Stiffness analysis of locally-buckled thin-walled continuous beams

A direct iterative numerical method is presented for predicting the post-local-buckling response of thin-walled continuous structures. Nonlinearities due to local buckling and non-linear material properties are accounted for by the nonlinear moment-curvature relations of the section derived with the aid of effective width concept. Since the effective width of the compression element decreases as the stress borne by the element edge increases, the effective flexural rigidity of the cross-section varies along the member length depending upon the magnitude of the moment at the section. In the post-buckling range, the member is treated as a nonprismatic section. For continuous thin-walled structures, it is further complicated by the fact that the bending moment distribution throughout the structure and the member stiffnesses are interdependent. The proposed direct iterative solution scheme includes a stiffness matrix method of analysis in conjunction with a numerical integration procedure for evaluating the member stiffnesses. The method is employed to analyze continuous beams in the post-buckling range. Using the moment distribution of an elastic prismatic continuous beam based on the nonbuckling analysis as a first approximation, it has been found that the iterative solution scheme converges rapidly.An excellent agreement has been obtained between the results based on the method presented and from an earlier study for continuous beams. The stiffness formulation is direct and is well suited for the analysis of continuous thin-walled structures.     

Lateral buckling of locally buckled beams using finite element techniques

This paper deals with lateral-torsional buckling of beams which have already buckled locally before the occurrence of overall buckling. Due to the weakening effects of local buckling, the stiffness of the beam is reduced. As a result, overall lateral buckling takes place at a lower load than the member would carry in the absence of local buckling. The effective width concept is used in this investigation to account for the post-buckling strength in the buckled compression plate elements of the beam section. A finite element formulation in conjunction with effective width concept is presented. Due to the nonlinearity involved because of local buckling, an iterative procedure is necessary. Search techniques are used to find the load factor. The method combined with an analysis on nonlinear bending moment distribution can be used to analyze the lateral stability problem of locally buckled continuous structure. In this case, both elastic stiffness matrix and geometric stiffness matrix must be revised at each load level. A computer program has been prepared for an IBM 370/165 computer.     

金属缠绕垫片结构参数对压缩-回弹性能的影响

为分析金属缠绕垫片结构参数对螺栓法兰连接系统密封失效的影响规律,以柔性石墨-不锈钢金属缠绕垫片为研究对象,采用试验与数值模拟相结合的方法研究垫片的压缩-回弹性能,并以垫片外缘最小压应力为指标分析螺栓法兰连接系统的密封性能,考核垫片结构参数对垫片外缘最小压应力和密封性能的影响.结果 表明:试验与有限元数值模拟得到的柔性石墨-不锈钢金属缠绕垫片的非线性压缩-回弹曲线吻合很好;与工程常用的柔性石墨-不锈钢金属缠绕垫片结构参数相比,适当减小金属带宽度、石墨带宽度、缠绕层数及金属带角度可以在不同程度上提高螺栓法兰连接系统的密封性能.     

A theoretical and experimental investigation into cold-formed channel sections in bending with the unstiffened flanges in compression

Twenty-six cold-formed channel sections were tested in four-point bending with the unstiffened flanges in compression. The width/thickness ratio for the web varied from 18.8 to 188 and the width/thickness ratio for the flange varied from 3 to 91.75. Post-buckling theories were derived using yield line theory considering both simply supported and partially fixed boundaries to predict the ultimate moment capacity and resulting midspan deflections. Good correspondence was observed between theory and experiment. The use of the Mathcad symbolic algebra package is described in the production of moment–curvature relationships.     

Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

Post-buckling strength of thin-walled members

The finite element displacement method considering both geometrical nonlinearity and material non-linearity has been used to investigate the post-buckling behaviour and the ultimate strength of thin-walled nonplanar (three-dimensional) structural members. The two types of nonlinearities are based on Lagrangian coordinates and the flow theory of plasticity, and the formulations are developed using the variational principle and the incremental variational principle. The tangent stiffness matrix which is derived explicitly up to a point prior to volume integration, has been found to be quite efficient. The cases of a hat-section beam under a concentrated load for a web crippling study and a channel section subjected to combined bending and torsion are used to show the capabilities of the computer program. Results indicate that the conventional linear, elastic analysis over-estimates the strength of thin-walled members and may not even be a useful approximation and that the structure may be excessively deformed when approaching the ultimate load. The study also demonstrates the merit of using the finite element method for detailed investigations of particular problems.     

The large deformation,post-buckling and catastrophic stability of slender cantilever columns

Several solutions, analytical and numerical, exist for the cases of nonlinear bending of beams and post-buckling of columns. However, the solution for the general case of an arbitrarily inclined load does not seem to exist. In this paper, the governing equation of equilibrium for a cantilever of variable cross section under an arbitrarily inclined load is formulated, by using the axial and transverse displacement quantities. The solution is obtained by means of a simple numerical iterative procedure. The results for the post-buckling and non-linear bending of the structure under concentrated tip load, uniformily distributed load and linearly varying load are presented, and compared with the available results, wherever possible. Further, a type of catastrophic behaviour in post-buckling region is observed and studied using the same numerical scheme. The method is general and can be used for any load variation and for any tapered beam-column.     

Non-linear GBT formulation for open-section thin-walled members with arbitrary support conditions

This paper presents the development and illustrates the application of a beam finite element based on Generalised Beam Theory (GBT) and intended to analyse the elastic post-buckling behaviour of thin-walled steel members exhibiting arbitrary open cross-section and with non-standard support conditions (e.g., in-span bracing systems). After briefly reviewing the main concepts and procedures required to obtain the GBT system of non-linear equilibrium equations, the paper describes the steps involved in the numerical implementation (incremental–iterative strategy) of a non-linear beam finite element that incorporates the influence of the non-standard support conditions. Finally, the application and capabilities of the proposed GBT-based beam finite element are illustrated by means of the presentation and discussion of numerical results concerning the post-buckling behaviour of lipped I-section columns and lipped channel columns and beams with and without localised displacement restraints. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the commercial code Ansys.     

Minimum cost design of reinforced concrete beams using continuum-type optimality criteria

This paper outlines a general procedure for obtaining, on the basis of continuum-type optimality criteria (COC), economic designs for reinforced concrete beams under various design constraints. The costs to be minimized include those of concrete, reinforcing steel and formwork. The constraints consist of limits on the maximum deflection, and on the bending and shear strengths. However, the formulation can easily cater for other types of constraints such as those on axial strength. Conditions of cost minimality are derived using calculus of variation on an augmented Lagrangian. An iterative procedure based on optimality criteria is applied to a test example involving a reinforced concrete propped cantilever beam whose cross-section varies continuously. Numerical examples are presented in which the design variables are both the width and the depth or the depth alone, and the optimal costs are compared. The solution of the test example with depth alone as the design variable is confirmed by an alternative approach using discretized continuum-type optimality criteria (DCOC).     

Post-buckling analysis of laminated stiffeners

An analytical-numerical procedure is extended to investigate the local buckling and post-buckling behaviour of thin-walled laminated stiffeners under uniaxial compression. Two stiffener sections are considered: I and channel sections. The stiffener section is treated as a plate assembly. The analysis uses a deflection-type perturbation technique to determine the buckling load and the post-buckling equilibrium path, ensuring compatibility at the plate junctions, adding extra in-plane terms in the strain-displacement relations, and including shear stresses along the flange-web junctions. Meanwhile, the effects of geometrical imperfections are taken into consideration.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. I: Stability formulation and closed-form solutions

For spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections, an improved analytical formulation is proposed. Firstly the displacement field is introduced considering the second order terms of semi-tangential rotations. Next an elastic strain energy is derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. And then the potential energy due to initial stress resultants is consistently derived with accurate calculation of Wagner effect. In addition, closed-form solutions for in-plane and lateral-torsional buckling loads of curved beams subjected to uniform compression and pure bending are newly derived. In the companion paper, FE procedures are developed by using curved and straight beam elements with arbitrary thin-walled sections. In numerical examples, to illustrate accuracy and validity of this study, closed-form solutions for in-plane and out-of-plane buckling loads are presented and compared with those obtained from analytical solutions by other researchers.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. II: F. E. solutions and parametric study

In the companion paper, an improved formulation for spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of semi-tangential rotations. Thus the elastic strain energy and the potential energy due to initial stress resultants are consistently derived. Also closed-form solutions for in-plane and lateral-torsional buckling of curved beams subjected to uniform compression and pure bending are newly derived for mono-symmetric thin-walled curved beams under simply supported and clamped end conditions. In this paper, F. E. procedures are developed by using curved and straight beam elements with non-symmetric cross-sections. Analytical and numerical solutions for spatial buckling of shear deformable thin-walled circular beams are presented and compared in order to illustrate the accuracy and the practical usefulness of this study. In addition, the extensive parametric studies are performed on spatial stability behavior of curved beams. Particularly transition and crossover phenomena of buckling mode shapes with change in curvature and length of beam on buckling for curved beams are investigated for the first time.     

Numerical solution of large deformation problems involving stability and unilateral constraints

The present study is concerned with the numerical treatment of large deformation beam problems where stability as well as post-buckling behaviour is coupled with frictional contact constraints. The flexible beams are described according to a nonlinear rod-type theory which accounts for both finite rotations and large deformations. The contact conditions are introduced via a penalty function method. From these conditions we obtain a linear complementary problem (LCP) resulting from the variational inequality formulation. For the examination of the post-buckling behaviour the displacement control method is applied. Particular attention is paid to the development of the linear complementary problem combining with the computational strategy for tracing limit points. Finally, the modification algorithms of the linear complementary problem, in which the penalty factors have been eliminated, are proposed. The numerical techniques not only allow some limit points to be passed, but also guarantee the computational stability characteristics during the Newton-Raphson's iterative process. Numerical examples are presented that illustrate the performances of the proposed algorithms.     

空间薄壁梁单元面向对象程序的实现

针对传统有限元分析软件主要面向过程设计,其可维护性和可扩展性等较差的问题,基于面向对象程序设计方法,建立具有内部节点的空间薄壁截面梁单元模型,给出线弹性空间薄壁梁单元的UML类图,介绍矩阵类、截面类、材料类、节点类、单元类和结构类等6种类成员的主要属性和方法.用C#编制相应的有限元程序,通过T形框架算例比较和验证其位移和弯曲转角计算值、理论解和ANSYS的BEAM 189梁单元的数值解,结果表明该程序精度良好,可用于空间薄壁结构的有限元分析.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

Equivalent post-buckling models for the flexural behaviour of steel connections

Analytical solutions for the evaluation of the behaviour of steel connections are presented which are able to reproduce their full non-linear behaviour. Because usual models for the analysis of steel connections consist of translational springs and rigid links whereby the springs exhibit a non-linear force–deformation response, usually taken as a bi-linear approximation, they require an incremental non-linear analysis. Using a substitute elastic post-buckling model where each bi-linear spring is replaced by two equivalent elastic springs in the context of a post-buckling stability analysis using an energy formulation, closed-form solutions are obtained for a connection loaded in bending. Application to a beam-to-column welded connection using the component (spring) characterisation of code regulations yields the same results in terms of moment resistance and initial stiffness, being additionally able to trace the full unstiffening response.     

On the optimal design of the shape of a buckled elastic beam

Analysis of stability, post-buckling bending and vibrations is performed for a beam (a spring element) having an optimal shape. A buckled pin-jointed spring element of a constant thickness and variable width is considered. The optimal shape of this beam is suggested to provide a uniform distribution of maximum bending stresses in its buckled equilibrium configuration for a given value of a supercritical axial force. Sensitivities of a critical force and a buckling mode to variations of the shape of a beam are calculated. A dependence of the static lateral deflection upon an axial force is analysed. Nonlinear equations of large-amplitude oscillations are derived by a use of the Hamilton principle. The natural frequencies of a spring element, compressed by a supercritical force are calculated. Received April 29, 1999     

变截面铁木辛柯梁振动特性快速计算方法

提出了一种快速计算变截面铁木辛柯梁横向振动特性的方法.基于铁木辛柯梁理论建立的变截面梁的横向振动方程,其梁的截面参数如有效剪切面积、密度、弯曲刚度、转动惯量等沿梁轴线连续或非连续变化；首先将变截面梁等效为多段均匀阶梯梁；然后基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)连续条件,建立相邻两段模态函数间相互关系,并递推出首段段与末段模态函数相互关系,利用边界条件得到相应特征方程,使用Newton-Raphson方法计算其固有频率；最后针对梁常见边界条件,得到计算变截面铁木辛柯梁横向振动固有频率特征     

Nonlinear analysis of steel frameworks through direct modification of member stiffness properties

This paper presents a computer-based method for nonlinear analysis of planar steel frameworks under monotonic loading that is directly based on the matrix displacement method of analysis. Stiffness degradation factors progressively deteriorate the post-elastic bending, shearing and axial stiffness properties of framework members over an incremental load history until failure of part or all of the structure occurs. The analytical procedure is based on Timoshenko beam theory to allow for the analysis of frameworks for which effects of shear deformation on elastic behavior may be significant, and employs transcendental stability functions to model the effect of axial force on the bending stiffness of members. Also accounted for is the influence of residual stresses on the initial-yield and full-yield capacities of members, and the effect of both combined moment plus axial force and combined moment plus shear force on the post-elastic behavior of members. The nonlinear analysis method is applied for two benchmark planar steel structures, and it is shown to give results comparable to both experimental and analytical results previously published in the literature. The method is readily implemented on a computer using conventional matrix structural analysis techniques, and is directly applicable for the efficient nonlinear analysis of frameworks of any scale or complexity.     

Computer simulation of the scatter in steel member strength

A Monte Carlo method for digital computer simulation of the strength of (steel) members and structures is presented and is applied to rolled steel beams and columns, and thin-walled cylinders. Input data are cumulative distribution functions (histograms) for the geometric and strength variables. The output (i.e. the scatter in structural strength) is printed as histograms and is statistically analysed.Each output histogram is compared with the Gaussian normal distribution. Using the nonparametric test of homogeneity a number of histograms may then be compared.The case studies presented deal with the plastic strength of steel beams and the maximum load of axially loaded steel columns and thin-walled cylinders. Mathematical models for beams subject to pure bending moment, moment and axial force, moment and shear, or uniform torsion are presented. For the initially straight, centrally loaded column a tangent modulus theory which considers residual stresses is used.The simulations have been carried out for one HEA beam, four HEB beams and three IPE beams. Comparison of the simulation results show that the scatter in load carrying capacity of the simulated beams and columns can be regarded as normally distributed, that the load carrying capacity of beams and columns of the same group (HEB or IPE) and beams and columns of the groups HEA and HEB have distributions which differ very little from each other, and that the scatter in simulated beam strength, and in simulated column strength for short and medium length columns, is much more affected by the variation in yield strength of the material than by the variation in cross sectional data. This conclusion holds for ordinary distributions in yield strength of structural carbon steel.Comparisons of simulation results and test results show good agreement for the beams. The agreement is not so good for the columns mainly because in the tangent modulus theory it is assumed that the columns are initially straight. For the cylinders excellent agreement was achieved.The experience gained with the simulation system presented here shows that a medium size computer can be economically used to simulate a relatively large number of plays.