Buckling of Columns with Intermediate Elastic Restraint

This paper presents a comprehensive set of exact stability criteria for Euler columns with an intermediate elastic restraint. A subset of this class of problem is the buckling problem of columns with an intermediate rigid support where the elastic restraint takes on an infinite stiffness. Also, this study reiterates the existence of a critical elastic restraint stiffness in which the buckled mode switches to a higher-buckling mode of the corresponding column without an intermediate support. It is clear that this critical stiffness value exists only when the restraint is placed at the node of the higher-buckling mode and the buckling load associated with this critical stiffness value is the maximum achievable value that can be attained with an intermediate elastic restraint.     

Influence of Shear Deformation on Buckling of Pultruded Fiber Reinforced Plastic Profiles

Theoretical studies of the influence of shear deformation on the flexural, torsional, and lateral buckling of pultruded fiber reinforced plastic (FRP)-I-profiles are presented. Theoretical developments are based on the governing energy equations and full section member properties. The solution for flexural buckling is consistent with the established solution based on the governing differential equation. The new solutions for torsional and lateral buckling incorporate a reduction factor similar to that for flexural buckling. The solution for lateral buckling also incorporates the influence of prebuckling displacements. Closed form solutions for a series of simply supported, pultruded FRP I-profiles, based on experimentally determined full section flexural and torsional properties, indicate the following conclusions. For members subjected to axial compression, shear deformation can reduce the elastic flexural and torsional buckling loads by up to approximately 15% and 10%, respectively. For members subjected to bending, prebuckling displacements can increase the buckling moments by over 20% while shear deformation decreases the buckling moments by less than 5%.     

Complementary Energy Based Formulation for Torsional Buckling of Columns

A unique formulation for the elastic torsional buckling analysis of columns is developed in this paper based on the principle of stationary complementary energy. It is well known that in displacement based numerical formulations, discretization errors lead to stiffer behavior; hence convergence from above. On the other hand, discretization errors in complementary energy based numerical formulations lead to softer behavior in linear elasticity problems, which is a desired feature from the engineering view point. However, complementary energy based formulations can only overpredict the buckling loads for the flexural buckling problems of columns unless the physical conditions are compromised. In this study a formulation based on the principle of stationary complementary energy is considered for the elastic torsional buckling analysis of columns. The complementary energy expression is obtained from the well known total potential energy functional by using Frederichs’ transformation. In contrast to flexural buckling analysis of columns, it is shown that when the principle of stationary complementary energy is used, the torsional buckling loads can be underpredicted. A mathematical proof is provided to elucidate this property. The convergence behavior of the approximate solutions is illustrated through numerical examples for several columns with different boundary conditions.     

Shear-Flexural Buckling of Cantilever Columns under Uniformly Distributed Load

Approximate buckling formulas for shear–flexural buckling of cantilever columns subjected to a uniformly distributed load are derived, based on Timoshenko’s energy method. In this method the deflection curve at buckling is approximated by a trial function. Instead of trying to describe all possible buckling modes with one trial function, two trial functions are used: one to describe shear dominated localized buckling, another to describe bending dominated global buckling. It is investigated whether the bending dominated global buckling modes can best be described using polynomial functions, trigonometric functions, or a function defined by the lateral (flexural and shear) deflection of the cantilever column under uniformly distributed lateral load. The results of the derived formulas are compared to the exact solution and other approximate buckling formulas found in the literature. Attention is drawn to the fact that the shear–flexural buckling load cannot exceed the shear buckling load.     

Static Stability of Beam-Columns under Combined Conservative and Nonconservative End Forces: Effects of Semirigid Connections

Stability criteria that evaluate the effects of combined conservative and nonconservative end axial forces on the elastic divergence buckling load of prismatic beam-columns with semirigid connections is presented using the classic static equilibrium method. The proposed method and stability equations follow the same format and classification of ideal beam-columns under gravity loads presented previously by Aristizabal-Ochoa in 1996 and 1997. Criterion is also given to determine the minimum lateral bracing required by beam-columns with semirigid connections to achieve “braced” buckling (i.e., with sidesway inhibited). Analytical results obtained from three cases of cantilever columns presented in this paper indicate that: (1) the proposed method captures the limit on the range of applicability of the Euler’s method in the stability analysis of beam-columns subjected to simultaneous combinations of conservative and nonconservative loads. The static method as proposed herein can give the correct solution to the stability of beam-columns within a wide range of combinations of conservative and nonconservative axial loads without the need to investigate their small oscillation behavior about the equilibrium position; and (2) dynamic instability or flutter starts to take place when the static critical loads corresponding to the first and second mode of buckling of the column become identical to each other. “Flutter” in these examples is caused by the presence of nonconservative axial forces (tension or compression) and the softening of both the flexural restraints and the lateral bracing. In addition, the “transition” from static instability (with sidesway and critical zero frequency) to dynamic instability (with no sidesway or purely imaginary sidesway frequencies) was determined using static equilibrium. It was found also that the static critical load under braced conditions (i.e., with sidesway inhibited) is the upper bound of the dynamic buckling load of a cantilever column under nonconservative compressive forces. Analytical studies indicate the buckling load of a beam-column is not only a function of the degrees of fixity (ρa and ρb), but also of the types and relative intensities of the applied end forces (Pci and Pfj), their application parameters (ci, ηj, and ξj), and the lateral bracing provided by other members (SΔ).     

Approximate Derivation of Critical Buckling Load

This paper proposes an approximate derivation for the critical buckling load of a column, based on the application of a uniformly loaded beam's midspan moment and deflection to the buckled column's rotational equilibrium. The curvature of a pin-ended member, when it buckles under axial load, is similar to the curvature assumed by the same member when it deflects under a uniformly distributed load applied transversely along its entire length. Euler's famous equation for critical buckling load is based, of course, on the former assumption, in which the deflected column assumes the shape of a sine curve. However, dividing a uniformly loaded beam's midspan moment by its deflection provides a conservative result for the critical buckling load, within 3% of Euler's value, that can be derived solely on the basis of these commonly used beam equations.     

Plastic Torsional Buckling of Cruciform Compression Members

In this paper the plastic torsional buckling of a cruciform column is revisited. The interest in this classical problem resurfaced from a practical application in the area of seismic protection of structures. The theoretical challenges associated with this problem emerge from the “paradoxial” differences between the plastic buckling strength that results from the total deformation and the incremental theories of plasticity. The paper shows that when the flanges of the column are not perfectly straight, the incremental theory of plasticity predicts that at the onset of plastic torsional buckling, the shear stress and the shear strain are related with the tangent shear modulus. The analysis presented herein involves a small-strain theory, examines the column at its slightly deformed configuration, and results are obtained with hand calculations. Experimental evidence supporting the theoretical findings is presented.     

抗弯刚度比对加劲板屈曲性能的影响

以一大型薄壁钢结构的加劲板为研究对象,采用有限元方法,考虑了13种不同的刚度比、多种不同的加劲肋布置方式以及边界条件等因素,分析了加劲板线性屈曲和非线性屈曲性能.抗弯刚度比对加劲板的屈曲性能影响显著,加劲板最佳抗弯刚度比将其线性屈曲模态划分为整体屈曲和局部屈曲,其值为10~20.加劲板非线性屈曲荷载随抗弯刚度比增大而提高.另外,在加载方向增加加劲肋布置可以提高加劲板局部屈曲荷载,在非加载方向增加加劲肋布置对加劲板的局部屈曲性能影响较小.     

Optimal Stabilization of Column Buckling

Linear buckling of column structures is an important design constraint in many structures, particularly where weight is a primary concern. Active strengthening is the application of feedback control to increase the critical buckling load of the structure. An important feature of this control problem is that the structure is inherently unstable when the axial load surpasses the critical buckling load. This research presents a design method for creating optimal buckling control systems using state or static output feedback. The primary feature of this method is the ability to select the designed closed loop, actively strengthened, critical buckling load. The stability of the resulting controllers is determined using Lyapunov methods. Simulation and experimental demonstration of this algorithm is performed using a column employing piezoelectric actuators, and MEMS-based strain sensors. The optimal buckling controllers developed are able to increase the critical buckling load by a factor of 2.9. The closed loop system is able to support lower axial loads indefinitely (>30 min).     

Closed-Form Solutions for Buckling of Long Anisotropic Plates with Various Boundary Conditions under Axial Compression

Closed-form solutions for buckling of long plates with flexural/twist anisotropy with the short edges simply supported and with the longitudinal edges simply supported, clamped, or elastically restrained in rotation under axial compression are presented. An energy method (Rayleigh–Ritz) is employed to obtain the critical buckling loads. The critical buckling loads are expressed in terms of minimum nondimensional buckling coefficients and stiffness parameters. The new closed-form solutions show an excellent agreement when compared to existing solutions and finite-element analysis. Due to their simplicity and accuracy, the new closed-form solutions can be confidently used as an alternative to computationally expensive structural analysis to assess buckling in the preliminary design phase of composite structures.     

Tension and Compression Stability and Second-Order Analyses of Three-Dimensional Multicolumn Systems: Effects of Shear Deformations

The stability and second-order analyses of three-dimensional (3D) multicolumn systems including the effects of shear deformations along the span of each column are presented in a condensed manner. This formulation is an extension to an algorithm presented recently by the writer in 2002 and 2003 by which the critical load of each column, the total critical load, and the second-order response of a 3D multicolumn system with semirigid connections can be determined directly. The proposed solution includes not only the combined effects of flexural deformations and shear distortions along the columns in their two principal transverse axes, but also the effect of the shear forces along each member induced by the applied end axial force as the columns deform and deflect (as suggested by Haringx in 1947 and explained by Timoshenko and Gere in 1961) in their two principal transverse axes. The extended characteristic transcendental equations (corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited) that are derived and discussed in this publication find great applications in the stability and second-order analyses of 3D multicolumn systems made of materials with relatively low shear stiffness such as orthotropic composite materials (fiber reinforced plastic) and multilayer elastomeric bearings used for seismic isolation of buildings. The phenomenon of buckling under axial tension in members with relatively low shear stiffness (observed by Kelly in 2003 in multilayer elastomeric bearings, and recently discussed by the writer in 2005) is captured by the proposed method. Tension buckling must not be ignored in the stability analysis of multicolumn systems made of columns in which the shear stiffness GAs is of the same order of magnitude as π2EI/h2.     

基于非协调F?ppl-von Kármán方程组的冷轧带材后屈曲分析

为了揭示冷轧带材前屈曲面内残余应力与后屈曲挠度、后屈曲残余应力的关系,引入非协调F?ppl-von Kármán方程组,建立了两边自由无限长带条后屈曲的非线性偏微分方程组边值问题模型。根据冷轧带材后屈曲挠度具有轧制方向单波长周期性变化的特点,将非线性偏微分方程组边值问题分离变量而形成非线性常微分方程组边值问题。将边值问题中涉及的各物理量无量纲化,并分析这些物理量的数量级,进而确定出带有待定系数的无量纲挠度函数的形式。然后将总势能写成只与无量纲挠度函数有关的形式,并利用Ritz法确定各待定系数。最后采用其他文献中的计算结果与本文提出方法的计算结果进行对比,发现较为吻合,并解释了产生误差的原因。同时针对某冷轧厂产品计算出后屈曲释放后的残余应力,并计算了使带钢保持平直的最小张应力,为板形仪的合理应用提供了参考。     

偏心框架结构采用扭转调谐液柱阻尼器的设计方法

扭转调谐液柱阻尼器（torsional tuned liquid column damper,TTLCD）是一种有效的扭转振动控制装置.本文对该阻尼器在偏心框架结构中的设计方法进行研究,针对不同的倾斜管道投影点和倾斜角,建立了三种形式的扭转调谐液柱阻尼器在地震作用下的运动方程和控制力方程,给出TTLCD参数优化的具体公式,其方法为对比扭转调谐液柱阻尼器-偏心结构和扭转调频质量阻尼器（torsional tuned mass damper,TTMD）-偏心结构,将两控制系统转化,利用Den Hartog或Ikeda给出的调频质量阻尼器（tuned liquid mass damper,TMD）参数优化公式来实现TTLCD参数优化,最后给出TTLCD的设计流程.通过对一个四层偏心框架结构进行模拟,验证了理论设计方法的合理性.     

Elastic Stability and Second-Order Analysis of Three-Dimensional Frames: Effects of Column Orientation

The elastic stability and second-order analysis of three-dimensional (3D) multicolumn systems including the effects of cross-section orientation of each column are presented in a condensed manner using the classical Timoshenko’s stability functions. This formulation is an extension of an algorithm recently presented by the writer in 2002 by which the effective length K-factor for each column, the total critical load, and second-order analysis of a 3D multicolumn system can be determined directly. Extended characteristic transcendental equations corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited with semirigid connections are derived and discussed. The proposed method is limited to 3D multicolumn systems made up with doubly symmetrical vertical columns with the principal axes of each column cross section oriented in any direction with respect to the floor global axes and with every column sharing the same interstory sidesways (i.e., two horizontal translations and a rotation about the vertical axis). Shear and axial deformations in all members are omitted. Three comprehensive examples are presented and the calculated results compared with those obtained using SAP2000 (Version 6.1, 1997) showing: (1) the effectiveness and simplicity of the proposed approach; (2) its validity to carry out stability and second-order analyses of 3D multicolumn systems; and (3) the importance of the orientation of the cross section of the columns on the lateral response of 3D multicolumn systems. Analytical results indicate that a frame reaches its maximum overall lateral stiffness and a dominant sidesway buckling (without overall frame torsional rotation or twist about its vertical axis) when all columns are oriented with their minor axis tangent to the circumscribed circle such that the multicolumn system acts as a tube offering its maximum twist stiffness.     

Buckling Mode Interaction in Fixed-End Column with Central Brace

The postbuckling behavior of an elastic fixed-end column with an elastic brace at the center is investigated. Attention is focused on those of brace stiffness near its threshold value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. We show that, for the brace stiffness greater than the threshold value, there are precisely two secondary bifurcation points on each primary postbuckling path bifurcating from one of the least two classical buckling loads, and the corresponding secondary postbuckling paths connect all of these secondary bifurcation points in a loop. For the brace stiffness less than the threshold value, no secondary bifurcation occurs. The asymptotic expansions of the primary and secondary postbuckling paths are constructed. The stability analysis indicates that, when the brace stiffness goes beyond its threshold value, the primary postbuckling path with a node in the center becomes unstable from stable by means of the secondary bifurcation (i.e., secondary buckling occurs).     

First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns

In this work, closed-form expressions for the buckling loads of a weakened column with different boundary conditions are presented. The cracked-column model is based on the well-known method consisting of dividing the column into two segments connected by a rotational linear spring whose flexibility is related to the crack size and the geometry of the cross section. For the formulation of closed-form expressions, the perturbation method is used and the results are compared with those found by directly solving the eigenvalue problem.     

Buckling of a Weakened Column

The buckling problem of a column weakened at an interior location is studied for the first time. The weakness is modeled by a rotationally restrained junction. Exact buckling load values are obtained for the weakened column with various end conditions. Depending on the end conditions of the column, the buckling loads show sensitivity (and insensitivity) to junction location and rotational stiffness. The optimum location of the junction could be at the midpoint, at the ends, or somewhere in between.     

Monitoring Steel Girder Stability for Safer Bridge Erection

The collapse of the State Route 69 Bridge over the Tennessee River near Clifton, Tennessee, is an example of how instability and lateral torsional buckling failure of a single steel bridge girder during erection might cause collapse of the whole steel superstructure. Close attention should be given to the stability of steel plate girders during erection when the lateral support provided to the compression flange might temporarily not be present. Rules of thumb in use today have been adopted by contractors/subcontractors to check the stability of cantilever or simply supported girders under erection using the L/b ratio, where L is the unbraced length and b is the compression flange width. For each girder section, a maximum L/b ratio exists beyond which lateral torsional buckling failure would occur under girder self-weight. Parametric studies were conducted following the latest AASHTO LRFD code in order to indentify the maximum L/b ratio for various girder sections and check the rules of thumb, as well as determine the dominating section parameters on girder stability under erection. Advanced nonlinear finite-element analyses were also conducted on a girder section for both the cantilever and the simply supported case in order to further understand the behavior of girder instability due to lateral torsional buckling under the self-weight, as well as to develop a trial-and-error methodology for identifying the maximum L/b ratio using computer analysis. At the same time, the effect of lateral bracing location on the cantilever free end has been investigated, and it turned out that bracing the top tension flange would be more effective to prevent lateral torsional buckling than bracing the bottom compression flange.     

Inelastic Cyclic Model for Steel Braces

A beam–column element that can accurately model the inelastic cyclic behavior of steel braces is presented. A bounding surface plasticity model in stress-resultant space coupled with a backward Euler algorithm is used to keep track of spread of plasticity through the cross section. Deterioration of cross-section stiffness due to local buckling is accounted for through a damage model. The proposed formulation has been implemented in a large deformation analysis program and is shown to be capable of predicting with reasonable accuracy the experimentally observed inelastic behavior of a variety of members subjected to reversed cyclic loading and a subassemblage under simulated seismic conditions.     

Asymptotic-Numerical Method to Analyze the Postbuckling Behavior, Imperfection-Sensitivity, and Mode Interaction in Frames

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an “internally constrained” beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to validate and illustrate the application and potential of the proposed method, several numerical results are presented, concerning (1) four validation examples (Euler column and three simple frames—two or three members), for which there exist some (perfect frame) analytical and numerical asymptotic results reported in the literature; (2) a single-bay pitched-roof frame with partially restrained column bases; and (3) a three-bay frame with two leaning columns. These results comprise (1) the initial postbuckling behavior of perfect frames (individual and coupled buckling modes) and (2) geometrically nonlinear equilibrium paths describing the behavior of frames containing initial geometrical imperfections or primary bending moments. In the latter case, most of the semianalytical results are compared with fully numerical values, yielded by finite element analyses performed in the commercial code ABAQUS.