Buckling of Variable Section Columns under Axial Loading

In this paper, the static stability of the variable cross section columns, subjected to distributed axial force, is considered. The presented solution is based on the singular perturbation method of Wentzel-Kramers-Brillouin and the column is modeled using Euler-Bernoulli beam theory. Closed-form solutions are obtained for calculation of buckling loads and the corresponding mode shapes. The obtained results are compared with the results in the literature to verify the present approach. Using numerous examples, it is shown that the represented solution has a very good convergence and accuracy for determination of the instability condition.     

Exact Solution for Buckling of Columns Including Self-Weight

In this technical note, analytical solutions for the elastic buckling of heavy columns with various combinations of end conditions are derived, for the first time, in terms of generalized hypergeometric functions. The benchmark solutions may be used to assess the accuracy of approximate formulas and numerical solutions.     

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.     

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.     

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.     

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.     

Buckling of GFRP Columns: An Empirical Approach to Design

This paper presents the results of an experimental study concerning the buckling characteristics of pultruded columns with a hollow circular cross section. Commercially available structural elements were selected from three different manufacturers. Resonant frequency and short-column strength were use to obtain the bending stiffness and the local buckling load of the material. Based on such experimental work, a new design method is proposed herein following Maquoi and Rondal’s formulation. The proposed method has been verified experimentally with test results of hollow rectangular closed section columns and also with experimental work gathered from the literature. In all cases, the method developed in this project proved to be successful. Finally, such method was compared with Barbero and Tomblin’s method, showing some differences in the values predicted with the two methods. Some experimental results, which were predicted accurately with the method proposed herein, were overestimated with Barbero and Tomblin’s method.     

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.     

Effect of Base Fixity on Buckling of Heavy Column with End Load

This paper studies the effect of a rotationally restrained base on the buckling of a standing column subjected to both its own weight and a tip load. The characteristic equation is derived analytically in terms of Bessel functions. The results show stability is greatly compromised when the base is not securely fixed. Rotational spring constants for some simple base constraints are estimated.     

Postbuckling of Axially Loaded Functionally Graded Cylindrical Panels with Piezoelectric Actuators in Thermal Environments

A compressive postbuckling analysis is presented for a functionally graded cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electrical, and thermal loads. The temperature field considered is assumed to be of uniform distribution over the panel surface and through the panel thickness and the electric field considers only the transverse component EZ. The material properties of the presently considered functionally graded materials (FGMs) are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, whereas the material properties of the piezoelectric layers are assumed to be independent of the temperature and the electric field. The governing equations are based on a higher-order shear deformation theory with a von Kármán-Donnell-type of kinematic nonlinearity. A boundary layer theory for shell buckling is extended to the case of hybrid FGM cylindrical panels of finite length. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect FGM cylindrical panels with fully covered piezoelectric actuators, under different sets of thermal and electrical loading conditions. The effects due to temperature rise, volume fraction distribution, applied voltages, panel geometric parameters, in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Performance of Axially Loaded Short Rectangular Columns Strengthened with Carbon Fiber-Reinforced Polymer Wrapping

This paper presents results of a comprehensive experimental investigation on the behavior of axially loaded short rectangular columns that have been strengthened with carbon fiber-reinforced polymer (CFRP) wrap. Six series, a total of 90 specimens, of uniaxial compression tests were conducted on rectangular and square short columns. The behavior of the specimens in the axial and transverse directions is investigated. The parameters considered in this study are (1) the concrete strength; (2) the aspect ratio of the cross section; and (3) the number of CFRP layers. The findings of this research can be summarized as follows: The CFRP wrapping enhances the compressive strength and the ductility of both square and rectangular columns, but to a lesser degree than that of circular columns. The ultimate strength and the ductility of the CFRP confined concrete increase with increasing number of confining layers. The increase in strength and ductility is more significant for lower strength concrete, representing poor or degraded concrete, than for normal-to-high strength concrete; that is, the maximum gain in strength that can be achieved for 3 ksi concrete wrapped columns is approximately 90%, as compared to only 30% for 6 ksi concrete wrapped columns. The CFRP confining jacket must be sufficiently stiff to develop appropriate confining forces at relatively low axial strain levels. The gain in compressive strength obtained by the CFRP confined concrete depends mainly on the relative stiffness of the CFRP jacket to the axial stiffness of the column.     

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.     

Classic Buckling of Three-Dimensional Multicolumn Systems under Gravity Loads

The elastic stability of three-dimensional (3D) multicolumn systems under gravity loads is analyzed in a condensed manner using the classical Timoshenko stability functions. The characteristic equations corresponding to multicolumn systems with sidesway uninhibited, partially inhibited, and totally inhibited are derived. Using the transcendental equations of the proposed method, the effective length K factor for each column and the total critical axial load of an entire story can be determined directly. The proposed method is applicable to 3D framed structures with rigid, semirigid, and simple connections. It is shown that the elastic stability of framed structures depends on: (1) the axial load pattern on the columns; (2) the variation in size and height among the columns; (3) the plan layout of the columns; (4) the overall floor-torsional sway caused by any asymmetries in the loading pattern, column layout, and column sizes and heights (all of which reduce the flexural-buckling capacity of multicolumn systems); (5) the end restraints of the columns; and (6) the bracings along the two horizontal and rotational directions of the floor plane. The proposed method solves the classical bifurcation stability of 3D frames directly without complex matrix solutions, however, it is limited to frames made up of columns of doubly symmetrical cross section with their principal axes parallel to the global axes. Examples are presented that show the effectiveness of the proposed method and the results compared with those obtained by complex matrix methods.     

Buckling and Vibration Analysis of Laminated Panels Using VICONOPT

The analysis aspects of the 23,000‐line FORTRAN program VICONOPT are described. Overall stiffness matrices assembled from the earlier exact VIPASA flat plate stiffnesses are optionally coupled by Lagrangian multipliers to find critical buckling loads, or natural frequencies of undamped vibration, of prismatic assemblies of anisotropic flat plates with arbitrarily located point supports or simple transverse supporting frames. The longitudinal continuity of typical wing and fuselage panels is closely approximated because the solutions are for the infinitely long structure obtained by repeating a bay and its supports longitudinally. Any longitudinally invariant in‐plane plate stresses are permitted, and very rapid solutions are guaranteed by numerous refinements, including multilevel substructuring and a method for repetitive cross sections that is exact for regular polygons used to represent cylinders. Modal displacements and stresses in or between plies of laminated plates are calculated and plotted, with values being recovered at all nodes of substructures. Comparison with usual approximate finite‐element methods confirms that, for comparably converged solutions, VICONOPT is typically between 100 and 104 times faster.     

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.     

Creep Analysis of Axially Loaded Fiber Reinforced Polymer-Confined Concrete Columns

An analytical model is developed to study the time-dependent behavior of concrete-filled fiber reinforced polymer (FRP) tubes (CFFT) and fiber-wrapped concrete columns (FWCC) under sustained axial loads. The model utilizes the double power law creep function for concrete in the framework of rate of flow method, and the linear viscoelastic creep model for FRP. It follows geometric compatibility and static equilibrium, and considers the effects of sealed concrete, multiaxial state of stresses, creep Poisson’s ratio, stress redistribution, variable creep stress history, and creep rupture. The model is verified against previous creep tests by the writers on FWCC and CFFT columns. It is then used to study the practical design parameters that may affect creep of FRP-confined concrete under service loads, or lead to creep rupture at high levels of sustained load. Creep of FWCC is shown to be close to that of sealed concrete of the same mix, as the effect of confinement on creep of concrete is not very significant. CFFT columns, on the other hand, creep much less than FWCC, mainly due to axial stress redistribution. As the stiffness of the tube increases relative to the concrete core, larger stress redistributions take place further reducing the creep. However, there is a threshold, beyond which, stiffer tubes would not significantly lower the creep of concrete. Creep rupture life expectancy of CFFT columns is shown to be quite acceptable.     

Spatial Rotation Kinematics and Flexural–Torsional Buckling

This paper aims to clarify the intricacies of spatial rotation kinematics as applied to three-dimensional (3D) stability analysis of metal framed structures with minimal mathematical abstraction. In particular, it discusses the ability of the kinematic relationships traditionally used for a spatial Euler–Bernoulli beam element, which are expressed in terms of transverse displacement derivatives, to detect the flexural–torsional instability of a cantilever and of an L-shaped frame. The distinction between transverse displacement derivatives and vectorial rotations is illustrated graphically. The paper also discusses the symmetry and asymmetry of tangent stiffness matrices derived for 3D beam elements, and the concepts of semitangential moments and semitangential rotations. Finally, the fact that the so-called vectorial rotations are independent mathematical variables are pointed out.     

Dynamic Thermal Buckling of Functionally Graded Spherical Caps

In the present work, dynamic buckling behavior of clamped functionally graded spherical caps suddenly exposed to a thermal field is studied using the finite-element procedure. The material properties are graded in the thickness direction. The temperature load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling temperature. Numerical study is carried out to highlight the influences of shell geometries and material gradient index on the critical buckling temperature.