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.     

Timoshenko Beam-Column with Generalized End Conditions and Nonclassical Modes of Vibration of Shear Beams

The main objective of this publication is to derive, in a classic manner, the characteristic equations for the undamped natural frequencies and the corresponding modes of vibration of a two-dimensional (2D) Timoshenko beam–column with generalized support conditions (i.e., with semirigid flexural restraints and lateral bracings as well as lumped masses at both ends) and subjected to a constant axial load along its span. The model includes the simultaneous effects (or couplings) of bending and shear deformations, translational and rotational inertias of all masses considered. The proposed model is general, showing that the natural frequencies and the corresponding modes of vibration of 2D beam–columns are highly sensitive to the coupling effects just mentioned. This is particularly true in members with low shear stiffness and with the end flexural restraints and lateral bracing approaching those of free–free and pinned–free conditions. A second objective of this paper is to show that the obtained solution reproduce, as a special case, the nonclassical modes of shearbeams, including the inversion of modes of vibration (i.e., higher modes crossing lower modes) in shear beams with pinned–free and free–free end conditions, and the phenomena of double frequencies at certain values of beam slenderness (r/L).     

Large Deflection Stability of Slender Beam-Columns with Semirigid Connections: Elastica Approach

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.     

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.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

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).     

Beam-Column Design Equations for Wide-Flange Pultruded Structural Shapes

A simple procedure is developed for the selection of pultruded structural shapes to be used as beam-columns in structural design. The design equations are then validated by comparison with experimental data gathered during beam-column testing of wide-flange and I-beam pultruded structural shapes. The design procedure accounts for axial load eccentricity and bending action induced by lateral loads and end-moments. The design equations are set in the context of load and resistance factor design, considering both strength and serviceability. This paper addresses the methodology to determine the resistance factors, which should be used with properly selected load-factors accounting for the variability and uncertainty of the loads. The design equations use section-properties, such as the bending stiffness (EI), which must be measured and supplied by industry. It is found that the section-properties used in the design of beams and columns are sufficient for the design of beam-columns. Therefore, the cost and time involved in testing structural shapes are minimized. This paper also addresses the means by which section-properties can be generated effectively and inexpensively.     

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.     

Second-Order Axial Deflections of Imperfect 3-D Beam-Column

Beam-columns, in general, undergo axial elongation not only from the applied axial forces but also from the transverse deflections. A practical method that takes into account the effects of these transverse deflections on the total axial deformation of a beam-column δt is by multiplying the first-order axial stiffness AE∕L by the geometrically nonlinear factor s1 [i.e., δt = P∕(s1AE∕L)]. A general solution for s1 is derived for the combined effects of end moments, a uniformly distributed load, a series of concentrated loads, sidesway, and out-of-straightness. This solution requires numerical integration and is limited to 3D elastic prismatic beam-columns with doubly symmetrical cross sections or singly symmetrical 2D beam-columns under small strains. The proposed solution can be applied to the second-order and stability analyses of frames and to the evaluation of the axial load induced by transverse loads in beams built into rigid supports. These effects are particularly important in long-span structures. An example is presented to show the validity of the proposed formulation.     

Buckling Reversals of Axially Restrained Imperfect Beam-Column

The stability and postbuckling analysis of an axially restrained prismatic beam-column with single symmetrical cross section and an initial imperfection (camber) is presented. The proposed model is that by Timoshenko but including the effects of small camber of any form and any transverse loading. This model can be used to (1) determine the prebuckling elastic response and initial buckling load; (2) explain the postbuckling elastic behavior including the phenomena of snap-through, snap-back, and reversals of deflections; and (3) determine the effects of high modes of buckling on the stability behavior of beam-columns with small camber. In addition, closed-form equations corresponding to the transverse and axial deflections caused by any transverse loads on a partially restrained beam-column are developed as well as the bending stress along its span. It is shown that the prebuckling, stability, and postbuckling behavior of a beam-column depends on (1) the cross section and material properties (area, inertia, and elastic modulus); (2) the magnitude of the end restraints; and (3) the type and lack of symmetry about the beam-column midspan of the applied transverse loads and initial camber or imperfection. For transverse loads that are not symmetrical with respect to the beam-column midspan, the pre- and postbuckling criterion given by Timoshenko might yield significant errors in both the critical load and deflections. Three examples are presented that show the effectiveness and validity of the proposed equations and the limitations of Timoshenko's criteria.     

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.     

Nonuniform Torsion of Composite Bars by Boundary Element Method

In this paper a boundary element method is developed for the nonuniform torsion of composite bars of arbitrary constant cross section. The composite bar consists of a matrix surrounding a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The bar is subjected to an arbitrarily concentrated or distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Because warping is prevented, besides the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary-value problems with respect to the variable along the beam angle of twist and to the warping function at the shear center are formulated and solved employing a boundary element method approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses caused by restrained warping is investigated.     

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.     

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.     

Signs of surface torsion and torsional dioptric power

Although there is agreement in the literature over the magnitude of torsion and torsional dioptric power, there is ambiguity over the signs of those quantities. The purpose of this paper is to define terms in such a way that the ambiguity is removed. Explicit equations are presented for torsion and torsional power along a meridian of a surface. In keeping with common practice in other disciplines, right-handed torsion is chosen to be positive. The components of the dioptric power matrix of thin systems and of the reduced vergence matrix are reinterpreted in terms of curvital and torsional power. In this reinterpretation the off-diagonal components of the matrices remain the torsional power and the reduced torsion along the meridian orthogonal to the reference meridian. However, they become the negatives of those quantities along the reference meridian. In particular, the top-right component can be interpreted as the reduced torsion or the torsional power along the meridian orthogonal to the reference meridian and the bottom-left as the negative of those quantities along the reference meridian. Torsion and torsional power along a meridian, as well as curvature and curvital power, are invariant under change of reference meridian and under spherocylindrical transposition.     

Transverse Natural Vibration of a Beam with an Internal Joint Carrying In-Plane Flexibilities

Existing research on semirigid jointed frame includes only the rotational flexibility of the joint without consideration of the flexibility in the direction of the transverse shear force. This omission would lead to inaccuracies in the dynamic response of structures, especially in the nonlinear analysis. This paper investigates the dynamic behavior of a bolted joint which has flexibility in both the tangential and rotational directions. The joint is prestressed with axial tension in the bolt shank. It is represented as a virtual connection spring element at the intersection between the beam and the supporting member. The formulation of the hybrid beam–column element including the end springs is presented, and the dynamic behavior of a cantilever beam with this nonlinear semirigid joint is studied. The natural frequencies and mode shapes remain relatively unchanged only for a limited range of the joint stiffnesses, and the eigenpair is influenced by the instantaneous stiffness of the joint defined at a point on the hysteretic loop, particularly when the initial moment stiffness is small.     

Monitoring and Analysis of a Bridge with Partially Restrained Bearings

This paper reports on the monitoring and analysis of a two-span bridge in which the bearings were partially restrained. In an earlier experimental study, it was shown that the natural frequencies changed in colder weather, and it appeared that this was due to restraints in the end bearings. This research was conducted to verify this initial conclusion and to develop an analytical approach based on the finite-element method to model this change. Additional field measurements were made. The nonlinear dynamic finite-element analysis is based on a planar model that includes the influence of both the deck cracking and the eccentric axial forces, which develop when the bearings are restrained. Both the flexural and the torsional modes are evaluated. Although the changes in the bearings and the overall structural behavior were relatively small, the results show that it was nevertheless possible to verify the changes with a nonlinear dynamic finite-element analysis calibrated with field measurements.     

Stability and Collapse of Semirigidly Connected Portal Frames

A numerical procedure for the nonlinear elastic‐plastic instability analysis and collapse of semirigidly connected portal frames, with elastic rotational restraints at the supports, is presented. The procedure is based on nonlinear kinematic relations and linearly elastic material behavior except at the plastic regions (concentrated plasticity). The nonlinear flexible connections are represented by polynomial models. A computational technique for incorporating the stability and strength into the analysis is described in detail. It is found that several important parameters affect the failure modes and consequently the critical loads. These parameters are the slenderness ratio, support restraints, type of connections, and the loading conditions. It is also demonstrated that the connection flexibility has considerable effect on the critical load and the deformation. It is further concluded that for design application the assumption of linear (instead of nolinear, polynomial) connection behavior is adequate for portal frames only if the loading conditions do not produce a significant amount of bending moment at the joints.     

Influence of Torque on Lateral Capacity of Drilled Shafts in Sands

As a result of recent changes in the requirements involving hurricane extreme events (e.g., wind velocities), the Florida Department of Transportation has moved away from cable-stayed signs, signals, and lights systems to mast arm/pole structures. Unfortunately, the newer systems develop significant lateral and torque loading on their foundations (e.g., drilled shafts). Current design practice for a mast arm/pole foundation is to treat lateral loading and torsion separately (i.e., uncoupled); however, recent field-testing suggests otherwise. This paper reports on the results of 91 centrifuge tests. 54 of the tests were conducted in dry sand and 37, in saturated sands. The tests varied the lateral load to torque ratios, shaft embedment depths, and soil strengths. The experiments revealed that even though the torsional resistances of the shafts were not influenced by lateral load, the shafts’ lateral resistance was significantly impacted by torsion. Reductions in lateral resistance of 50% were recorded for shafts under high torque to lateral load ratios. Using the free earth support assumption and the ultimate soil pressure the soil pressure distribution along the shaft was developed. Using force and moment equilibrium, as well as the applied torque, maximum shear, and moments were computed. The predicted values were found to be within 25% (10% on average, except for the tests in saturated dense sand with polymer slurry) of the experimental results.     

Experimental Study of the Biaxial Cyclic Behavior of Thin-Wall Tubes of NiTi Shape Memory Alloys

Combined torsion-tension cycling experiments were performed on thin-wall tubes (with thickness/radius ratio of 1:20, similar to that found for stents) of nearly equiatomic NiTi shape memory alloys (SMAs). Experiments were controlled by axial displacement and torsional angle with step loading involving torsional loading to a maximum strain, followed by tensile loading, and reverse-order unloading. The superelasticity of the material is confirmed by pure torsion and tension experiments at the test temperature. The evolution of equivalent stress-strain curves as well as the separated tensile and torsional stress-strain curves during cycling is analyzed. Results show that the equivalent stress increases greatly with a small amount of applied axial strain, and the equivalent stress-strain curves have negative slopes in the phase transformation region. The shear stress drops when the torsional strain is maintained at its maximum value and the tensile strain is increased. The shear stress increases with decreasing tensile strain, but it cannot recover to the original value after the complete unloading of the tensile strain. Attention is also paid to dissipated energy density and characteristic stress evolutions during cycling.