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.     

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.     

Stability of composite thin-walled beams with shear deformability

In this paper, a theoretical model is developed for the stability analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and non-uniform warping), featured in a consistent way by means of a linearized formulation based on the Reissner’s Variational Principle. The model is developed using a non-linear displacement field, whose rotations are based on the rule of semi-tangential transformation. This model allows to study the buckling and lateral stability of composite thin-walled beam with general cross-section. A finite element with two-nodes and fourteen-degrees-of-freedom is developed to solve the governing equations. Numerical examples are given to show the importance of the shear flexibility on the stability behavior of this type of structures.     

Effect of tangent track buckle on vehicle derailment

In order to investigate the effect of a tangent track buckle on the dynamic derailment of a railway vehicle, a coupled vehicle/track dynamics model is developed, in which the vehicle is modeled as a 35 D.O.F. multibody system and the track is modeled as a 3-layer discrete elastic support model. Rails are assumed to be Timoshenko beams supported by discrete sleepers, and the effects of vertical and lateral motions and rolling of the rail on the wheel/rail creepages are taken into account. The sleepers are treated as Euler beams on elastic foundation for the vertical vibration, while as lumped masses in the lateral direction. A moving sleeper support model is developed to simulate the effect of the periodical discrete sleepers on the vehicle/track interaction. The vehicle and the track are coupled by wheel/rail contacts whereas the normal forces and the creep forces are calculated using the Hertzian contact theory and the nonlinear creep theory by Shen et al., respectively. The equations of motion of the coupled vehicle/track system are solved by means of an explicit integration method. A tangent track buckle is simulated with a cosine function, which describes the misalignment of the track with different lengths due to its buckling. In the analysis the effects of the buckle wavelength and amplitude and of the vehicle speed on the dynamic behavior of the coupled vehicle/track system are considered. The present paper analyzes in detail the conventional derailment coefficients which include the ratio of the wheel/rail lateral force to the vertical force, the wheel load reduction, and the new criteria indicating the wheel/rail contact point traces and the wheel rise with respect to the rail. These criteria are simultaneously used to evaluate the risk of derailment of the whole vehicle. The numerical results obtained indicate that the track misalignment caused by the buckle and the vehicle speed have a great influence on the whole vehicle running safety when the vehicle passes through the buckled tangent track.     

Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.     

Lateral buckling analysis of beams by the fem

Two new finite element formulations for the calculation of the lateral buckling load for elastic straight prismatic thin-walled open beams under conservative static loads, are presented. The stability criterion used is based on the positive definiteness of the second variation of the total potential energy. One formulation is suitable for sections where the initial bending is about a dominant major axis. The other finite element formulation takes account of initial bending curvature and essentially takes the form of a quadratic eigenvalue problem. Both formulations are tested with problems that have classical solutions or experimentally determined results and are shown to be accurate.     

Lateral buckling of tee beams under moment gradient

This paper investigates the elastic lateral buckling of simply supported tee beams under moment gradient. Solutions are obtained via the energy approach, assuming a general Fourier sine series for the buckled shapes. It is found that buckling capacity of tee beams is greatly influenced by the inability of the web to resist compressive bending stresses. Although uniform moment is the most severe loading condition for inverted tee beams, this is not true in the case of tee beams. This study shows that a high moment gradient is generally a more critical form of loading for tee beams. Formula for the buckling moment modification factor for doubly symmetric I-beams found in design specifications and standard texts could be unsafe when applied to tee beams.     

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.     

Review and comparison of finite element flexural–torsional models for non-linear behaviour of thin-walled beams

The authors have developed a beam finite element model in large torsion context for thin-walled beams with arbitrary cross sections [1]. In the model, the trigonometric functions of the twist angle θ_x (c = cos θ_x − 1 and s = sin θ_x) were included as additional variables in the whole model without any assumption. In the present paper, three other 3D finite element beams are derived according to three approximations based on truncated Taylor expansions of the functions c and s (cubic, quadratic and linear). A finite element approach of these approximations is carried out. Finally, it is worth mentioning that the promising results obtained in [1], [2] encourage the authors to extend the formulation of the model in order to include load eccentricity effects. Solution of the non-linear equations is made possible by Asymptotic Numerical Method (ANM) [3]. This method is used as an alternative to the classical incremental iterative methods. Many comparison examples are considered. They concern the non-linear behaviour of beams under twist moment and the post buckling behaviour of struts under axial loads or the beam lateral buckling under eccentric bending loads. The obtained results highlight the discrepancies between the various approximations often employed in thin-walled beams literature for the geometrically non-linear analysis of beams in flexural–torsional behaviour.     

Inelastic distortional buckling of I-beams

A finite element method of analysis is developed for the inelastic distortional buckling of determinate, hot-rolled I-beams. The method permits an economic computer analysis to be made of inelastic distortional buckling of members under various conditions of loading, end support and restraint. Following studies of the accuracy and convergence of the method, its scope of application is demonstrated by studing the inelastic buckling of simply supported beams, beams on seats and beams with complete and continuous tension flange restraint.     

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.     

On the stability of plates under combined compression and in-plane bending

The conventional stability analysis of plates under combined compression and in-plane bending is based on the assumption that the plate is free to move laterally and, hence, the restraints imposed by the attached elements against this motion are ignored. The paper explores the influence of these restraints on the plate under this type of loading. The unloaded edges are assumed to be partially restrained against in-plane translation while remaining straight and the distributions of the resulting forces acting on the plate are shown. The stability analysis is done numerically using the Galerkin method and various strategies that economize the numerical implementation are presented. The results are obtained showing the variation of the buckling load, from free edge translation to fully restrained, for simply supported and clamped unloaded edges for various plate aspect ratios and stress gradient coefficients. An apparent decrease in the buckling load is observed due to these destabilizing forces acting in the plate and changes in the buckling mode are observed by increasing the intensity of the lateral restraint. A comparison is made between the buckling loads predicted from various formulae in stability standards based on free edge translation and the values derived from the present investigation. A difference of about 34% in the predicted buckling load and different buckling load were found.     

A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint

In this paper, a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity. For various end boundary conditions, we transform the governing equation with varying coefficients to linear algebraic equations; then a characteristic equation in critical buckling loads will be obtained. Several examples of estimating buckling loads under typical end supports are discussed. By comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams, it can be found that our method has fast convergence and the obtained numerical results have high accuracy. Moreover, the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties. The effects of gradient parameters on the critical buckling loads are elucidated. Finally, we give an example to illustrate the enhancement of the load-carrying capacity of tapered beams for admissible shape profiles with constant volume or weight. The proposed method is of benefit to optimum design of beams against buckling in engineering applications.     

A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded

A coupled torsional-bending finite element with shear deformations and rotatory inertia for vibration of nonsymmetric thin walled beams axially loaded is developed. The equations of motion are based on Vlasov’s theory of thin-walled beams, which are modified to include an axial load. The formulation is also applicable to solid beams. The Hermite cubic polynomials are adopted as shape functions. Mass, elastic stiffness and geometrical stiffness matrices of unsymmetrical cross-section beams are presented. In order to verify the accuracy of this theory and the corresponding beam element developed, a numerical study is presented and compared with the literature and experimental tests.     

Thermo-elastoviscoplastic snapthrough behavior of cylindrical panels

The thermo-elastoviscoplastic snapthrough behavior of simply supported cylindrical panels is investigated. The analysis is based on nonlinear kinematic relations and nonlinear rate-dependent unified constitutive equations which include both Bodner-Partom's and Walker's material models. A finite element approach is employed to predict the inelastic buckling behavior. Numerical examples are given to demonstrate the effects of several parameters which include the temperature, thickness and flatness of the panel. Comparisons of buckling responses betweeen Bodner-Partom's model and Walker's model are given. The creep buckling behavior, as an example of time-dependent inelastic deformation, is also presented.     

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.     

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.