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     

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.     

Prediction of deflection of reinforced concrete shear walls

Reinforced concrete shear walls are used in tall buildings for efficiently resisting lateral loads. Due to the low tensile strength of concrete, reinforced concrete shear walls tend to behave in a nonlinear manner with a significant reduction in stiffness, even under service loads. To accurately assess the lateral deflection of shear walls, the prediction of flexural and shear stiffness of these members after cracking becomes important. In the present study, an iterative analytical procedure which considers the cracking in the reinforced concrete shear walls has been presented. The effect of concrete cracking on the stiffness and deflection of shear walls have also been investigated by the developed computer program based on the iterative procedure. In the program, the variation of the flexural stiffness of a cracked member has been evaluated by ACI and probability-based effective stiffness model. In the analysis, shear deformation which can be large and significant after development of cracks is also taken into account and the variation of shear stiffness in the cracked regions of members has been considered by using effective shear stiffness model available in the literature. Verification of the proposed procedure has been confirmed from series of reinforced concrete shear wall tests available in the literature. Comparison between the analytical and experimental results shows that the proposed analytical procedure can provide an accurate and efficient prediction of both the deflection and flexural stiffness reduction of shear walls with different height to width ratio and vertical load. The results of the analytical procedure also indicate that the percentage of shear deflection in the total deflection increases with decreasing height to width ratio of the shear wall.     

Stability of frames with semirigid joints

This paper presents a modified stiffness matrix method for finding the elastic buckling load of semirigid frames. The method, besides accounting for the partial rigidity of the joints, also considers the effects of flexure on axial stiffness, geometric changes and the P-Δ effect. An experimental test has been devised to check the validity of the theoretical analysis and it is found that the method gives results of reasonable accuracy. A computer program has been written to carry out the doubly iterative process of finding the buckling load and is used to make a parametric study on the effect of rigidity of various joints and also bracings on a single-bay double-storey frame. The results of the parametric studies are presented in charts.     

The effect of material and thickness variability on the buckling load of shells with random initial imperfections

The effect of material and thickness imperfections on the buckling load of isotropic shells is investigated in this paper. For this purpose, the concept of an initial ‘imperfect’ structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as the thickness of the shell. The initial geometric imperfections are described as a two-dimensional uni-variate (2D-1V) stochastic field with statistical properties that are either based on an available data bank of measured initial imperfections or assumed, in cases where no experimental data is available. In order to describe the non-homogeneous characteristics of the initial imperfections, the spectral representation method is used in conjunction with an autoregressive moving average model with evolutionary power spectra based on a statistical analysis of the experimentally measured imperfections. In cases where no experimental results is available, the initial imperfections are assumed to be homogeneous and their impact on the buckling load is investigated on the basis of ‘worst’-case scenarios with respect to the correlation length parameters of the stochastic fields. The elastic modulus and the shell thickness are described as 2D-1V non-correlated homogeneous stochastic fields, while the stochastic stiffness matrix of the shell elements is formulated using the local average method. The Monte Carlo Simulation method is used to calculate the variability of the buckling load, while for the determination of the limit load of the shell, a stochastic formulation of the elastoplastic and geometrically non-linear TRIC facet triangular shell element is implemented.     

Vibration and local instability of thermally stressed plates

The vibration and buckling of a double wedge square cantilever plate has been investigated. It is shown that the free vibration modes, which occur at ΔT_ref = 0, transition into the buckled modes which occur at ΔT_ref = ΔT_refcr for the respective mode. ΔT_refcr for a particular mode is defined as the magnitude of thermal load at which the frequency of the particular mode vanishes. The analysis, in which no assumption whatsoever is made about the shape of the vibration modes, about the vibration frequencies, about the shape of the buckled modes, or about the magnitude of the critical loads, yields the same number of buckling eigenvalues and buckling modes as there are vibration eigenvalues and vibration modes. Gradual application of the load in the analysis permits the change in each vibration frequency of interest and its associated mode to be followed up to the load at which the frequency of the mode becomes zero. This constitutes the limit of linear theory. Only linear theory is used in this paper; thus, no post buckled behavior is considered. As the load is increased, the thin edges of the plate begin to duform during vibration. This local deformation, which begins in the vibration mode, is shown to transition into the phenomena of local edge buckling at ΔT_refcr for the mode.     

Application of a refined plate bending element to buckling problems

The buckling stiffness matrix of a refined plate bending element is derived for various continuous and non-uniform distributions of in-plane forces. Elements of this matrix are expressed in an explicit form and can, therefore, be readily used in a finite element computer program for solving plate buckling problems with various loading and edge conditions. A number of problems are solved and the results obtained are compared with analytical and/or numerical results obtained by using other procedures. It is shown that the refined plate bending element is superior to simpler elements when used for solving plate buckling problems.     

Inelastic stability of laterally unsupported i-beams

A computer method to study the inelastic stability of laterally unsupported steel I-beams and based on a general non-linear theory is presented.Traditionally, the problem of flexural-torsional stability of beams is treated as a lateral buckling problem. Some of the draw-backs of these earlier studies are given below:The classical theory assumes that the deformations are small. In addition the deformation field is linearized. This theory is therefore valid only when the major axis flexural rigidity is much greater than its minor axis rigidity, so that deformations before the onset of lateral buckling are negligible.The lateral buckling theory is valid for straight beams, with loads applied rigorously in the plane of symmetry. Practical beams have initial imperfections and unavoidable load eccentricities. So the true behavior is better described by the stability phenomenon.For beams of intermediate length for which buckling occurs in the inelastic range, the tangent modulus theory is generally used. For ideally straight beams the tangent modulus theory provides an estimate for the collapse load which is slightly conservative. However, for practical beams with initial deformations, this need not be the case.In the majority of existing studies on inelastic lateral buckling, the differential equations for beams under uniform moment are used without modification for beams under moment gradient. In the later case the shear center line is inclined to the centroidal and geometrical axes. The differential equations for beams under uniform moment should therefore be modified by adding additional terms.The majority of the existing studies are limited to the behavior of isolated beams with simple end-conditions and so the beneficial effect of adjacent members on the beam collapse load cannot be studied accurately.A general non-linear theory to describe the spatial behavior of beams and that doesn't have the deficiencies mentioned above, is developed in the present paper.The paper also presents a computer method of solving these non-linear equations using the method of finite differences. Several numerical examples presented and comparison with the existing theoretical and experimental results show the applicability of the theory to a wide range of problems.     

Post buckling behavior of transverse cracked columns

Post buckling behavior of a column with a transverse surface crack on the one side is studied, considering the local flexibility because of the crack. This flexibility, also called compliance, is known to be related to the stress intensity factor. This relation is generalized and expressed in the form of a complete local stiffness matrix of the cracked section of the beam. The Paris equation for the deflection of cracked members is extended for this purpose to give the generalized influence coefficients, being considered as incremental deflections because of the presence of the crack.Eigenvalue solutions for the buckling load are developed which do not differ for appreciably slender columns from known solutions based on some only of the local flexibility coefficients reported in the literature.Moreover, two distinct buckling modes have been identified to closing cracks, because of the different behavior of the cracked region in compression and tension or positive and negative bending modes.The post buckling behavior has been studied, for both buckling modes solving numerically the nonlinear equation for the elastica with the local flexibilities because of the crack.As expected, this behavior of the column is, in general, stable with positive slopes. However, due to the character of the closing cracks, jumping phenomena are governing the transition from the zero equilibrium to the post buckling equilibrium paths.The postbuckling behavior is finally tabulated for a simply supported column as a function of the crack depth.     

Compressive buckling analysis and design of stiffened flat plates with multilayered composite reinforcement

This paper describes an analysis and its application in design for compressive buckling of flat stiffened plates considered as an assemblage of linked orthotropic flat plate and beam elements. Plates can be multilayered, with possible coupling between bending and stretching. Structural lips and beads are idealized as beams. The plate and the beam elements are matched along their common junctions for displacement continuity and force equilibrium in an exact manner. Buckling loads are found as the lowest of all possible general and local failure modes. The mode shape is used to determine whether buckling is a local or general instability and is particularly useful to the designer in identifying the weak elements for redesign purposes. Typical design curves are presented for the initial buckling of a hat stiffened plate locally reinforced with boron fiber composite.     

On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.     

不同脱层位置下脱层屈曲梁振动实验研究

通过实验对一端固定一端夹支脱层屈曲梁在轴向周期激励作用下的非线性动力响应进行了实验研究.利用位移时间历程图,相图和频谱图,对多组不同脱层位置下脱层屈曲梁的非线性动力响应进行了分析.实验表明脱层梁结构存在倍周期以及混沌运动等非线性动力学行为.同时实验还表明,在相同的脱层长度下,脱层位置对脱层梁的动力学特性有明显影响,即脱层区域中心越靠近梁结构的中心位置,脱层梁的一阶自然频率越低,且越容易在较低的激励频率和激励荷载下发生周期分叉和混沌等行为.     

隔振对象重量变化对准零刚度隔振器隔振性能的影响

本文主要研究隔振对象重量变化对一类准零刚度隔振器隔振性能的影响,并给出了新的研究结果.文中使用欧拉屈曲梁构建负刚度调节结构并设计了隔振系统的平衡位置可调机构.假设系统有轻微的过载和超载,推导了系统的动力学方程并进行求解,定义了非线性隔振系统的力传递率及位移传递率来评价系统的隔振性能.对线性隔振系统,超载会让隔振频率略微降低,共振放大峰略微增大.对于准零刚度隔振系统,力传递率和线性系统类似,但是对于位移传递率,过载会导致系统固有频率和共振放大峰均升高,反而不利于隔振.研究结果可以对此类隔振系统的使用场合以及对过载和轻载的选择有工程指导意义.     

An improved numerical scheme for characterizing dynamic behavior of high-speed rotating elastic beam structures

The demand for designing high-speed turbomachinery has led to intensive research in dynamic modeling of rotating elastic mechanisms in recent years. Such a demand in design setting can be addressed more effectively with the development of a more efficient computational scheme. In this paper we present an improved numerical method with three new features. First of all, the time separation concept is introduced to allow time independent terms being computed separately and assembled with time dependent terms in each time marching cycle to form global system equations. Second, the Timoshenko beam with nonlinear geometric stiffness is modeled with exact tangent matrix as opposed to conventional pseudo-tangent matrix approximation. Third, the computational scheme is implemented in homogeneous coordinates that provide a more natural and efficient vector representation. Kane's classic rotating beam problem is used to test for accuracy and computer time. The result matches very well with Kane's solution. However, the computer time needed for the present approach is reduced by more than 70%.     

Elastic stability and buckling loads of multi-span nonuniform beams,shafts and frames on rigid or elastic supports by finite element method using planar uniform line elements

A numerical computer method using planar flexural finite line element for the determination of buckling loads of beams, shafts and frames supported by rigid or elastic bearings is presented. Buckling loads and the corresponding mode vectors are determined by the solution of a linear set of eigenvalue equations of elastic stability. The elastic stability matrix is determined as the product of the bifurcation sidesway flexibility matrix and the second order bifurcation sidesway stiffness matrix which is formed using the element bifurcation sidesway stiffness matrices. The bifurcation sidesway flexibility matrix is determined by partitioning the inverse of the global external stiffness matrix of the system which is formed from the element data using the element stiffness matrices. The method is directly applicable to the determination of the buckling loads of beams and frames partially or fully supported by elastic foundations where the foundation stiffness is approximated by a discrete set of springs. The method of the article provides means to consider complex boundary conditions in buckling problems with ease. Four numerical examples are included to illustrate the industrial applications of the contents of the article.     

Buckling of sandwich beams with compliant interfaces

Buckling of elastic sandwich beams is analyzed accounting for the compliance of the interfaces between the skin and core. A relation between tractions and displacement jumps across the interfaces characterizes the interfacial compliance. Timoshenko co-rotational beam elements are used to discretize each layer of the sandwich. The dependence of the bifurcation load on the stiffness of the core and on the interfacial compliance are illustrated by considering examples of a sandwich beam with two sets of boundary conditions. It is shown that the load at bifurcation buckling is sensitive to the compliance of the interfaces and that a sufficiently large interfacial compliance can significantly decrease the bifurcation load.     

The localization of buckling modes in nearly periodic trusses

The global buckling problem of nearly periodic trusses, including intermediate weak members and disordered panels, is considered. The possibility of local buckling or plastification of individual bars is neglected. The problem is solved numerically using a finite element program. The results show the buckling load factor as well as the buckled mode shapes as functions of a disorder related parameter. These relations are found to be of a nonlinear nature for both the critical loads and the normalized displacements of selected nodes of the studied structures for a given mode shape. Thus, small changes in the disorder related parameter result in a sharp decrease of normalized displacements everywhere but at a certain region of the trusses presented. These results suggest that the phenomenon of buckling mode localization may be utilized in order to passively control structural elastic instability.     

Bifurcation, pre- and post-buckling analysis of frame structures

A procedure is developed for investigating the stability of complex structures that consist of an assembly of stiffened rectangular panels and three-dimensional beam elements. Such panels often form one of the basic structural components of an aircraft or ship structure. In the present study, the stiffeners are treated as beam elements, and the panels between them as thin rectangular plate elements, which may be subject to membrane and/or bending and twisting actions.

The main objective of the study is the determination of the critical buckling loads and the generation of the complete force-deformation behavior of such structures within a specified load range, based on the use of a computer program developed for this purpose. The present formulation can trace through the postbuckling or post limit behavior whether it is of an ascending or descending type. A limit load extrapolation technique is automatically initiated within the computer program, when the stability analysis of an imperfect or laterally loaded structure is being carried out.

The general approach to the solution of the problem is based on the finite element method and incremental numerical solution techniques. Initially, nonlinear strain-displacement relations together with the assumed displacement functions are utilized to generate the geometric stiffness matrices for the beam and plate elements. Based on energy methods and variational principles, the basic expressions governing the behavior of the structure are then obtained. In the incremental solution process, the stiffness properties of the structure are continuously updated in order to properly account for large changes in the geometry of the structure.

The computer program developed during the course of this study is referred at as GWU-SAP, or the George Washington University Stability Analysis Program.     

Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.