Static Stability Formulas of a Weakened Timoshenko Column: Effects of Shear Deformations

The static stability analysis of two-dimensional Timoshenko columns weakened at an arbitrary section is derived in a classic manner. The effects of shear deformations along the column, influenced by the additional shear force induced by the applied axial load as the member deforms according to the modified shear equation proposed by Haringx, are presented and studied in detail. The proposed model also captures: (1) the influence on the buckling load capacity of the column when an arbitrary weakened section is formed at any location; (2) the tension buckling phenomenon due to the low shear stiffness of columns made of composite materials or elastomeric rubbers; and (3) the beneficial effects of an additional lateral bracing located at the weakened section to alleviate the buckling load reduction of the column. Seven classical and nonclassical cases of columns mostly used in civil and mechanical engineering are summarized in condensed formulas which allow the straightforward determination of buckling loads and shapes.     

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.     

Stability Criteria for Euler Columns with Intermediate and End Axial Loads

Presented herein are exact stability criteria for Euler columns under intermediate and end concentrated axial loads. The stability criteria cover all combinations of classical boundary conditions, arbitrary location of the intermediate concentrated load, and ratios of the magnitude of the intermediate load to the end load. Also included is the buckling problem of a new class of Euler columns where one segment is in tension while the other segment in compression.     

Nonlinear Analysis of Uniform Pantographic Columns in Compression

The linear analysis of a uniform pantographic deployable column shows that, in bending, its behavior is very similar to that of an equivalent solid column, whereas under axial loading the two columns display distinct differences in their force and deformation distributions. The total change in the height of a particular pantographic unit in the deployable structure consists of two parts, one due to relative rotation of bars in the unit, the other to their bending. To account for configuration changes, the internal forces must satisfy the equilibrium of each unit “after rotation.” The additional pantographic unit deformation due to bending of bars is found to be based on these forces. The set of equilibrium and nonlinear deformation equations is solved iteratively. The “deformation-controlled” approach for solving this system of equations shows the load maximum in the equilibrium paths that corresponds to the snap-through buckling of the top pantographic unit. It is found that the change in the number of units in the column introduces only minor differences in the equilibrium paths as long as the column height and degree of deployment are kept constant. The axial stiffness of the pantographic column is greatly increased and the snap-through buckling considerably postponed if just one additional constraint is introduced, namely the horizontal link between the two nodes at a particular unit interface. The optimal location of the link is found.     

Inelastic Plate Buckling

Current models to determine the local buckling stress of inelastic plates under in-plane loading are based on plastic deformation theory and semirational or empirical relationships. A successful J2 flow theory describing inelastic local buckling of initially perfect plates needs to avoid two well-known pitfalls known as the “inelastic column buckling paradox” and the “plastic buckling paradox.” While the former problem, which found its origin in 1895 in Engesser’s double modulus approach, was resolved by Shanley in the late 1940s, a convincing solution of the plastic buckling paradox has not yet been presented. This paper proposes a modification to the J2 flow theory which hinges on the determination of the shear stiffness from second-order considerations. A differential equation is derived which describes the incremental plate deformations at the inelastic local buckling load. The differential equation is studied for two cases of boundary conditions: a plate simply supported along four edges and a plate simply supported along three edges with one longitudinal edge free.     

Buckling of Circular Mindlin Plates with an Internal Ring Support and Elastically Restrained Edge

This paper is concerned with the elastic buckling problem of circular Mindlin plates with a concentric internal ring support and elastically restrained edge. In solving this problem analytically, the circular plate is first divided into an annular segment and a core circular segment at the location of the internal ring support. Based on the Mindlin plate theory, the governing differential equations for the annular and circular segments are then solved exactly and the solutions brought together by using the interfacial conditions. New exact critical buckling loads of circular Mindlin plate with an internal ring support and elastically restrained edge are presented for the first time. The optimal radius of the internal ring support for maximum buckling load is also found. An approximate relationship between the buckling loads of such circular plates based on the classical thin plate theory and the Mindlin plate theory is also explored.     

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).     

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.     

Euler Critical Force Calculation for Laced Columns

The paper presents a method of solving the buckling problem of laced column as a statically indeterminate structure without analyzing determinants of high order. The flexural and torsional buckling problems of laced column are reduced to the two-point boundary value problem for a difference equation system. The value of Euler critical load is determined as a result of analyzing the fourth order determinant for column with any degree of static indeterminacy. The solution is based on the method of initial values. Stability of columns with any types of lattice (crosswise, serpentine, with batten struts); with any number of lattice panels and with variable lattice spacing can be examined by this manner. The analogy between the flexural and torsional buckling of the laced column is established. It enables one to use the same relations for consideration of both kinds of buckling. The obtained numerical results show that the Euler critical loads calculated by this method can be substantially differed from those based on the approximated Engesser’s approach. A PC program for checking stability of laced column by designer can be developed on the basis of the present method.     

Slender Steel Columns Strengthened Using High-Modulus CFRP Plates for Buckling Control

This paper presents the results of an experimental investigation into the behavior of slender steel columns strengthened using high-modulus (313?GPa), carbon fiber-reinforced polymer (CFRP) plates. Eighteen slender hollow structural section square column specimens, 44×44×3.2?mm, were concentrically loaded to failure. The effectiveness of CFRP was evaluated for different slenderness ratios (kL/r), namely, 46, 70, and 93. The maximum increases in ultimate load ranged from 6 to 71% and axial stiffness ranged from 10 to 17%, respectively, depending on kL/r. As kL/r reduced, the effectiveness of CFRP plates also reduced, and failure mode changed from CFRP plate crushing after occurrence of overall buckling, to debonding prior to, or just at, buckling. A simplified analytical model is proposed to predict the ultimate axial load of FRP-strengthened slender steel columns, based on the ANSI/AISC 360-05 provisions, which were modified to account for the transformed section properties and a failure criteria of FRP derived from the experimental results. It was shown that for a given FRP reinforcement ratio, there is a critical kL/r at the low end, below which FRP may not enhance the strength of the column.     

Buckling of a Weakened Infinite Beam on an Elastic Foundation

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.     

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.     

Moment-Inelastic Rotation Behavior of Longitudinally Stiffened Beams

The significance for inelastic design of moment-inelastic rotation behavior with respect to interior pier sections of steel girder bridges is experimentally investigated. Under center span loading conditions, 12 welded, built-up, simply supported beams with various slenderness ratios of the flange and web plates are tested. In this test, lengths and locations for partial longitudinal stiffeners on the web plates are varied, and the results are then compared with the inelastic deformation capacity of beams without longitudinally stiffened web plates. The results are also compared with the inelastic design code in AASHTO LRFD bridge design specifications. It is concluded that (1) the ultimate strength of stiffened beams is governed by the local buckling at the compression flange of the far end from the loading point due to the presence of a partial longitudinal stiffener; and (2) the inelastic rotation capacity and ultimate strength of a beam with a stiffened web plate are remarkably improved. The optimum length and location of stiffeners on the plates are given.     

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).     

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).     

Analytical Study of Bifurcation and Nonlinear Behavior of Sandwich Columns

Buckling behavior of axially compressed sandwich columns exhibits a variety of interesting phenomena. The column can buckle in a long wave “overall mode” or a short-wave “wrinkling mode,” the latter involving severe bending of the facings and transverse deformation of the core. The full-range nonlinear behavior of these structures is investigated in the elastic range taking typical examples. Columns can be, though not always, imperfection sensitive when wrinkling is the principal mode of buckling. The columns buckling in the overall mode collapse by the formation of localized wrinkles in the postbuckling range. An interesting new finding of this study is that, for certain combinations of core compliance and facing thickness, there can be a bifurcation from the prebuckling path at a load smaller than that predicted by the linear stability analysis. In this scenario, if wrinkling is the dominant mode of buckling, the column turns out to be imperfection sensitive.     

Ductility Evaluation of Concrete-Encased Steel Bridge Piers Subjected to Lateral Cyclic Loading

An experimental and analytical study was conducted to investigate the ductility of concrete-encased steel piers, referred to as “steel-reinforced concrete (SRC) construction.” Based on the cyclic lateral loading tests of SRC column specimens, the restorable and ultimate limit states are defined as the point when concrete cover spalling occurs (equivalent to longitudinal bar buckling) and the point when flange buckling of the H-shaped steel occurs, respectively. To estimate the lateral displacement capacity at both the restorable and ultimate limit states, the curvature distribution of the column was calculated based on the buckling analysis of the longitudinal bar, which was restrained by a concrete cover and transverse reinforcement, and of the steel flange encased in concrete. The lateral displacement was obtained by integrating the curvature distribution. Comparison of the computed results with experimental results, including other writers’ reports, confirmed that the proposed method can appropriately estimate the lateral displacement at the restorable and ultimate limit states, and it can accurately evaluate the buckling characteristics of the longitudinal bar and steel flange components of SRC column specimens.     

减宽侧压板坯变形分析和压下规程改进

板坯调宽侧压过程中常出现失稳弯曲及头尾部鱼尾过长现象,降低了产品的成材率.对减宽侧压过程的板坯金属变形流动情况进行了分析,并结合外端与变形区的相互影响,采用塑性失稳理论对板坯的失稳弯曲现象和头尾部鱼尾变形进行了研究.在理论分析的基础上,提高了压紧辊的间隙,减少了失稳弯曲.提出了新的减宽压下规程,对减宽技术进行了改进,减宽精度明显提高,带钢头尾部的切损减少.     

Energy Capacity Criterion for the Design of Columns against Collapse

Current practice in buckling design is based on the ultimate strength of a member such that excessive lateral deflections do not occur. This ultimate strength is often dictated by empirically derived formulae which correspond to postulated element imperfections including residual stresses. In this paper, the postbuckling behavior of columns is considered, and the limit state is considered to be total collapse. The load versus end displacement relationship is derived for a simplified elastic–perfectly plastic column model. Using this curve, the energy absorbed and released in going from one load/deflection state to another can be easily quantified. With this, the energy absorption capacity of an already loaded column is considered to be the critical performance metric. This “energy capacity criterion” is then used to develop alternative design curves for different levels of energy absorption capacity.     

小方坯连铸末端电磁搅拌综合控制模型

 为解决连铸生产过程中因拉速、过热度等工艺条件频繁波动而导致铸坯凝固末端发生变化,使得末端电磁搅拌(FEMS)难以产生稳定良好的搅拌效果的问题,提出一种FEMS综合控制模型。此模型通过在线凝固传热模型计算得到FEMS安装处的坯壳厚度SF,然后采用基于目标坯壳厚度控制的二冷模型调节二冷水量使SF保持稳定,并根据不同的SF调节FEMS的电流和频率,使FEMS的使用效果达到最优值。采用射钉试验验证了数学模型的计算精度。计算机模拟及现场应用的结果表明,在160mm×160mm小方坯连铸机上,此综合控制模型能够使FEMS安装处的坯壳厚度保持稳定,目标坯壳厚度设定为52mm时,有效提高铸坯的内部质量。