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).     

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.     

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.     

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.     

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).     

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.     

Compression and Shear Behavior of Mudstone Aggregates

In this paper, a staged compression–immersion–direct shear test was conducted on the compacted samples of crushed mudstone aggregates, and its compressive and shear behavior are discussed with attention to cementation effects. Compression behavior of the compacted samples was influenced significantly by the compaction degree as expected. So were the shear behavior and shear strength. Immersion caused an additional compression and a reduction in mobilized shear stress and in the dilatant nature during shear at low applied pressure levels. Moreover, immersion reduced significantly the peak shear strength parameter c with only a little change in ?. The compression lines and critical state lines of the nonimmersed and immersed specimens seem to parallel each other, and the compression line of the nonimmersed and the critical state line of the immersed form the upper and lower bounds, respectively. A gap between the shear stress–void ratio lines of the specimens with and without immersion can be considered to represent a combined effect of cementation retained in a crushed mudstone aggregate itself and an interlocking effect of aggregates.     

Free Vibration of Three-Dimensional Orthotropic Uniform Shear Beam-Columns with Generalized End Conditions

The free vibration analysis of asymmetrical three-dimensional (3D) uniform shear beam-columns with generalized boundary conditions (semirigid flexural and torsional restraints, lateral bracings, and lumped masses at both ends) subjected to an eccentric end axial load in addition to a linearly distributed eccentric axial load along its span is presented in a classic manner. The five coupled governing equations of dynamic equilibrium (i.e., two shear equations, two bending moment equations, and the pure torsion moment equation) are sufficient to determine the natural frequencies and modal shapes. The proposed model which is an extension of a 2D model presented previously by the writer includes the simultaneous 3D coupling effects among the lateral deflections, deformations of the cross section along the member (shear, torsional and rotational), the translational, rotational and torsional inertias of all masses considered, an eccentric end axial load in addition to a linearly distributed axial load along its span, and the end restraints. Deformations caused by shear forces and pure torsion are considered. The effects of axial deformations, warping torsion and torsional stability are not included. The proposed model shows that the dynamic behavior of 3D shear beam-columns is highly sensitive to the coupling effects just mentioned, particularly in members with both ends free to rotate. Analytical results indicate that except for doubly symmetric members with concentric axial loads and with perfectly clamped ends, the natural frequencies and modal shapes of 3D shear beam-columns are determined from the eigenvalues of a full 8×8 matrix, rather than from the uncoupled equations of transverse (or shear-wave equations) and torsional moment equilibrium. Two comprehensive examples are presented that show the effectiveness of the proposed method.     

Directional Effects of Shear Combined with Compression on Bridge Elastomeric Bearings

Elastomeric bearings are widely used in bridge supports to accommodate thermal and other movements. The study presented in this paper extends an earlier investigation of two-dimensional bearing performance to three dimensions. Large-deformation rubber hyperelasticity is reviewed and a theoretical model is described with the steel-reinforced bearing subjected to compression in the direction through the thickness followed by shear in various lateral directions, including bridge longitudinal and transverse directions. Computations are carried out using the general-purpose, finite-element analysis computer program, ABAQUS. Conclusions are drawn regarding the effects of shear direction on bearing behavior.     

Buckling and Postbuckling of Antisymmetrically Laminated Cross-ply Shear-Deformable Cylindrical Shells under Axial Compression

The generalized Donnell-type equations governing large deflection of antisymmetrically laminated cross-ply cylindrical shells counting for transverse shear deformations are derived and presented. An asymptotic series solution is constructed by regular perturbation technique for postbuckling behaviors of the cylindrical shells with simply supported edges subjected to axial compression. Boundary layer influence at both ends of the shells on overall buckling and postbuckling are considered, and for consistency of the boundary valued problem, the boundary layer solutions are also designed to match the out-of-plane edge conditions by singular perturbation approach. Effects of transverse shear deformation, Batdorf’s parameter, elastic moduli ratio, and initial geometric imperfection on buckling and postbuckling performance of the shells are examined. Some numerical examples are taken for comparison of the present results of buckling loads and load–deflection curves of the shells with corresponding theoretical predictions to show effectiveness and accuracy of the present asymptotic perturbation solution.     

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.     

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.     

Hyperelasticity Model for Finite Element Analysis of Natural and High Damping Rubbers in Compression and Shear

Rate-independent monotonic behavior of filled natural rubber and high damping rubber is investigated in compression and shear regimes. Monotonic responses obtained from tests conducted in both regimes demonstrate the prominent existence of the Fletcher–Gent effect, indicated by high stiffness at low strain levels. An improved hyperelasticity model for compression and shear regimes is proposed to represent the rate-independent instantaneous and equilibrium responses including the Fletcher–Gent effect. A parameter identification scheme involving simultaneous minimization of least-square residuals of uniaxial compression and simple shear data is delineated. The difficulties of identifying a unique set of hyperelasticity parameters that hold for both compression and shear deformation modes are thus overcome. The proposed hyperelasticity model has been implemented in a general purpose finite element program. Finite element simulations of experiments have shown the adequacy of the proposed hyperelasticity model, estimated parameters, and employed numerical procedures. Finally, numerical experiments were conducted to further explore the potential of the proposed model, and estimated parameters in analyzing rubber layers of a base isolation bearing subjected either to compression or to a combination of compression and shear.     

Effects of Cross Anisotropy on Three-Dimensional Behavior of Sand. I: Stress–Strain Behavior and Shear Banding

The significance of material cross anisotropy in sands is underscored and experimentally evaluated in a series of true triaxial tests on Santa Monica beach sand in a cubical device. Failure patterns, initiation and development of shear banding, and complete stress–strain behavior are described for the entire range of the Lode angle under general three-dimensional loading conditions. Localized failure was found to govern the ultimate resistance of the sand for intermediate values of parameter b = (σ2?σ3)/(σ1?σ3) in each of the three sectors of the octahedral plane. Variations of the friction angle are fully described and show its significant dependence on the inherent cross-anisotropic material structure.     

Effects of Cross Anisotropy on Three-Dimensional Behavior of Sand. II: Volume Change Behavior and Failure

The volume change behavior of cross-anisotropic sand is studied using results of a series of cubical triaxial tests. The relationships between the volumetric response, failure, and shear localization are addressed. Rates of dilation under various three-dimensional stress conditions are evaluated in conjunction with the peak shear resistance and initiation of shear banding in specimens of dense Santa Monica beach sand. The location of the line in principal stress space along which the tendency to deform changes from compressive to dilative (the characteristic line) is determined using two different methods. The uniqueness of this characteristic line for cross-anisotropic materials is analyzed.