Optimal Control of Earthquake Response Using Semiactive Isolation

The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal semiactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler–Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.     

Site-Specific Validation of Random Vibration Theory-Based Seismic Site Response Analysis

Seismic site response analysis is typically performed using a suite of rock acceleration-time histories prescribed at the base of a soil column and propagated to the ground surface. To develop statistically stable estimates of the site response, a large number of input motions are required. Alternatively, random vibration theory (RVT) can be used to predict statistically stable estimates of the surface response spectrum in one analysis without the need to prescribe the input rock motion in the time domain. Thus, the critical and time consuming activity of choosing appropriate input ground motions and fitting them to a target spectrum is avoided. This paper describes the RVT approach, its analytical background and input requirements, and provides a site-specific validation of the procedure against traditional site response predictions. The single-corner frequency Brune source spectrum is used in the RVT procedure to describe the input motion in the frequency domain. RVT site response predictions using the Brune spectrum as input are compared with those from traditional site response analyses that incorporate different suites of input rock motions. Results indicate that RVT site response analysis can provide a response spectrum that is similar to the median response spectrum from analyses performed using a suite of input rock motions. However, the favorable comparison is obtained only when the seismological parameters used to describe the RVT input motion are carefully chosen to be consistent with the suite of input rock motions.     

Seismic Simulation of Inelastic Soils via Frequency-Dependent Moduli and Damping

Although soils are known to exhibit nonlinear behavior even at small strains, evaluations of the response of sedimentary basins to strong seismic motions are almost always based on linear, elastic solutions incorporating frequency-independent damping. The principal reasons for this relate to the robustness of the linear algorithm and the ease with which the required parameters can be determined experimentally in engineering practice. Most often, but not always, attempts are made in these analyses to compensate for the inelastic behavior by adjusting the material parameters for the representative levels of strain by means of an iterative method. However, both the standard iterative method and the direct linear solution without iterations suffer from two important shortcomings. First, they do not account for the effect of high confining pressures on inelastic behavior. However, it is known from experiments with sands subjected to cyclic shearing strains under confining pressures of up to 5 Mpa, that in highly confined samples, the material remains nearly elastic for a larger range of strains than do those samples subjected to a lesser pressure. Second, the amplification analyses disregard the fact that small-amplitude, high-frequency components of deformation involve hysteresis loops with little modulus degradation or damping (i.e., nearly elastic secondary loops). Thus, motions computed at the surface of the basin with the standard method usually exhibit unrealistically low amplitudes at high frequencies. This article presents the results obtained with a series of “true” nonlinear numerical analyses with inelastic (Masing-type) soils and layered profiles subjected to broadband earthquake motions, taking into account the effect of the confining pressure. These show that it is possible to simulate closely the actual inelastic behavior of rate-independent soils by means of linear analyses in which the soil moduli and damping change with frequency. It is emphasized that the variation in the linear model of the material parameters with frequency arises solely because the strains have broad frequency content, and not because the materials exhibit any rate dependence when tested cyclically. The proposed new model is successfully applied to a 1-km-deep model for the Mississippi embayment near Memphis, Tenn. The seismograms computed at the surface not only satisfy causality (which cannot be taken for granted when using frequency-dependent parameters), but their spectra contain the full band of frequencies expected.     

Dynamics of SDOF Systems with Nonlinear Viscous Damping

The effects of nonlinear viscous damping on the dynamic response of single-degree-of-freedom (SDOF) structural systems are analyzed. This kind of damping characterizes a special class of fluid viscous dampers recently utilized in the field of vibration control as base-isolation devices or viscoelastic elements included in steel braces of framed structures. The analytical relationship adopted to reproduce the mechanical behavior of the fluid viscous dampers is a fractional power-law of the velocity, the exponent of which ranges between 0.1 and 0.2. This function had been previously calibrated on the results of a special experimental survey carried out at the University of Florence. The dynamics of the classical linear-viscous SDOF oscillator is herein reformulated on the basis of the above-mentioned fractional viscous damping (FrVD) relationship. In particular, the transient and steady-state responses are examined in both free and forced vibration conditions. The magnification and transmissibility factors are analytically determined for different damping levels. Moreover, the relation between the viscous damping coefficient and the frequency ratio (i.e., the ratio of the dynamic load to the oscillator frequencies) is defined. The diagrams describing these functions provide direct correlations between the damping as well as the elastic properties of the system and the frequency content of the dynamic action.     

Dynamic Ratcheting in Elastoplastic Single-Degree-of-Freedom Systems

In dynamic analysis, hysteretic damping often provides a reasonable model of the inelastic behavior of a structure. Nonlinearity presented by hysteretic damping, however, introduces the possibility of developing complicated motions not expected in linear dynamics. In this study, motions of a single-degree-of-freedom system with hysteretic damping under dual-frequency sinusoidal excitations are investigated through numerical simulation. Hysteretic damping behavior is represented by three different plasticity models: the elasto-perfectly-plastic model; the linear kinematic hardening model; and the two-surface model. Under certain conditions, the resultant motions from the elasto-perfectly-plastic model and the two-surface model exhibit a continual increment of plastic deformation in successive cycles. Parametric study shows that this dynamic ratcheting develops when applied frequencies are commensurable (i.e., related to each other with integer ratio), and the product of terms comprising the ratio is an even number. In the Poincaré section, motion from commensurable frequencies shows limit cycle behavior, whereas the boundedness of motion for incommensurable frequencies is depicted by having quasi-periodicity. On the other hand, the response of the linear kinematic hardening model is qualitatively different and, in particular, dynamic ratcheting does not develop, irrespective of the frequency commensurability. These findings suggest that model selection may have unanticipated consequences for the analysis and design of structural systems subjected to severe dynamic loadings, such as major earthquakes.     

Damping of Cables by a Transverse Force

A general asymptotic format is presented for the effect on the modal vibrations of a transverse damper close to the end of a cable. Complete locking of the damper leads to an increase of the natural frequencies, and it is demonstrated that the maximum attainable damping is a certain fraction of the relative frequency increase, depending on the type of damping device. The asymptotic format only includes a real and a complex nondimensional parameter, and it is demonstrated how these parameters can be determined from the frequency increase by locking and from an energy balance on the undamped natural vibration modes. It is shown how the asymptotic format can incorporate sag of the cable, and specific results are presented for viscous damping, the effect of stiffness and mass, fractional viscous damping, and a nonlinear viscous damper. The relation of the stiffness component to active and semiactive damping is discussed.     

1D Time-Domain Solution for Seismic Ground Motion Prediction

A full time-domain solution for predicting earthquake ground motion based on the 1D viscoelastic shear-wave equation is presented. The derivation results in a time-domain equation in the form of an infinite impulse response filter. A solution in the time domain has several advantages including causality, direct modeling of impulsive and transient processes, and ease of inclusion of nonlinear soil behavior. The method is applicable to any arbitrarily layered silhouette presented as SH-wave velocity, damping coefficient, and mass density profiles for designated soil intervals. For nonlinear evaluations, an equivalent-linear formulation is incorporated and the standard modulus and damping degradation curves become part of the input set. Input motion can be either rock-outcrop or body-wave motions measured or estimated at the bottom of the geologic profile, and the output is the estimated ground motion time history. Application of the method to vertical array strong motion records from Garner Valley, and Wildlife Site, Calif., shows that predicted surface (and interval) ground motion is virtually identical to that measured. The differences between the results of linear and nonlinear analyses are negligible for most cases. A comparison of the time-domain model with SHAKE shows that SHAKE fails to accurately predict time histories in some situations, whereas the time-domain solution always yields satisfactory predicted surface ground motions.     

Influence of Input Motion and Site Property Variabilities on Seismic Site Response Analysis

Seismic site response analysis evaluates the influence of local soil conditions on earthquake ground shaking. There are multiple sources of potential uncertainty in this analysis; the most significant pertaining to the specification of the input motions and to the characterization of the soil properties. The influence of the selection of input ground motions on equivalent-linear site response analysis is evaluated through analyses performed with multiple suites of input motions selected to fit the same target acceleration response spectrum. The results indicate that a stable median surface response spectrum (i.e., within ±20% of any other suite) can be obtained with as few as five motions, if the motions fit the input target spectrum well. The stability of the median is improved to ±5 to 10% when 10 or 20 input motions are used. If the standard deviation of the surface response spectra is required, at least 10 motions (and preferably 20) are required to adequately model the standard deviation. The influence of soil characterization uncertainty is assessed through Monte Carlo simulations, where variations in the shear-wave velocity profile and nonlinear soil properties are considered. Modeling shear-wave velocity variability generally reduces the predicted median surface motions and amplification factors, most significantly at periods less than the site period. Modeling the variability in nonlinear properties has a similar, although slightly smaller, effect. Finally, including the variability in soil properties significantly increases the standard deviation of the amplification factors but has a lesser effect on the standard deviation of the surface motions.     

Vibration and Damping in Three-Bar Tensegrity Structure

The most interesting examples of tensegrity structures are underconstrained and display an infinitesimal flex. In the direction of that flex the force-displacement relationship is highly nonlinear, resulting from geometric stiffening and influenced by the effect of prestress at equilibrium. A tensegrity structure would therefore display nonlinear vibrations when excited in the direction of the infinitesimal flex, the “frequency” decreasing with amplitude. Movement in the direction of the flex occurs with only infinitesimal change in member length, and therefore under conventional models of material damping in members the motion would not vanish as rapidly as it would for a conventional oscillator. We study one particular tensegrity geometry for which we present the force-displacement relationship in analytical form and then examine the nonlinear vibrations. We observe the role of damping and we discuss those implications for the design of tensegrity structures in space applications.     

Seismic Site Response for Near-Fault Forward Directivity Ground Motions

Forward directivity effects in the near-fault region produce pulse-type motions that differ significantly from ordinary ground motions that occur at greater distances from the causative fault. Current code site factors are based on empirical observations and analyses involving less intense nonpulse ordinary ground motions. Nonlinear site response analyses with bidirectional shaking are performed using representative site profiles to quantify seismic site response effects for intense near-fault motions resulting from forward directivity. Input rock motions are represented with simplified velocity pulses that characterize the amplitude and period of forward directivity motions. Results indicate that site response affects both the amplitude and period of forward directivity pulses, and hence, local site conditions should be considered when evaluating seismic designs in the near-fault region. Stiff soil sites tend to amplify the peak ground velocity and increase the period of pulse-type motions, particularly, when the period of the rock motion coincides with the degraded period of the site. Amplification is limited at soft soil sites by the dynamic strength of the weak soil, so attenuation occurs for intense input motions. This nonlinearity is not reflected in the site factors in current building codes. Guidance is provided for estimating the amplitude and pulse period for velocity pulses at soil sites.     

Fault Rupture Propagation through Sand: Finite-Element Analysis and Validation through Centrifuge Experiments

The three notorious earthquakes of 1999 in Turkey (Kocaeli and Düzce) and Taiwan (Chi-Chi), having offered numerous examples of surface fault rupturing underneath civil engineering structures, prompted increased interest in the subject. This paper develops a nonlinear finite-element methodology to study dip–slip (“normal” and “reverse”) fault rupture propagation through sand. The procedure is verified through successful Class A predictions of four centrifuge model tests. The validated methodology is then utilized in a parametric study of fault rupture propagation through sand. Emphasis is given to results of engineering significance, such as: (1) the location of fault outcropping; (2) the vertical displacement profile of the ground surface; and (3) the minimum fault offset at bedrock necessary for the rupture to reach the ground surface. The analysis shows that dip–slip faults refract at the soil–rock interface, initially increasing in dip. Normal faults may keep increasing their dip as they approach the ground surface, as a function of the peak friction angle φp and the angle of dilation ψp. In contrast, reverse faults tend to decrease in dip, as they emerge on the ground surface. For small values of the base fault offset, h, relative to the soil thickness, H, a dip–slip rupture cannot propagate all the way to the surface. The h/H ratio required for outcropping is an increasing function of soil “ductility.” Reverse faults require significantly higher h/H to outcrop, compared to normal faults. When the rupture outcrops, the height of the fault scrap, s, also depends on soil ductility.     

On-Site Nonlinear Hysteresis Curves and Dynamic Soil Properties

Strong motion records at five vertical array sites in Japan are used to examine soil shear modulus and material damping as a function of shear strain during large earthquakes. Acceleration data from the sites are processed directly for evaluation of site shear stress-strain hysteresis curves for different time windows of the record. Results of the analysis demonstrate a significant nonlinear ground response at the sites with surface peak ground accelerations exceeding 90 gal. The results of shear stress-strain hysteresis curves are also used to estimate variation of soil shear modulus and material damping characteristics with shear strain amplitude at each site. The identified shear modulus-shear strain and damping ratio-shear strain relationships are in general agreement with published laboratory results. These response interpretations are also compared with the results of a frequency-domain analysis by using the spectral ratio (uphole∕downhole) technique. There is general agreement between the time- and frequency-domain results. The results illustrate the significance of the site nonlinearity during strong ground motions as well as the accuracy of the dynamic soil properties obtained from laboratory tests.     

Simplified Procedure for Estimating Earthquake-Induced Deviatoric Slope Displacements

A simplified semiempirical predictive relationship for estimating permanent displacements due to earthquake-induced deviatoric deformations is presented. It utilizes a nonlinear fully coupled stick-slip sliding block model to capture the dynamic performance of an earth dam, natural slope, compacted earth fill, or municipal solid-waste landfill. The primary source of uncertainty in assessing the likely performance of an earth/waste system during an earthquake is the input ground motion. Hence, a comprehensive database containing 688 recorded ground motions is used to compute seismic displacements. A seismic displacement model is developed that captures the primary influence of the system’s yield coefficient (ky), its initial fundamental period (Ts), and the ground motion’s spectral acceleration at a degraded period equal to 1.5Ts. The model separates the probability of “zero” displacement (i.e., ? 1?cm) occurring from the distribution of “nonzero” displacement, so that very low values of calculated displacement do not bias the results. The use of the seismic displacement model is validated through reexamination of 16 case histories of earth dam and solid-waste landfill performance. The proposed model can be implemented rigorously within a fully probabilistic framework or used deterministically to evaluate seismic displacement potential.     

Relating Damping to Soil Permeability

Published comparisons of complex moduli in dry and saturated soils have shown that viscous behavior is only evident when a sufficiently massive viscous fluid (like water) is present. That is, the loss tangent is frequency dependent for water saturated specimens, but nearly frequency independent for dry samples. While the Kelvin–Voigt (KV) representation of a soil captures the general viscous behavior using a dashpot, it fails to account for the possibly separate motions of the fluid and frame (there is only a single mass element). An alternative representation which separates the two masses, water and frame, is presented here. This Kelvin–Voigt–Maxwell–Biot (KVMB) model draws on elements of the long standing linear viscoelastic models in a way that connects the viscous damping to permeability and inertial mass coupling. A mathematical mapping between the KV and KVMB representations is derived and permits continued use of the KV model, while retaining an understanding of the separate mass motions.     

Experimental Verification of Smart Cable Damping

Stay cables, such as are used in cable-stayed bridges, are prone to vibration due to their low inherent damping characteristics. Transversely attached passive viscous dampers have been implemented in many bridges to dampen such vibration. However, only minimal damping can be added if the attachment point is close to the bridge deck. For longer bridge cables, the relative attachment point becomes increasingly smaller, and passive damping may become insufficient. A recent analytical study by the authors demonstrated that “smart” semiactive damping can provide increased supplemental damping. This paper experimentally verifies a smart damping control strategy employing H2/linear quadratic Gaussian (LQG) clipped optimal control using only force and displacement measurements at the damper for an inclined flat-sag cable. A shear mode magnetorheological fluid damper is attached to a 12.65?m inclined flat-sag steel cable to reduce cable vibration. Cable response is seen to be substantially reduced by the smart damper.     

Viscous Effect on the Roll Motion of a Rectangular Structure

The viscous effect on the roll motion of a rectangular structure was investigated in a two-dimensional wave tank. The structure was used to simulate a simplified barge in the beam sea condition. The structure with a draft one-half of its height was hinged at the center of gravity and free to roll (1 degree of freedom) by waves. The dynamic characteristics of the structure were first identified, including its roll natural period. The dynamic response of the barge-like structure under wave actions was then tested with regular waves with a range of wave periods that are shorter, equal to, and longer than its roll natural period. Particle image velocimetry was used to obtain the velocity field in the vicinity of the structure. The coupled interactions between the incident waves and the structure were demonstrated by examining the vortical flow fields to elucidate the effect of viscous damping (also called the eddy making damping) to the roll motion of the structure over wave periods. For incoming waves with a wave period same as the roll natural period, the structure roll motion was, as expected, greatly reduced by the viscous-damping effect. At wave periods shorter than the roll natural period, the structure roll motion was slightly reduced by the viscous effect. However, at wave periods longer than the roll natural period, the viscous effect due to flow separation at structure corners indeed amplified the roll motion. This indicates that not only can the viscous effect damp out the roll motion, it can also amplify the roll motion.     

Semianalytical Solution of Wave-Controlled Impact on Composite Laminates

Based on a structural model for wave-controlled impact, a modified Hertzian contact law was used to investigate the impact responses of composite laminates. The original nonlinear governing equation was transformed into two linear equations using asymptotic expansion. Closed-form solution can be derived for the first linear homogeneous equation, which is the equation of motion for single degree of freedom system with viscous damping. The second linear nonhomogeneous equation was solved numerically. The overall impact responses for wave-controlled impacts can be obtained semianalytically and agree well with the numerical solutions of nonlinear governing equations. The proposed methodology is useful for providing guidance to numerical simulation of impact on complex composite structures with contact laws fitting from experimental data.     

Optimal Vibration Absorber with Nonlinear Viscous Power Law Damping and White Noise Excitation

A tuned mass damper with a nonlinear power law viscous damper excited by white noise is considered. The system is analyzed by statistical linearization and stochastic simulation with the objective of minimizing the standard deviation of the response. It is shown that the optimal parameters for the tuned mass damper are unaffected by the magnitude of the structural damping in the linear case. However, in the nonlinear case the structural damping influences the equivalent parameters obtained by statistical linearization and thereby indirectly the optimal values for the damper parameters. Results from stochastic simulation show good agreement with results from statistical linearization in terms of the standard deviation of the response. It is shown that the optimal damping, which can be obtained by the passive device, is the same for the linear and nonlinear damper. However, for the nonlinear tuned mass damper the optimal parameters will depend on both structural damping and excitation intensity (or vibration amplitude). The results are presented in such a way that they can be used directly for the design of a tuned mass damper with damping governed by a nonlinear viscous power law.     

Optimal Control: Basis for Performance Comparison of Passive and Semiactive Isolation Systems

Passive damping in shock and vibration isolation systems reduces the deformation of the isolation system but can increase the acceleration sustained by the isolated object. Semiactive (i.e., controllable) damping systems offer a solution to the problem of increased vibration transmissibility at high frequencies. Semiactive damping is especially relevant to protecting acceleration-sensitive components to the effects of large impulsive earthquakes. In this paper, we compare three semiactive control policies, i.e., pseudonegative-stiffness control, continuous pseudoskyhook-damping control, and bang-bang pseudoskyhook-damping control, in terms of their effectiveness in addressing the deficiencies of passive isolation damping. In order to establish a performance goal for these suboptimal semiactive control rules, we present a method for true optimization of the response of dynamically excited, semiactively controlled structures subjected to constraints imposed by the dynamics of a particular semiactive device. The optimization procedure involves solving Euler–Lagrange equations. The closed-loop dynamics of structures with semiactive control systems are nonlinear due to the parametric nature of the control actions. These nonlinearities preclude an analytical evaluation of Laplace transforms. In this paper, frequency response functions for semiactively controlled structural systems are compiled from the computed time history responses to sinusoidal and pulse-like base excitations. For control devices with no saturation forces, the closed-loop frequency response functions are independent of the excitation amplitude. We make use of this homogeneity of the solution of semiactive control systems and present results in dimensionless form.