Column Stability and Minimum Lateral Bracing: Effects of Shear Deformations

Stability equations that evaluate the elastic critical load of columns in any type of construction with sidesway uninhibited, partially inhibited, and totally inhibited including the effects of bending and shear deformations are derived in a classical manner. The “modified” shear equation proposed by Timoshenko and Gere is utilized in the derived equations which can be applied to the stability of frames (“unbraced,” “partially braced,” and “totally braced”) with rigid, semirigid, and simple connections. The complete column classification and the corresponding three stability equations overcome the limitations of current methods. Simple criteria are presented that define the concept of minimum lateral bracing required by columns and plane frames to achieve nonsway buckling mode. Four examples are presented that demonstrate the effectiveness and accuracy of the proposed stability equations and the importance of shear deformations in columns with relatively low shear stiffness AsG such as in built-up metal columns or columns made of laminated composites (fiber-reinforced polymers).     

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.     

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.     

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.     

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 of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

Shear Failure of Pultruded Fiber-Reinforced Polymer Composites under Axial Compression

When structural elements are subjected to compressive loads, the shear forces and stresses induced by second-order effects may lead to shear failure prior to compressive failure. This is particularly likely to occur in the case of pultruded glass fiber-reinforced polymer profiles, which normally exhibit low shear strength in relation to compressive strength. This paper analyzes the effects of initial imperfection, slenderness, shear-to-compressive strength ratio, shear coefficient, and type of shear failure criterion on ultimate load and failure mode (shear or compressive failure). A formulation for predicting ultimate load based on shear failure and second-order deformation is proposed. The results obtained compare well with similar results obtained using other methods and experimental data available in literature. The proposed method is based strictly on mechanics and thus requires no fitting to experimental data.     

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).     

Postbuckling of Shear Deformable Laminated Cylindrical Shells

A postbuckling analysis is presented for a shear deformable laminated cylindrical shell of finite length subjected to compressive axial loads. The governing equations are based on Reddy’s higher-order shear deformation shell theory with a von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, cross-ply laminated cylindrical shells. The effects of transverse shear deformation, shell geometric parameters, total number of plies, fiber orientation, and initial geometric imperfections are studied.     

Fracture Analysis of Shear Deformable Bi-Material Interface

Based on a novel split bi-layer shear deformable beam model capable of capturing the local deformation at the crack tip, the explicit closed-form solutions of bi-material interface fracture are presented in this paper. A recently developed novel shear deformable bi-layer beam theory is briefly reviewed, from which the deformation at the crack tip is explicitly derived. A new expression for the energy release rate is then obtained using the J integral, in which several new terms associated with the transverse shear force are present; this represents an improved solution compared to the one from the classical beam model. By exploiting the two concentrated crack tip forces, the general loadings acting at the crack tip are decomposed into two groups which produce only the mode I and mode II energy release rates, respectively; the total energy release rate is thus decomposed into the mode I and II components in a global sense. The stress intensity factor referred to as local decomposition is also obtained including the transverse shear effect. The difference between the global and local mode decompositions is clarified, and a simple relationship between them is provided. The effect of the existence of a thin layer of adhesive on the stress intensity factor is further studied by an asymptotic analysis. A simple and improved expression for the T stress, the nonsingular term of stress at the crack tip, is also given. The fracture parameters of several commonly used interface fracture specimens are summarized. The present fracture analysis including the transverse shear effect is in better agreement with finite element analyses and shows advantages and improved accuracy over the available classical solutions.     

Measurement of the Timoshenko Shear Stiffness. I: Effect of Warping

Fiber-reinforced polymer (FRP) composite beams are increasingly finding use in construction. Due to their lower stiffness relative to steel sections, the design of FRP structures is usually deflection controlled. Furthermore, shear deformation can be significant in FRP beams, thus, requiring the use of the Timoshenko beam theory to estimate deflections. However, the Timoshenko shear stiffness can be difficult to measure. Part of the measurement error has been attributed to shear warping effects. It has been hypothesized that warping restraints at loading points and supports increase the apparent shear stiffness to a degree that is significant at relatively short spans, e.g., L/h<10 to 15. In this study, the influence of warping on short to moderate length FRP beams under various types of loading and boundary conditions is considered using finite-element analysis. In particular, a commercially available thin-walled FRP beam was investigated. The results suggest that warping has a negligible effect for thin-walled beams at reasonable spans, i.e., L/h>5. On the contrary, the effective shear stiffness is found to decrease at shorter span lengths. This is the first of two papers in a series.     

On the Comparison of Timoshenko and Shear Models in Beam Dynamics

The classical Timoshenko beam model and the shear beam model are often used to model shear building behavior both for stability or dynamic analysis. This technical note questions the theoretical relationship between both models for large values of bending to shear stiffness parameter. The simply supported beam is analytically studied for both models. Asymptotic solutions are obtained for large values of bending to shear stiffness parameter. In the general case, it is proven that the shear beam model cannot be deduced from the Timoshenko model, by considering large values of bending to shear stiffness parameter. This is only achieved for specific geometrical parameter in the present example. As a conclusion, the capability of the shear model to approximate Timoshenko model for large values of bending to shear stiffness parameter is firmly dependent on the material and geometrical characteristics of the beam section and on the boundary conditions.     

Measurement of the Timoshenko Shear Stiffness. II: Effect of Transverse Compressibility

This is the second of two papers devoted to the issue of measuring the Timoshenko shear stiffness of thin-walled composite beams. In the first paper, the effect of warping on the effective Timoshenko shear stiffness, as measured through bending tests, was studied. The bending test was simulated using finite-element analysis, and the results indicated that the warping effect was minimal. On the other hand, the evidence suggests that transverse flexibility may have a significant influence on the effective Timoshenko shear stiffness, decreasing the effective shear stiffness at shorter test spans. The purpose of the present study is to further investigate this effect and to explore the use of a sandwich theory to predict the measurement error. A higher-order sandwich theory, which captures the transverse strain at concentrated loads and supports, is applied to a commercially available thin-walled composite beam. The results indicate that the sandwich model does capture the decrease in the effective shear stiffness at short spans, and the dependence of the shear stiffness on span-to-depth ratio is similar to that calculated in the first paper, using the finite-element method.     

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.     

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%.     

剪切带与金矿成矿研究进展

剪切带是发育在地壳内部的一条线性高应变带,20多年来,与之有关的金矿成矿的研究成果颇丰,我们从剪切带与金矿的时间、空间、元素迁移富集、矿化蚀变类型、成矿动力学等五个方面概略介绍相关研究成果,以期引起业内同行对剪切带与金矿关系研究的重视。     

Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium

The dynamic modal analysis (i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors) and static stability (i.e., critical loads and buckling modes) of two-dimensional (2D) cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base (i.e., zero rotation of the cross sections), the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium “ignoring” the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: (1) the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness (L/r); and (2) the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer (2005) in columns made of elastomeric materials.     

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.     

Shear Resistance of FRP RC Beams: Experimental Study

This study investigates the shear behavior of concrete beams reinforced with fiber-reinforced polymer (FRP) reinforcement. Six beams were subjected to two successive phases of testing. Half of the beams were reinforced in flexure with conventional steel reinforcement, while the other half were reinforced with glass fiber bars. Different shear span to depth ratios, ranging from 1.1 to 3.3, were analyzed in order to study the variation in the shear behavior of beams characterized by different types of shear failure. No shear reinforcement was provided in the first phase of testing, while in the second phase, just enough glass and carbon shear reinforcement was provided to enable failure due to shear. The results of these tests are presented and compared to predictions according to the design recommendations proposed by the ACI and the Institution of Structural Engineers, U.K. The results of this study show that these approaches, which are based on modifications of equations derived for steel reinforcement, underestimate the contribution of the concrete and the shear reinforcement to the total shear capacity of FRP RC beams. It is shown that both approaches can be modified to become less conservative.