C-, Hat- and rack upright-sections are compared with the predictions given by EC3. The procedure in EC3 was modified by determining the distortional buckling stress using GBT, taking into account the actual column length and the end boundary conditions. This lead to better agreement between the experimental results and the theoretical predictions.     

冷弯薄壁型钢构件畸变屈曲研究综述与分析

对冷弯薄壁型钢构件的畸变屈曲进行了较全面的综述,介绍了畸变屈曲的特点和性能,总结了国内外畸变屈曲的研究成果,最后对畸变屈曲尚待研究的问题进行了归纳和探讨。     

Application of surfaces of ultimate strength for thin-walled beams

This paper deals with the problem of ultimate load-carrying capacity of thin-walled sections subject to combined load. That has direct implementation in sizing and design of thin-walled structures. It is solved using the ultimate strength method based on the theory of plastic analysis of structures. It is assumed that the elastic strains are negligible in comparison to the plastic strains and that justifies the application of a fully plastic model. The following problems have to be analyzed before the sizing and design is completed:

• Load vectors and load components

• Locations of the plastic neutral axes

• The surface of ultimate strength

The most important achievement presented in this paper is an improvement for the location of the plastic neutral axis. Until now, the position of the plastic neutral axis has been localized by iterations, starting with the position from the elastic model. That led, in some cases, to a statically inadmissible model and lack of equilibrium in case of asymmetric sections or asymmetric loads.A successful solution to the problem consists in covering the whole section with a mesh of plastic neutral axes and a cluster of corresponding points on the surface of ultimate strength.One of the points on the surface has load components in proportion with the load vector. The corresponding location of the plastic neutral axis is precisely the one we are seeking.The load vector may be extended to pierce one of the triangles that the surface is made of. The coordinates of the point where the load vector is piercing the triangle is a weighted-average of the coordinates of three vertices of the triangle. The same weight is used to localize the plastic neutral axis corresponding to the piercing point of the surface.     

Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections

In this paper, the analysis of the behavior of thin-wa lled beams, derived from Proki ’s work, is carried out by using beam theory with a single warping function valid for arbitrary form of cross sections, without any distinction between open and closed profiles and without using sectorial coordinates. The finite element method is used and numerical examples show the accuracy of the solution by comparison with other numerical or analytical results. For the stability analysis, analytical and numerical calculations of critical loads are given for beams submitted to bending moment and centrally applied forces. Equilibrium equations are established from the principle of virtual work. Critical loads are calculated by considering that a structure already in equilibrium reaches instability if there is one or more than one equilibrium position for the same loading. Results with this formulation are compared to those obtained with classical warping functions.