Seismic Analysis of Structures with Viscoelastic Dampers

Viscoelastic dampers are being used in structures to mitigate dynamic effects. The models of varying complexities from simple maxwell element to differential models with fractional and complex order derivatives have been used to represent their frequency-dependent force deformation characteristics. More complex models are able to capture the frequency dependence of the material properties better, but are difficult to use in analyses. However, the classical models consisting of assemblies of Kelvin and/or Maxwell elements with an adequate number of parameters can be formed to capture the frequency dependence as accurately as the more sophisticated fractional derivative models can do. The main advantage in adopting these classical models is a simpler and smaller system of equations, which can be conveniently analyzed for nonlinear and linear systems. In this study, the two classical mechanical models consisting of Kelvin chains and Maxwell ladder are used. It is shown that these mechanical models are as effective as the fractional derivative model in capturing the effect of the frequency dependence of the material properties in response calculations and are more convenient to use in dynamic response analyses.     

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.     

Seismic Energy Dissipation of Inelastic Structures with Tuned Mass Dampers

The energy transfer process of using a tuned mass damper (TMD) in improving the ability of inelastic structures to dissipate earthquake input energy is investigated. Inelastic structural behavior is modeled by using the force analogy method, which is the backbone of analytically characterizing the plastic energy dissipation in the structure. Numerical simulations are performed to study the energy responses of structures with and without TMD installed. The effectiveness of TMD in reducing energy responses is also studied by using plastic energy spectra for various structural yielding levels. Results show that the use of TMD enhances the ability of the structures to store larger amounts of energy inside the TMD that will be released at a later time in the form of damping energy when the response is not at a critical state, thereby increasing the damping energy dissipation while reducing the plastic energy dissipation. This reduction of plastic energy dissipation relates directly to the reduction of damage in the structure, and TMD is therefore concluded to be quite effective in protecting structures from suffering major damage during an earthquake. However, storing energy in the TMD is restricted if the structure becomes plastic at a small displacement level. In this case, the effectiveness of TMD diminishes, and the structural response becomes practically the same as those without TMD installed.     

Use of Exact Solutions of Wave Propagation Problems to Guide Implementation of Nonlinear Seismic Ground Response Analysis Procedures

One-dimensional nonlinear ground response analyses provide a more accurate characterization of the true nonlinear soil behavior than equivalent-linear procedures, but the application of nonlinear codes in practice has been limited, which results in part from poorly documented and unclear parameter selection and code usage protocols. In this article, exact (linear frequency-domain) solutions for body wave propagation through an elastic medium are used to establish guidelines for two issues that have long been a source of confusion for users of nonlinear codes. The first issue concerns the specification of input motion as “outcropping” (i.e., equivalent free-surface motions) versus “within” (i.e., motions occurring at depth within a site profile). When the input motion is recorded at the ground surface (e.g., at a rock site), the full outcropping (rock) motion should be used along with an elastic base having a stiffness appropriate for the underlying rock. The second issue concerns the specification of viscous damping (used in most nonlinear codes) or small-strain hysteretic damping (used by one code considered herein), either of which is needed for a stable solution at small strains. For a viscous damping formulation, critical issues include the target value of the viscous damping ratio and the frequencies for which the viscous damping produced by the model matches the target. For codes that allow the use of “full” Rayleigh damping (which has two target frequencies), the target damping ratio should be the small-strain material damping, and the target frequencies should be established through a process by which linear time domain and frequency domain solutions are matched. As a first approximation, the first-mode site frequency and five times that frequency can be used. For codes with different damping models, alternative recommendations are developed.     

Seismic Control of a Nonlinear Benchmark Building Using Smart Dampers

This paper addresses the third-generation benchmark problem on structural control, and focuses on the control of a full-scale, nonlinear, seismically excited, 20-story building. A semiactive design is developed in which magnetorheological (MR) dampers are applied to reduce the structural responses of the benchmark building. Control input determination is based on a clipped-optimal control algorithm which employs absolute acceleration feedback. A phenomenological model of an MR damper, based on a Bouc–Wen element, is employed in the analysis. The semiactive system using the MR damper is compared to the performance of an active system and an ideal semiactive system, which are based on the same nominal controller as is used in the MR damper control algorithm. The results demonstrate that the MR damper is effective, and achieves similar performance to the active and ideal semiactive system, while requiring very little power.     

Predictive Optimal Linear Control of Elastic Structures during Earthquake. Part I

A predictive optimal linear control (POLC) algorithm is proposed for controlling the seismic responses of elastic structures. This algorithm compensates for time delay that occurs in real control application by predicting the structural response in the classical optimal linear control equation. The unique feature of this proposed POLC algorithm is that it compensates for time delay very effectively over a very wide range of time delay magnitudes. Numerical examples of single-degree-of-freedom structures are presented to study the performance of the proposed POLC system for various time delay magnitudes. Results show that a time delay always causes degradation of control efficiency, and POLC can greatly reduce this degradation. The effects of natural periods and damping of the structure, different earthquake characteristics and numerical integration schemes, and choices of control gains on the degradation induced by time delay are carefully studied in the analysis. Results show that using a larger time delay magnitude may give smaller structural responses, and this magnitude is independent of earthquake characteristics but dependent on the control gains. Finally, practical application of POLC is performed on a six-story moment-resisting steel frame. It is demonstrated that POLC maintains stability in multi-degree-of-freedom structures and at the same time it has a satisfactory control performance.     

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.     

Simplified Method to Include Cumulative Damage in the Seismic Response of Single-Degree-of-Freedom Systems

In this paper a consistent method including damage criteria in the seismic response of single-degree-of-freedom systems is proposed. The method allows the determination of suitably modified strength or displacement inelastic spectra through the introduction of an equivalent damage factor pdam that accounts for earthquake damage potential; analogously capacity spectra could be obtained. Three types of damage indices are considered (Park and Ang index DP&A, energy index DE, and low-cycle fatigue index DF) and derivation of pdam is pursued for all these cases. Moreover approximate simplified expressions in the function of Cosenza and Manfredi seismological ID index, which accounts for cyclic damage potential of an earthquake, are also proposed. In this way damage capacity spectra are obtained to improve the seismic assessment of existing structures including damage effect.     

Effect of Nonlinear Soil Behavior on Inelastic Seismic Response of a Structure

The characteristics of the earthquake motions at the base of a structure are affected by the properties of the underlying soil through the soil amplification and soil–structure interaction phenomena. In this paper the effect of nonlinear soil behavior on the elastic and inelastic response spectra of the motions that would be recorded at the free surface of a soft soil deposit or at the base of each structure is investigated. The analyses are conducted for a soil layer by itself and for a complete soil structure system using a finite element discretization of the soil in cylindrical coordinates and an approximate linear iterative procedure to simulate nonlinear behavior. Studies are conducted for structures, with a constant base and variable height modeled as equivalent linear or nonlinear single degree of freedom systems and an input motion at the base of the soil deposit representative of rock outcrop motions. Both mat and pile foundations are considered. The results illustrate clearly the importance of the nonlinear soil behavior.     

Nonlinear Response of Deep Immersed Tunnel to Strong Seismic Shaking

Critical for the seismic safety of immersed tunnels is the magnitude of deformations developing in the segment joints, as a result of the combined longitudinal and lateral vibrations. Analysis and design against such vibrations is the main focus of this paper, with reference to a proposed 70?m-deep immersed tunnel in a highly seismic region, in Greece. The multisegment tunnel is modeled as a beam connected to the ground through properly calibrated interaction springs, dashpots, and sliders. Actual records of significant directivity-affected ground motions, downscaled to 0.24 g peak acceleration, form the basis of the basement excitation. Free-field acceleration time histories are computed from these records through one-dimensional wave propagation equivalent-linear and nonlinear analyses of parametrically different soil profiles along the tunnel; they are then applied as excitation at the support of the springs, with a suitable time lag to conservatively approximate wave passage effects. The joints between the tunnel segments are modeled realistically with special nonlinear hyperelastic elements, while their longitudinal prestressing due to the great (7?bar) water pressure is also considered. Nonlinear dynamic transient analysis of the tunnel is performed without ignoring the inertia of the thick-walled tunnel, and the influence of segment length and joint properties is investigated parametrically. It is shown that despite ground excitation with acceleration levels exceeding 0.50 g and velocity of about 80?cm/s at the base of the tunnel, net tension and excessive compression between the segments can be avoided with a suitable design of joint gaskets and a selection of relatively small segment lengths. Although this research was prompted by the needs of a specific project, the dynamic analysis methods, the proposed design concepts, and many of the conclusions of the study are sufficiently general and may apply in other immersed tunneling projects.     

Seismic Response Control of Building Structures with Superelastic Shape Memory Alloy Wire Dampers

This paper presents a simulation-based benchmark control study in which shape memory alloy (SMA) wire dampers are utilized to control the seismic response of a three-story nonlinear steel frame building. The SMA wire damper uses superelastic Nitinol wires for energy dissipation because of its high fatigue life and large recoverable strain. An analytical model which considers the training effect of SMA wires is used to describe the stress-strain relationship of superelastic SMA wires. The performance of SMA wire dampers is investigated in the framework of the third-generation benchmark problem on structural control. A comparative study of the seismic response control performance of SMA wire dampers with either unprestrained or prestrained SMA wires was also conducted. The results of this simulation-based benchmark control study show that SMA wire dampers, as a passive structure control measure, can effectively reduce the seismic responses of the three-story nonlinear benchmark building structure and has the potential to withstand several design earthquakes without the need for repair.     

Seismic Energy Dissipation of Inelastic Structures with Multiple Tuned Mass Dampers

The ability to use multiple tuned mass dampers (TMDs) in improving inelastic structural performance to dissipate the earthquake input energy is investigated. Inelastic structural behavior is modeled using the force analogy method, which is the backbone of analytically characterizing the plastic energy dissipation in the structure. Both tuning period and placement of the multiple TMDs are studied to give the best structural performance in terms of plastic energy dissipation. Numerical simulations are performed to study the energy responses of structures with and without TMD installed, and the effectiveness of TMDs in the reduction of energy responses is also studied by using tuned mass spectra. Results show that the installation of TMDs gives the structure additional capability of dissipating a large amount of damping energy and at the same time reducing the amount of plastic energy demand and therefore reducing damage in the structure. More important, TMDs have the ability to draw the plastic energy dissipation at the lower stories and release it to the upper stories. This is particularly beneficial for structures that would otherwise suffer more damage at the lower stories than the upper stories. However, the reduction in plastic energy dissipation is quite sensitive to the earthquake vibration characteristics, and TMDs should not be used for structures with weak upper stories.     

Identification of the Soil–Structure Interaction System Using Earthquake Response Data

This paper demonstrates how system identification techniques can be successfully applied to a soil–structure interaction system using the earthquake response data. The parameters identified are the shear moduli of several near-field soil regions and Young’s moduli of the shell sections of the structure. The soil–structure interaction system is modeled by the finite element method combined with the infinite element formulation for the unbounded layered soil medium. The simulated earthquake responses using the identified parameters are shown to be in excellent agreement with the observed response data. Prediction of the responses is also carried out for a larger earthquake event using the identified parameters as the initial properties in the equivalent linearization procedure. It has been found that the predicted responses are also compared very well with the measured responses.     

Nonlinear Response Analysis Considering Dynamic Stiffness with Both Frequency and Strain Dependencies

The time domain evaluation of the frequency-dependent dynamic stiffness was studied and some transform methods are being proposed. In this paper, the nonlinear response analysis method of the dynamic stiffness with both frequency and strain dependency was studied. First, the frequency-dependent complex stiffness was calculated at each strain level, and then they were transformed to the impulse response in the time domain. The characteristics of the complex stiffness and the impulse response of both two-layered soil and viscoelastic dampers were investigated. Then, the nonlinear time-history response analysis method considering both dependencies using the impulse response of each strain level was proposed. The earthquake response analysis of a structure on the two-layered soil was carried out as an example. The efficiency of the method was confirmed through these investigations.     

Stress Reduction Coefficient and Amplification Factor for Seismic Response of Ground

Incorporating the increase of shear modulus with depth (z) by using a depth exponent p, the response of linear visco-elastic ground under steady state harmonic vibration was examined. The soil mass lying above the rigid base was discretized into a large number of horizontal layers. For a given vertical thickness (Hr) of the upper ground material in the first mode of resonance, it was noted that the resonant frequency (fr) as well as the values of amplification factor (Mf) and the stress reduction coefficient (Cd) at a given z/Hr can be computed as a function of p and D, where D is the damping ratio of the material. An increase in D leads to (1) an increase in Cd; (2) a very little increase in fr; and (3) a decrease in Mf. On the other hand, an increase in p causes: (1) an increase in fr; (2) a marginal increase in Mf; and (3) a decrease in Cd. It was evident that the values of Mf and Cd at a given depth will be affected significantly by changes in the chosen thickness of the soil deposit in resonance.     

Nonlinear System Modeling and Velocity Feedback Compensation for Effective Force Testing

Effective force testing (EFT) is a test procedure that can be used to apply real-time earthquake loads to large-scale structural models. The implementation of the EFT method requires velocity feedback compensation for the actuators in order to apply forces accurately to test structures. Nonlinearities in the servosystem have a significant impact on the velocity feedback compensation and test results when large flow demands are present, which can be caused by large structural velocities and/or large forces applied to the test structure. This paper presents a nonlinear servosystem model, upon which a nonlinear compensation scheme is proposed. The model and compensation scheme are experimentally verified. The results indicate that the proposed model accurately describes the servosystem behavior, and with the nonlinear velocity feedback compensation, real-time dynamic testing can be conducted using the EFT method.     

Seismic Response of Liquid Storage Tanks Incorporating Soil Structure Interaction

A frequency domain method is presented to compute the impulsive seismic response of circular surface mounted steel and concrete liquid storage tanks incorporating soil-structure interaction (SSI) for layered sites. The method introduces the concept of a near field region in close proximity to the mat foundation and a far field at distance. The near field is modeled as a region of nonlinear soil response with strain compatible shear stiffness and viscous material damping. The shear strain in a representative soil element is used as the basis for strain compatibility in the near field. In the far field, radiation damping using low strain soil response is used. Frequency dependent complex dynamic impedance functions are used in a model that incorporates horizontal displacement and rotation of the foundation. The focus of the paper is on the computation of the horizontal shear force and moment on the tank foundation to enable foundation design. Significant SSI effects are shown to occur for tanks sited on soft soil, especially tanks of a tall slender nature. SSI effects take the form of period elongation and energy loss by radiation damping and foundation soil damping. The effects of SSI for tanks are shown to reverse the trend of force and moment reduction under earthquake loading as is usually assumed by designers. The reasons for this important effect in tank design are given in the paper and relate to the very short period of most tanks, hence, period lengthening may result in load increase. A comparison is made with SSI effects evaluated using the code SEI/ASCE 7-02. Period elongation is found to be similar for relatively stiff soils when assessed by the code compared with the results of the dynamic analysis. For soft soils, the agreement is not as good. Code values of system damping are found to agree reasonably well with an assessment based on the dynamic analyses for the range of periods covered by the code. Energy loss by material damping and radiation damping is discussed. It is shown that energy loss may be computed using the complex dynamic impedance function associated with the viscous dashpot in the analytical model. The proportion of energy loss in the translation mode compared to that dissipated in the rotational mode is addressed as a function of the slenderness of the tank. Energy loss increases substantially with the volume of liquid being stored.     

ONDA: Computer Code for Nonlinear Seismic Response Analyses of Soil Deposits

This paper describes a newly developed computer code for performing one-dimensional nonlinear dynamic analysis (ONDA) of soil deposits. The code has been developed by revisiting the 1982 work by Ohsaki with the purpose of simulating the ground response to an earthquake of moderate intensity (i.e., values of peak ground acceleration on stiff soil on the order of 0.15 to 0.25g, which are typical of many sites in Italy). In the Ohsaki model a horizontally stratified soil deposit is idealized as a discrete mechanical system composed of a finite number of lumped masses connected with a series of springs and dashpots. Nonlinearity is modeled by assuming (1) a “backbone” curve that describes the initial monotonic loading of the stress-strain curve, and (2) a “rule” that simulates the unloading-reloading paths and stiffness degradation undergone by soil as seismic excitation progresses. Typically, the backbone curve is obtained from conventional cyclic undrained loading laboratory tests. The rule generally used is the so-called Masing criterion, which assumes that the unload-reload branches of the stress-strain curve have the same shape as the initial loading curve but are affected by a scale factor (n) equal to 2. In this work, the Masing criterion has been modified by assuming a scale factor (n) not necessarily equal to 2. It turns out that a factor n greater than 2 allows the simulation of cyclic hardening, while cyclic softening can be modeled by assuming decreasing values of n even smaller than 2. Pyke proposed in 1979 to use a scale factor (n) lower than 2 to simulate cyclic degradation. According to Pyke, the n parameter is a function of the mobilization factor. The generalization of the Masing criterion allows ONDA to properly simulate the phenomena of soil hardening and soil degradation, giving it the capability to compute the permanent strains developed during a seismic event. The procedure required to evaluate the model parameters is also described in the paper. Note that the laboratory tests examined gave values of n between 2 and 6 for a strain level not greater than 0.3%. In ONDA the numerical solution of the nonlinear equations of motion is obtained using the unconditionally stable Wilson θ algorithm (with θ ≥ 1.37). The new method has been used to predict the seismic response at two sites in Italy. For these case studies, the maximum input acceleration was not greater than 0.3g and the computed shear strains were less than 0.2%. The ONDA results have been compared with those computed with SHAKE, EERA (equivalent-linear analysis), and NERA (nonlinear analysis).     

Nonlinear Earthquake Response Modeling of a Large-Scale Two-Span Concrete Bridge

A quarter scale model of a two-span RC bridge was tested using the Network for Earthquake Engineering Simulation (NEES) multiple shake table system at the University of Nevada, Reno, Nev. The project was funded through a National Science Foundation-NEES demonstration grant. The bridge system was tested from a preyield state until column failure. In depth analytical modeling was conducted to determine the effectiveness of current structural analysis software and methodology in predicting the bridge model response. Both SAP2000 v.9 and Drain-3DX were used for this purpose. Both models produced reasonable results up to column failure, however, the Drain-3DX model was determined to be most effective to predict the nonlinear bridge model response. Parametric studies were conducted to investigate optimal element discretization and integration parameters. Existing equations for pre and postyield column shear stiffness showed good correlation when compared with the measured data.     

Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.