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The paper focuses on the development of a general method for extreme response estimation of dynamical systems subjected to random excitations. One of the most important elements in structural safety is an assessment of the exceedance or failure probability. The frequently used Poisson assumption tends to overestimate the exceedance level if the response spectrum has narrow- banded features since the so-called clustering effects occur. The latter inaccuracy may be corrected by using an alternative approach studied in this paper, which enables accurate prediction of extremes irrespective of clustering effects. DOI: 10.1061/(ASCE)0733-9399(2010)136:3(290) CE Database subject headings: Engineering mechanics; Dynamics response; Structural reliability; Nonlinear analysis; Random processes; Predictions; Statistics. Author keywords: Extreme response; Narrow banded process; Clustering; Mean up-crossing rate; Nonlinear dynamic systems; Monte Carlo simulation.

The failure probability of a series structural system theoretically involves multidimensional integration and is usually difficult to calculate. The search for efficient computational procedures for estimating system reliability has resulted in several approaches, including bounding techniques and efficient Monte Carlo simulations. For the narrow bound method, the joint failure probability of every pair of failure modes needs to be calculated. In the present paper, in order to improve the accuracy of the narrow bound estimation method, a computationally effective point estimation method for calculating the joint probability of every pair of failure modes is proposed and examined for series systems. Based on the computational results of several illustrative examples, it can be seen that the results by the present method are in good agreement with those obtained through integration. DOI: 10.1061/(ASCE)0733-9399(2007)133:5(588) CE Database subject headings: Structural reliability; System reliability; Joints; Failures; Integrated systems.

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 [[pi].sup.2]EI/[h.sup.2]. DOI: 10.1061/(ASCE)0733-9399(2007)133:1(106) CE Database subject headings: Buckling; Building codes; Columns; Computer applications; Frames; Shear deformation; Compression.

Gravity loads play an important role in the linear and nonlinear behavior of buildings during earthquakes. They can also be the cause of ultimate collapse of the structures. In this study, the governing equation of continuous uniform shear buildings under the effects gravity loads is derived and eigenfrequencies, displacement, and drift mode shapes are obtained by eigenanalysis. It is shown that how the geometric properties of the structure affect the fundamental oscillation period and response of the building. Inclusion of the effects of the gravity loads makes the solution of the governing differential equation dependent on the Bessel functions of the first and second kind. The modal load and mass equations are solved using the orthogonality relations of Bessel functions. Effects of gravity on displacement and drift behavior of shear buildings on soft soils and rock subjected to limited near-fault earthquake excitations are shown. DOI: 10.1061/(ASCE)0733-9399(2007)133:1(48) CE Database subject headings: Shear walls; Drift; Lateral displacement; Structural stability; Frequency analysis; Eigenvalues; Eigenvectors; Gravity loads.

The ductility of an elastic structure with a growing crack may be defined as the ratio of the additional load-point displacement that is caused by the crack at the moment of loss of stability under displacement control to the elastic displacement at no crack at the moment of peak load. The stability loss at displacement control is known to occur when the load-deflection curve of the whole structural system with the loading device (characterized by a spring) reaches a snapback point. based on the known stress intensity factor as a function of crack length, the well-known method of linear elastic fracture mechanics is used to calculate the load-deflection curve and determine the states of snapback and maximum loads. An example of a notched three-point bend beam with a growing crack is analyzed numerically. The ductility is determined and its dependence of the structure size, slenderness, and stiffness of the loading device is clarified. The family of the curves of ductility versus structure size at various loading device stiffnesses is found to exhibit at a certain critical stiffness a transition from bounded single-valued functions of D to unbounded two-valued functions of D. The method of solution is general and is applicable to cracked structures of any shape. The flexibility (force) method can be adapted to extend the ductility analysis to structural assemblages provided that the stress intensity factor of the cracked structural part considered alone is known. This study leads to an improved understanding of ductility, which should be useful mainly for design against dynamic loads.

A new [C.sub.0] finite element for accurate analysis of laminated composite beam structures is developed. The element formulation is based on a quadratic zig-zag layerwise theory developed previously by the writers. The theory assumes a zig-zag distribution of the in-plane displacement field through the thickness and satisfies the interlaminar shear stress continuity across the layer interfaces. In developing the finite-element formulation, the shear strain fields are made field consistent, and thus the shear locking phenomenon is eliminated. A new transverse normal strain is derived by assuming the transverse normal stress to be constant through the thickness of the laminate. This assumption is shown to remove Poisson's ratio stiffening. The results obtained from the present finite element are found to be in good agreement with exact elasticity solutions available for simply supported beams. A multisublaminate approach that is simple to implement with the present element is shown to improve the predictions of the present model for complex laminated structures.

The postbuckling behavior of an elastic fixed-end column with an elastic brace at the center is investigated. Attention is focused on those of brace stiffness near its threshold value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. We show that, for the brace stiffness greater than the threshold value, there are precisely two secondary bifurcation points on each primary postbuckling path bifurcating from one of the least two classical buckling loads, and the corresponding secondary postbuckling paths connect all of these secondary bifurcation points in a loop. For the brace stiffness less than the threshold value, no secondary bifurcation occurs. The asymptotic expansions of the primary and secondary postbuckling paths are constructed. The stability analysis indicates that, when the brace stiffness goes beyond its threshold value, the primary postbuckling path with a node in the center becomes unstable from stable by means of the secondary bifurcation (i.e., secondary buckling occurs).

The dynamic stability of electrically conducting beam-plates in transverse magnetic fields is considered in this paper. A concise theory of flexural vibration of magnetoelastic plates immersed in transverse magnetic fields is presented first. The dynamic stability of a beam-plate is then examined without making a priori assumptions about its magnetic property. In contrast to the writer's earlier work, where a beam plate is subjected to an external axial load in the presence of a longitudinal magnetic field, an axial force arises naturally from the magnetoelastic interaction in the form of surface traction. The magnetic damping due to the induced eddy current is also included in the analysis. Two nondimensional parameters are introduced and the stability of a simply supported beam-plate is examined as a function of these parameters. It is shown that three stability regions exist. Critical fields are obtained explicitly and discussed in some detail.

This paper summarizes results concerning the capacity of tubular members under combined loading due to external pressure and bending. A nonlinear finite-element technique, outlined in the companion paper is used. Three-dimensional analyses are performed towards an investigation of the effects of localized deformation. To verify the validity of the analytical procedure, computational results are obtained and compared with data from long-column and stub-column tests. Two-dimensional (cross-sectional) calculations furnish accurate estimates of the bending capacity of long, unstiffened tubes. However, three-dimensional effects must be taken into account when dealing with short tubes. Both experiments and computations show that short tubes undergo substantial inelastic deformation prior to buckling at a level of moment higher than the plastic moment. The presence of external pressure reduces the bending capacity and ductility of tubular members. Finally, in agreement with experimental data, the analytical results demonstrate the beneficial effects of capped-end compression on the pressure capacity of stub tubes.

This paper presents a finite-element technique for the analysis of tubular member stability under combined external pressure and structural loads. A tube element is developed for the purposes of this research. While polynomial (quadratic) interpolation is used in the longitudinal direction, Fourier series expansions of the displacement components are adopted at the nodal cross sections. The formulation accounts for large inelastic deformation and recognizes initial imperfections and residual stresses. To trace unstable equilibrium paths, arc-length procedures are implemented. A simple, yet effective estimate of the contribution of external pressure to the tube element stiffness matrix, particularly significant in the analysis of slender tubes, is included. Preliminary results regarding the behavior of tubular members subjected to pressure and bending are reported and discussed. The effects of initial imperfections and residual stresses on the response to pressure along with bending are summarized. Finally, the influence of residual stresses on thrust-moment interaction in tubular beam-columns is examined briefly.

This paper proposes a failure path approach for time-variant system reliability estimation of brittle structures. The significant failure sequences are identified by the branch-and-bound method, and the damage states in these sequences are grouped together into discrete damage levels ranging from intact structure to system collapse. The grouping is based on the number of degrees of redundancy in the structure as it accumulates different amounts of damage. The Markov chain assumption is used to model the cumulative damage under time-varying loads. The expressions for transition probabilities from one damage level to another are derived. The effect of load overlapping under multiple load processes is investigated. The method is applicable to large structures due to the systematic failure sequence identification strategy employed. The resultant probability transition matrix is found to be more compact and to represent the accumulation of damage better than existing methods. Two numerical examples are used to illustrate and examine the proposed method.

A new analytical solution is developed for steady-state lateral response of earth and rockfill dams built in semielliptical canyons and subjected to obliquely incident harmonic SH waves. The dam is idealized as a linearly hysteretic elastic body deforming only in shear, while the supporting canyon is modeled as an elastic rock half-space. The solution accounts in a rigorous way for the wave reflection and diffraction phenomena associated with the presence of the dam-filled canyon. The response of the dam and the canyon is expressed in terms of prolate radial and angular wave spheroidal functions. Representative results elucidate the importance of the effects of the dam-to-canyon material impedance, the canyon geometry, the angle of incidence, and the excitation frequency on the dam response. Comparisons are made with the response of the dam to 'rigid-base' synchronous excitation. The developed solution offers a useful practical tool for parametric studies and preliminary design calculations, as the semielliptical shape can approximate reasonably well the geometry of several actual canyons.

A closed-loop complete-feedback control algorithm is proposed for the control of a building modeled as a single-degree-of-freedom (SDOF) system using an active tuned mass damper (ATMD). The control force is calculated from the acceleration, velocity, and displacement feedbacks of the SDOF system and the active mass damper. The passive-control-device properties, including the stiffness and damping constants of the tuned mass damper and the three gain coefficients of the actuator, are derived by minimizing the displacement variance of the SDOF system. Simulations are performed to evaluate the performance of the active-tuned-mass-damper design on the examples of a SDOF system and a 10-story three-bay building frame. The results show that the control efficiency of the ATMD based on velocity feedback depends on the passive-control-device properties assumed. Also, the ATMD with optimal passive properties using complete feedback results in further reduction in the structural displacement. The results also show that for the same level of reduction in structural displacements, the control force required is smaller using complete feedback.

A semianalytical approach for the computation of the local buckling of stiffened plates under any combination of applied biaxial compression, in-plane bending, and shear stress is presented. The plate is treated as partially restrained against rotation and in-plane translation. In the first stage, the plate is treated as infinitely long and the buckling mode is idealized by straight lines with arbitrary parameters. The energy method is then used to formulate the buckling coefficient, K, in terms of general functions that describe the longitudinal and transverse displacement profiles. In the second stage, sequential quadratic programming (SQP) is used to find the critical combinations of the parameters in the idealized buckling mode that minimize the coefficient, K. Modification factors are then suggested to compute the buckling stress for plates of finite length. Using the derived formulations, a closed-form expression for the K factor is determined by choosing approximate displacement functions. Validation, accuracy, and comparison of the derived K factor is shown for the limiting conditions of simply supported and clamped edges. Finally, results are presented for the more general case of a plate partially restrained against rotation. The transition from rotationally free to rotationally clamped boundaries is shown by modifying the stiffener torsional rigidities. The destabilizing effect of the lateral restraint on the buckling stress is also shown.

This work gives the exact solutions for the coupled flexural-torsional buckling loads of variable and open cross-section columns, loaded by variable axial force. Both the cross-section dimensions and the axial load can vary along the column as polynomial expressions. The proposed solution is based on a new finite-element method for getting the exact stiffness matrix for the member, including the effects of the axial loading. The buckling load is found as the load that causes the stiffness matrix determinant to become zero. Several examples are given and compared to published results to demonstrate the accuracy and flexibility of the method. New, exact results are given for several other cases.

Experimental results of flutter derivatives for sectional models of long-span bridges are presented. The tests were carried out in the Water Channel Laboratory of the Center for Applied Stochastics Research of Florida Atlantic University, which has an 8-m-long test section and a maximum flow speed of 0.5 m/s. The experiments were conducted in the forced-oscillation mode. Specifically, each model is forced to execute a sinusoidal motion, either torsional or vertical, one at a time. In each test, both lift force and torsional moment are measured to obtain the noncoupled flutter derivatives as well as the coupled ones. The advantages of using water as the fluid medium, instead of air, are described. The flutter derivatives obtained in the water-channel tests are then compared with some published data obtained in the wind-tunnel tests using the free-vibration mode. The comparison shows that the water-channel data and the respective wind-tunnel data share the same trends. Finally, the measured flutter derivatives are converted to the corresponding impulse response functions required for the stochastic stability analysis in the time domain.

Motion stability of a long-span bridge in turbulent wind is studied. The bridge motion is represented by a torsional mode and a bending mode, and the new wind turbulence model proposed in an earlier paper is used in the analysis. This turbulence model is capable of matching closely a target spectral density, such as the well-known von-Karman or Dryden spectrum. It is shown that the presence of turbulence changes the combined structure-fluid system is determined by the largest Lyapunov exponent, and the motion is asymptotically stable if the largest Lyapunov exponent is negative. In this sense, the turbulence has a stabilizing or a destabilizing effect, depending on whether it increases or decreases the critical mean wind velocity at which the largest Lyapunov exponent vanishes. For a particular bridge model is investigated, it is found that the peak location of the spectral density of the turbulence is crucial to the stability condition. By changing the peak location of the spectrum, a stabilizing turbulence can become destabilizing, even when the mean-square value remains the same.

A method for computing probabilities of failure of structural components and systems is proposed. The approach is based on a radial-space division technique in conjunction with automatic generation of unit centerline vectors. Based on this technique, the standard normal space is divided into active and passive subdomains. The probability contribution of each active subdomain to the total probability of failure is estimated by approximating the actual limit-state hypersurface with a hypersphere segment centered on the actual hypersurface. The proposed method can solve with relatively high accuracy both component and system reliability problems involving nonlinearities and multiple extremum points of the probability density on the limit-state hypersurface. A number of numerical examples involving linear and nonlinear performance functions of components and systems are presented. The results are in good agreement with exact solutions, probability bounds, multinormal approximations, and/or Monte Carlo simulations.

Nonlocal continuum damage mechanics and random field theory are used to model the stochastic damage behavior of a strain-softening material with random field properties. The randomness in the damage process is introduced by considering the initial damage threshold and the local strain-softening behavior as a bivariate random Nataf field, defined by the marginal distributions and correlation matrix. A key component of the model is the introduction of two different length parameters: the characteristic length of the nonlocal damage model and the correlation distance for the random field. The probabilistic nonlocal damage model is illustrated by presenting finite-element analyses of direct-tension tests. It is found that the specimen exhibits a structural behavior representing a nonsymmetrical deformation and nonlinear stress-displacement curve. The influence of the cross-correlation coefficient between the initial damage and the strain-softening behavior is discussed. The probabilistic nonlocal damage model is also capable of describing the deterministic and probabilistic size effect of structures.

Analytic three-dimensional thermoelasticity solutions are presented for static problems of simply supported sandwich panels and cylindrical shells subjected t mechanical and thermal loads. The panels and shells have laminated composite face sheets of arbitrary thickness separated by a core. Each of the individual layers of the face sheets and the core is modeled as a three-dimensional continuum. Analytic first-order sensitivity coefficients are evaluated to asses the sensitivity of the responses to variations in material parameters of the face sheets and the core, as well as to variations in the curvatures and thicknesses of the sandwich and face sheets. Also, the strain energy associated with various stress components in the face sheets and core are calculated and compared. The information obtained in the present study can aid the development and assessment of two-dimensional models for sandwich structures and illuminate the role of particular material parameters in an equivalent model for the core.

Time-dependent distributions for temperature and moisture content in a concrete wall subjected to fire are determined by a model of heat and mass transfer in porous media. Free-water flow, bound-water diffusion, gas-phase flow, water-vapor diffusion. and thermal energy arc accounted for in the model. The governing transient differential equations are discretized by a control-volume formulation reported by Patankar in 1980, and solved in time using a fully implicit scheme. Two different fire exposure curves (namely, standard curve according to ASTM E-119 and a more realistic curve reported by Ellingwood in 1991) are employed in the analysis. The assessment of two simplified models, in which all pore water is assumed to evaporate at 100 [degrees] C, is performed b comparing their predictions to the full analysis. For temperature and saturation, the time developments predicted by the two simplified models and by the full analysis present similar trends; however, the simplified models seem t be appropriate only for very preliminary fire safely analyses, since they tend to underestimate temperatures.

In this investigation a nonlinear dynamic-stability analysis is performed on a two-bar geometrically imperfect frame subjected to an axial displacement of its joint, either suddenly applied or time dependent. The dynamic response of the frame is governed by a coupled system of two one-dimensional partial differential equations for the axial and lateral motion of each bar. One and two-mode solutions are thoroughly discussed for various geometric configurations of the frame. Dynamic buckling occurs when the corresponding frame under static loading loses its stability through a limit point. This happens for initial bar curvatures above a certain critical value; below this value the frame is dynamically stable. Numerical results are obtained by using Galerkin's method in connection with the seventh-order Runge-Kutta-Verner scheme with appropriate step size. The results of the one-mode solution are found to be in excellent agreement with those of previous analyses.

In this study, a passive control system has been proposed to protect structures from earthquake damages. In particular, the influence of an absorber on the dynamics of base-isolated structures has been investigated in detail. Two different types of base-isolation systems have been considered. These include rubber-bearing and sliding-friction isolators. The base isolators are represented by hysteretic models. The absorber consists of a linear spring mass attached to the base. Simulation studies were conducted for various models subjected to the 1971 San Fernando earthquake excitation. An optimization subroutine based on the simplex method was utilized to determine the best absorber parameters for a given situation. It is concluded that the base isolators can be much more effective in protecting conventional structures from earthquake damages if they are used in conjunction with an absorber. The optimum mass of these absorbers is found to be quite small as compared to the mass of the ground structures. It is anticipated that the results obtained in this study could be put in practice in the near future and serve as a meaningful design technique in preventing earthquake damages.

Under general conditions reliability analyses of elaborate structural models can be replaced by much simpler equivalent reliability analyses of corresponding idealized structural models such as those based on rigid-plastic theory. The equivalence is in the sense of being with respect to probability of failure and is obtained by applying a random factor correction of the strength variables of the simple model. This model-correction factor is called the effectivity factor, and it is approximated to the zeroth order by a constant or to the first order by an inhomogeneous linear function of all the basic variables in the realistic model. The coefficients of this function are calculated from the elaborate model (e.g., by finite element calculations) with values of the input variables chosen on the basis of a first-order reliability analysis of the simple model (FORM). The model-correction-factor method is a special type of response-surface method based on mechanical principles. It is illustrated quantitatively on a yield hinge frame model with strain softening mutual rotation-moment behavior in the hinges upon the onset of yielding.

The stability analysis of natural draught cooling tower shell subjected to nonaxisymmetric wind pressure has been carried out using finite ring elements. In this investigation, concept of a local enhancement of wind pressure coefficients at the top - in addition to the usual variation of wind pressures with height - has been newly introduced. Justification for the use of such modified wind pressure coefficients has been presented. By considering the modified distribution of wind pressure coefficients, a parametric study has also been carried out to assess the influence of top ring stiffener, gradual thickening at shell bottom edge, and vertical cracking due to thermal gradient on the buckling behavior of natural draught cooling tower shells. Guidelines are furnished for choosing dimensions of the top ring stiffener to avoid top edge buckling.

Based on wave theory and the concept of the equivalent foundation for the soil, a simplified method for the dynamic analysis of tapered structures during earthquakes is presented in this paper. The merit of this simplified method is its inclusion of important factors that may affect the responses of tapered structures. Further, the method is sufficiently simple that the dynamic analysis of a tapered structure can be accomplished with a microcomputer or a calculator.

This paper investigates both experimentally and theoretically the free rocking of a prismatic block supported by a stationary, horizontal foundation: the block is tilted, almost to the point of overturning, and released from this position. It is shown that the standard mathematical model for this problem is often inaccurate. A critical review of the implicit assumptions behind the standard model reveals that the free-rocking response of short blocks depends crucially on bouncing after each impact; out-of-plane effects are significant in very short blocks. The response of slender blocks is found to be easier to predict. Rocking has been observed during earthquakes in structures that consists of fairly rigid, unbonded elements, e.g. stacks of graphite blocks in nuclear reactors, and ancient Greek columns, and also in slender structures with foundations unable to resist uplift.

The effect of prebuckling deformation on the elastic buckling of circular plates is studied. Based on Trefftz's initial stress theory, the energy functional is obtained from which the governing differential equation for equilibrium can be derived via calculus of variations. Exact and simple approximate closed-form critical-load expressions are obtained after solving the two-point boundary value problem. The effect of the nonlinear terms involving in-plane displacements in the Green-Lagrange strain expressions on the solutions is also investigated. For circular plates with relatively large thickness-to-radius ratios, the allowance for prebuckling deformation affects the critical load of clamped plates more than simply supported plates. This occurs because the prebuckling deformation is greater in clamped plates, since they undergo higher loads before buckling.

Stability of a newly proposed structural system, namely the prebuckled (prestressed) sandwich arch is investigated. Stability of prestressed elastica arches with sandwich or homogeneous sections are compared through a parametric study of various geometric and mechanical factors by using a nonlinear finite element model. Results indicate that for prestressed sandwich elastica arches the relative thickness and shear modulus of the core layer not only affect the critical load of the arch, but may also change its governing mode of buckling. On the other hand, flexural rigidity of the core layer does not have a significant impact on the buckling load. Also, the bifurcation loads of prestressed sandwich arches increase almost proportionally with the rise parameter of the arch. The category of prestressed elastica arches can be optimized by using sandwich construction since prestressed sandwich arches are shown to be less vulnerable to loading imperfections leading to sidesway buckling. The effects are more pronounced for sandwich sections with thicker and softer cores.

Triaxial compression tests on sand have been performed to study stability and plastic yielding in the postpeak softening regime. Softening occurs in dense sand sheared at low confining pressure. The experiments show that the stability postulates by Drucker and by Hill are not applicable to sand in the softening regime. Plastic yielding, as well as elastic behavior, can occur along the same stress paths, and plastic strains may be obtained along stress paths with increasing, constant, and decreasing stress differences. Granular materials, in which softening occurs, can go through peak failure more than once. The behavior is entirely controlled by the type of volume change exhibited by the granular material and by the drainage conditions imposed on the material. A yield surface of the conventional type cannot be employed in the softening regime. The yield surface is more likely to take the form of a barrier or fence.

Theoretical explanation is provided on why turbulence in the wind flow can sometimes stabilize an essentially single-degree-of-freedom motion of a flexible bridge. The theory is based on a new turbulence model, which has a finite mean-square value and a versatile spectral shape, and is capable of matching closely a target spectrum. It is shown that, without turbulence, the onset of flutter instability is associated with a critical wind velocity and an eigenvector describing a combined critical structure-fluid mode. This particular composition strikes a balance between the energy inflow from fluid to structure, and the energy outflow from structure to fluid, so that an undamped structural motion becomes sustainable. At the introduction of turbulence, the combined structure-fluid mode is changed continuously and randomly, and the new energy flow balance, in the sense of statistical average, renders the stabilizing or destabilizing effect of turbulence possible. Numerical results are presented for the boundary of asymptotic sample stability for a bridge model undergoing a single-degree-of-freedom torsional motion.

An analysis is given of the statistical and probabilistic properties of the random time at which a piece of essential equipment, such as a computer, may topple under base excitations. The equipment is modeled as a rigid block, and the base accelerations are modeled as white noises in horizontal and vertical directions. The cases of a free-standing block and an anchored block are both considered. The white-noise idealization resembles the most intense portion of ground acceleration in a seismic event, which is an acceptable simplification provided that the correlation times of actual base excitations are short compared with the dominant quasi-periods of the induced rocking motion. It is shown that the presence of vertical excitation about one-half the level of horizontal excitation may increase the probability of toppling by 30-40%, and that larger blocks are more stable than the smaller ones of the same geometrical proportion. The analysis is also applicable to such objects as monuments, storage tanks, and certain types of tall buildings.

The buffeting response of long-span bridges in turbulent wind is dealt with in the present investigation. Herein the analysis is based on linear models describing the structural behavior and the fluid-structure interaction. Accounting for both parametric excitation (motion-induced loads) and buffeting (flow-induced) loads the bridge response is described by differential equations in the time domain. The basic concept of the method of analysis presented in the first paper is now extended to multidegree-of-freedom systems. First, a critical mode, which is significantly affected by parametric excitation, is isolated. Utilizing Markov process theory a statistical description of this critical mode in terms of probability density functions is given. Classical spectral analysis constitutes the basis for the analysis of the remaining (noncritical) modes. Employing the so-called Nataf model statistical information is obtained for the response quantities including exceedance rates of torsional and vertical bending displacements. Thus a most valuable basis for engineering decisions is provided by the present investigation. The suggested approach is then exemplified in the numerical examples showing among others the role of parametric excitation for the bridge response behavior in the region of stable motion.

The behavior of the response of long-span bridges under turbulent wind in the region of stable motion is investigated theoretically. Utilizing a single-degree-of-freedom structural model the basic concept of the method of analysis is introduced. Based on the assumption that both the structural behavior and the fluid-structure interaction can adequately be accounted for by linear models the bridge response is described by differential equations with time-varying coefficients. Assuming the turbulent components of the oncoming wind to be weakly stationary wideband random processes, which is considered to be justified when carried by high mean wind velocities, a modified stochastic averaging procedure is employed to obtain the probability density function for the response quantities. Furthermore expressions for the expected rate of threshold crossings from below and for the exceedance probability are evaluated. The influence of the parametric excitation (motion-induced loads) on the response behavior is clearly shown in the numerical example. A subsequent paper extends the concept to more realistic MDOF models.

This paper discusses the problem of flexural-torsional stability of columns with open thin-walled cross sections. It introduces an analytical solution technique applicable to the differential equation system describing the current problem. The approach derived is more comprehensive than those presented traditionally in the literature and is applicable in determining the value of critical load due to combined buckling modes. The system considered consists of three separate differential equations that are coupled in a general case. Various symmetries of the cross section release these couplings. This presentation is limited, for simplicity, only to columns having singly symmetric cross sections, or at least cross sections of point symmetry. Thus, there are couplings between two of the governing equations. The elastic restraints having varying stiffnesses with respect to translation, rotation, torsion and warping, separately, have been studied as boundary conditions through numerical examples. The stiffnesses of the restraints applied are chosen to cover the range from zero to infinity.

Using a theoretical approach, stability criteria for X-bracing Systems are formulated. For two different end constraints, simple closed-form relationships are presented for calculating the critical compressive load. As a design aid, curves of the effective length factor versus the ratio T/P, where T and P are tension and compression brace axial forces, respective y, are given. Criteria are formulated for the general case where the tension and compression braces may have different material and geometrical properties. Some important characteristics are revealed: the ratio T/P increases as the effective length factor k decreases; after a certain value at which the compression brace buckles in the second mode, it will remain unchanged. As the flexural rigidity, [E.sub.t][I.sub.t], of the tension brace increases, the ratio T/P at which buckling occurs in the second mode decreases, and the effective length factor k also decreases. Therefore, it is possible for the compression brace to snap through in the first buckling mode due to a total loss of the tension brace flexural rigidity. However, when the tension brace flexural rigidity reaches a certain value, the compression brace buckles in the second mode.

Preliminary design of a structure controlled by active bracing or mass dampers to withstand seismic motion or high winds requires rapid means of analysis and estimates. A response spectrum technique applied to simplified models of complex structures can provide such means. The spectra required for preliminary evaluation of a controlled structure include, besides the usual response parameters (displacement, acceleration, base shear), the control resource parameters, such as the size and capacity of hydraulic actuators; power requirements; and energy consumption. This paper presents an approach to develop design spectra based on the analogy of behavior of wall frame structures with the behavior of shear-flexural cantilever beams treated by the continuum approach. This approach is verified numerically and experimentally using simplified harameters representing the structures. The same approach is further used to develop typical response and resource spectra using simulated ground motions according to a current seismic code. Several recommendations on efficiency of control of wall-frame structures are presented using information from the design spectra developed herein., A model for the design spectra for actively controlled wall-frame buildings is presented. The model uses a continuum approach in the prediction of structural response and control requirements. The model is numerically viable. Seismic design spectra with this type of active control is presented. Results from analysis indicate that the efficiency of the system is affected by the deforming pattern of the structure and the location of control force. Frame-type buildings are more amenable to this type of system. Control of buildings on soft soil require more force for the reduction of structure response.

The seismic performance of fixed-base and base-isolated concentrically braced and special moment-resisting steel frames are determined. The fixed-base frames are designed in accordance with the 1990 Structural Engineers Association of California (SEAOC) Recommended Lateral Force Requirements and Commentary, while the base-isolated frames are designed to 100%, 50%, and 25% of the SEAOC-recommended lateral force levels. Nonlinear time-history analyses, for 54 ground motion records, are then conducted. The number of yielded or collapsed frames, number of yielded elements, total and relative roof displacement, and isolation-bearing displacement are statistically evaluated. Comparable or better performance is achieved for the base-isolated braced frame designed to 50% of the SEAOC-recommended lateral force. Comparable performance is achieved for the base-isolated moment frame at various design force levels, depending on the performance criteria., A study is conducted to obtain the seismic response of fixed-base and base-isolated structures. Fixed-based and base-isolated frames were designed using the 1990 Structural Engineers Assn of California specifications. Base-isolated frames also had lateral force levels of 100%, 50% and 25%. Results show that the base isolated structure designed with 50% lateral force level performs better compared to fixed-base structures in terms of the occurence of yielding or collapse, roof displacement, amount of yielded superstructure elements and first yield time.

A comment on a study on the stability of built-up columns is presented. In the study, problems of stability of initially stressed solids under mulitaxial stress may only be solved when analysis is grounded in the theory of finite strain. Two schools' formulas, thought to be related are actually equivalent, differing only in the physical meanings of corresponding shear moduli.