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.     

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.     

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.     

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.     

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 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.     

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.     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

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 Analysis of Nonuniform and Axially Graded Columns with Varying Flexural Rigidity

In this paper, we present a novel analytic approach to solve the buckling instability of Euler-Bernoulli columns with arbitrarily axial nonhomogeneity and/or varying cross section. For various columns including pinned-pinned columns, clamped columns, and cantilevered columns, the governing differential equation for buckling of columns with varying flexural rigidity is reduced to a Fredholm integral equation. Critical buckling load can be exactly determined by requiring that the resulting integral equation has a nontrivial solution. The effectiveness of the method is confirmed by comparing our results with existing closed-form solutions and numerical results. Flexural rigidity may take a majority of functions including polynomials, trigonometric and exponential functions, etc. Examples are given to illustrate the enhancement of the load-carrying capacity of tapered columns for admissible shape profiles with constant volume or weight, and the proposed method is of benefit to optimum design of columns against buckling in engineering applications. This method can be further extended to treat free vibration of nonuniform beams with axially variable material properties.     

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.     

Hierarchical High-Fidelity Analysis Methodology for Buckling Critical Structures

A hierarchical high-fidelity analysis methodology for predicting the critical buckling load of compression-loaded thin-walled isotropic shells is described. This hierarchical procedure includes three levels of fidelity for the analysis. Level 1 assumes that the buckling load can be predicted by the classical shell solution with simply supported boundary condition, and with a linear membrane prebuckling solution. Level 2 includes the effects of a nonlinear prebuckling solution and the effects of traditional clamped or simply supported boundary conditions. Level 3 includes the nonlinear interaction between nearly simultaneous buckling modes and the effects of boundary imperfections and general boundary conditions. Various deterministic and probabilistic approaches are used to account for the degrading effects of unavoidable shell-wall geometric imperfections. The results from the three solution levels are compared with experimental results, and the effects of the assumptions and approximations used for the three solution levels are discussed. This hierarchical analysis approach can be used in the design process to converge rapidly to an accurate prediction of the expected buckling load of a thin-shell design problem.     

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.     

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.     

Theoretical Evaluation of Flange Local Buckling for Horizontally Curved I-Girders

Curvature greatly complicates the behavior of horizontally curved steel plate girders used in bridge superstructures. The warping stress gradient across the width of I-girder flange plates reduces the vertical bending stress at which the flange plate buckles. The 2007 AASHTO Load and Resistance Factor Design Specifications eliminate the shortcomings of the 2003 AASHTO Guide Specifications for Horizontally Curved Bridges by unifying the flexural design of tangent and curved I-girder bridges. This paper evaluates flange local buckling resistance based upon theoretical and analytical models that consider the effect of stress gradient across the flange coupled with the influence of rotational resistance provided by the web. The developed equations are verified using the finite element method, and the potential impact is demonstrated using the design example presented in the Guide Specifications.     

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 Micropiles

Building codes and common design practices generally assume that the lateral support provided by the soil is sufficient to prevent buckling of fully embedded piles. As small diameter grouted piles (micropiles) have evolved from relatively low capacity friction piles to current applications that include high capacity elements, there is a need to revisit the issue of potential buckling of these very slender piles when embedded in soft soils. Pile buckling loads obtained from a semiempirical relationship are compared to the allowable loads permitted by current and proposed codes and design guidelines. It is concluded that buckling is generally not a concern for the most common types of micropile design, but there are some designs permitted under the codes and design guidelines for which buckling may be a controlling design factor.     

Exact Solutions for Buckling of Multispan Rectangular Plates

This paper presents exact solutions for buckling of multispan rectangular plates having two opposite edges simply supported and the other two edges being either free, simply supported, or clamped. The Levy solution procedure is employed to develop an analytical approach for buckling analysis of multispan plates. The Levy solution for each span is derived and the continuity along the interface of two spans is ensured through the implementation of the essential and natural boundary conditions at the interface. Extensive buckling factors, most of which are first-known exact solutions, are given in tabular and design chart forms for two- and three-unequal-span square plates subjected to uniaxial in-plane load in the x or y directions and biaxial in-plane load. The influence of the span ratios and plate boundary conditions on the buckling factors is discussed. Buckling factors are also obtained for two-, three-, and four-equal-span rectangular plates with various edge support conditions. The exact buckling solutions presented in this paper are of benchmark values for such plates.     

Assessment of Current Load Factors for Use in Geotechnical Load and Resistance Factor Design

Load and resistance factor design (LRFD) is the standard structural design practice. In order for foundation design to be consistent with current structural design practice, the use of the same loads, load factors, and load combinations would be required. In this paper, we review the load factors presented in various LRFD codes from the United States, Canada, and Europe. A simple first-order second-moment (FOSM) reliability analysis is presented to determine appropriate ranges for the values of the load factors. These values are compared with those proposed in the codes. The comparisons between the analysis and the codes show that the values of load factors given in the codes generally fall within ranges consistent with the results of the FOSM analysis. However, it would be desirable for the successful development and adoption of the geotechnical component of LRFD codes to have uniformity of load-factor values across different codes for the loads that are common for virtually all civil structures.     

State-Specific LRFR Live Load Factors Using Weigh-in-Motion Data

The LRFR Manual, within commentary Article C6.4.4.2.3, contains provisions for development of site-specific live load factors. In Oregon, truck weigh-in-motion (WIM) data were used to develop live load factors for use on state-owned bridges. The factors were calibrated using the same statistical methods that were used in the original development of LRFR. This procedure maintains the nationally accepted structural reliability index for evaluation, even though the resulting state-specific live load factors were smaller than the national standard. This paper describes the jurisdictional and enforcement characteristics in the state, the modifications used to described the alongside truck population based on the unique truck permitting conditions in the state, the WIM data filtering, sorting, and quality control, as well as the calibration process, and the computed live load factors. Large WIM data sets from four sites were used in the calibration and included different truck volumes, seasonal and directional variations, and WIM data collection windows. Finally policy implementation for actual use of the factors and future provisions for maintenance of the factors are described.